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Computing eigenvalues and Hermite interpolation for Dirac systems with eigenparameter in boundary conditions

Mohammed M Tharwat

Author Affiliations

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

Boundary Value Problems 2013, 2013:36  doi:10.1186/1687-2770-2013-36

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/36


Received:8 November 2012
Accepted:5 February 2013
Published:21 February 2013

© 2013 Tharwat; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. In this paper we use the derivative sampling theorem ‘Hermite interpolations’ to compute approximate values of the eigenvalues of Dirac systems with eigenvalue parameter in one or two boundary conditions. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Using computable error bounds, we obtain eigenvalue enclosures. Examples with tables and illustrative figures are given. Also numerical examples, which are given at the end of the paper, give comparisons with the classical sinc-method in Annaby and Tharwat (BIT Numer. Math. 47:699-713, 2007) and explain that the Hermite interpolations method gives remarkably better results.

MSC: 34L16, 94A20, 65L15.

Keywords:
Dirac systems; eigenvalue problems with eigenparameter in the boundary conditions; Hermite interpolations; truncation error; amplitude error; sinc methods

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M2">View MathML</a> be the Paley-Wiener space of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M3">View MathML</a>-entire functions of exponential type σ. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M4">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M5">View MathML</a> can be reconstructed via the Hermite-type sampling series

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M6">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M7">View MathML</a> is the sequences of sinc functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M8">View MathML</a>

(1.2)

Series (1.1) converges absolutely and uniformly on ℝ, cf.[1-4]. Sometimes, series (1.1) is called the derivative sampling theorem. Our task is to use formula (1.1) to compute eigenvalues of Dirac systems numerically. This approach is a fully new technique that uses the recently obtained estimates for the truncation and amplitude errors associated with (1.1), cf.[5]. Both types of errors normally appear in numerical techniques that use interpolation procedures. In the following we summarize these estimates. The truncation error associated with (1.1) is defined to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M9">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M10">View MathML</a> is the truncated series

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M11">View MathML</a>

(1.4)

It is proved in [5] that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M5">View MathML</a> is sufficiently smooth in the sense that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M14">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M15">View MathML</a>, then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M17">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M18">View MathML</a>

(1.5)

where the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M20">View MathML</a> are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M21">View MathML</a>

(1.6)

The amplitude error occurs when approximate samples are used instead of the exact ones, which we cannot compute. It is defined to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M22">View MathML</a>

(1.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M24">View MathML</a> are approximate samples of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M26">View MathML</a>, respectively. Let us assume that the differences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M29">View MathML</a>, are bounded by a positive number ε, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M30">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M31">View MathML</a> satisfies the natural decay conditions

(1.8)

(1.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M34">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M35">View MathML</a>, we have, [5],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M36">View MathML</a>

(1.10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M37">View MathML</a>

(1.11)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M38">View MathML</a> is the Euler-Mascheroni constant.

The classical [6] sampling theorem of Whittaker, Kotel’nikov and Shannon (WKS) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M39">View MathML</a> is the series representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M40">View MathML</a>

(1.12)

where the convergence is absolute and uniform on ℝ and it is uniform on compact sets of ℂ, cf.[6-8]. Series (1.12), which is of Lagrange interpolation type, has been used to compute eigenvalues of second-order eigenvalue problems; see, e.g., [9-15]. The use of (1.12) in numerical analysis is known as the sinc-method established by Stenger, cf. [16-18]. In [10,12], the authors applied (1.12) and the regularized sinc-method to compute eigenvalues of Dirac systems with a derivation of the error estimates as given by [19,20]. In [12] the Dirac system has an eigenparameter appearing in the boundary conditions. The aim of this paper is to investigate the possibilities of using Hermite interpolations rather than Lagrange interpolations, to compute the eigenvalues numerically. Notice that, due to Paley-Wiener’s theorem [21], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M41">View MathML</a> if and only if there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M42">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M43">View MathML</a>

(1.13)

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M44">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M45">View MathML</a> also has an expansion of the form (1.12). However, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M45">View MathML</a> can be also obtained by the term-by-term differentiation formula of (1.12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M47">View MathML</a>

(1.14)

see [[6], p.52] for convergence. Thus the use of Hermite interpolations will not cost any additional computational efforts since the samples <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M48">View MathML</a> will be used to compute both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M45">View MathML</a> according to (1.12) and (1.14), respectively.

Consider the Dirac system which consists of the system of differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M51">View MathML</a>

(1.15)

and the boundary conditions

(1.16)

(1.17)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M56">View MathML</a>, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M57">View MathML</a>

(1.18)

The eigenvalue problem (1.15)-(1.17) will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M59">View MathML</a>. It is a Dirac system when the eigenparameter λ appears linearly in both boundary conditions. The classical problem when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M60">View MathML</a>, which we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M61">View MathML</a>, is studied in the monographs of Levitan and Sargsjan [22,23]. Annaby and Tharwat [24] used Hermite-type sampling series (1.1) to compute the eigenvalues of problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M61">View MathML</a> numerically. In [25], Kerimov proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58">View MathML</a> has a denumerable set of real and simple eigenvalues with ±∞ as the limit points. Similar results are established in [26] for the problem when the eigenparameter appears in one condition, i.e., when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M65">View MathML</a> or equivalently when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M67">View MathML</a>, where also sampling theorems have been established. These problems will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M69">View MathML</a>, respectively. The aim of the present work is to compute the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68">View MathML</a> numerically by the Hermite interpolations with an error analysis. This method is based on sampling theorem, Hermite interpolations, but applied to regularized functions hence avoiding any (multiple) integration and keeping the number of terms in the Cardinal series manageable. It has been demonstrated that the method is capable of delivering higher-order estimates of the eigenvalues at a very low cost; see [24]. In Sections 2 and 3, we derive the Hermite interpolation technique to compute the eigenvalues of Dirac systems with error estimates. We briefly derive some necessary asymptotics for Dirac systems’ spectral quantities. The last section contains three worked examples with comparisons accompanied by figures and numerics with the Lagrange interpolation method.

2 Treatment of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58">View MathML</a>

In this section we derive approximate values of the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58">View MathML</a>. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58">View MathML</a> has a denumerable set of real and simple eigenvalues, cf.[25]. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M75">View MathML</a> be a solution of (1.15) satisfying the following initial:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M76">View MathML</a>

(2.1)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M77">View MathML</a> denotes the transpose of a matrix A. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M78">View MathML</a> satisfies (1.16), then the eigenvalues of the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58">View MathML</a> are the zeros of the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M80">View MathML</a>

(2.2)

Similarly to [[22], p.220], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M81">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M82">View MathML</a> satisfy the system of integral equations

(2.3)

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M87">View MathML</a>, are the Volterra operators defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M88">View MathML</a>

(2.5)

For convenience, we define the constants

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M89">View MathML</a>

(2.6)

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M91">View MathML</a> to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M92">View MathML</a>

(2.7)

As in [12] we split <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M93">View MathML</a> into two parts via

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M94">View MathML</a>

(2.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M95">View MathML</a> is the known part

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M96">View MathML</a>

(2.9)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M97">View MathML</a> is the unknown one

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M98">View MathML</a>

(2.10)

Then the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M97">View MathML</a> is entire in λ for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M100">View MathML</a> for which, cf.[12],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M101">View MathML</a>

(2.11)

The analyticity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M97">View MathML</a> as well as estimate (2.11) are not adequate to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M97">View MathML</a> lies in a Paley-Wiener space. To solve this problem, we will multiply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M97">View MathML</a> by a regularization factor. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M107">View MathML</a>, be fixed. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108">View MathML</a> be the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M109">View MathML</a>

(2.12)

We choose θ sufficiently small for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M110">View MathML</a>. More specifications on m, θ will be given latter on. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108">View MathML</a>, see [12], is an entire function of λ which satisfies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M112">View MathML</a>

(2.13)

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M113">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M114">View MathML</a>

(2.14)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M115">View MathML</a>

What we have just proved is that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108">View MathML</a> belongs to the Paley-Wiener space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M117">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M118">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M119">View MathML</a>, then we can reconstruct the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108">View MathML</a> via the following sampling formula:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M121">View MathML</a>

(2.15)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M123">View MathML</a>, and approximate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108">View MathML</a> by its truncated series <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M125">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M126">View MathML</a>

(2.16)

Since all eigenvalues are real, then from now on we restrict ourselves to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M127">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M128">View MathML</a>, the truncation error, cf. (1.5), is given for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M129">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M130">View MathML</a>

(2.17)

where

(2.18)

The samples <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M133">View MathML</a>, in general, are not known explicitly. So, we approximate them by solving numerically <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M134">View MathML</a> initial value problems at the nodes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M135">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M137">View MathML</a> be the approximations of the samples of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M139">View MathML</a>, respectively. Now we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M140">View MathML</a>, which approximates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M125">View MathML</a>

(2.19)

Using standard methods for solving initial problems, we may assume that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M143">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M144">View MathML</a>

(2.20)

for a sufficiently small ε. From (2.13) we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M108">View MathML</a> satisfies the condition (1.9) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M107">View MathML</a> and therefore whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M147">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M148">View MathML</a>

(2.21)

where there is a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M149">View MathML</a> for which, cf. (1.10),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M150">View MathML</a>

(2.22)

Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M151">View MathML</a>

In the following, we use the technique of [27], where only the truncation error analysis is considered, to determine enclosure intervals for the eigenvalues; see also [24,28]. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M152">View MathML</a> be an eigenvalue with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M153">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M154">View MathML</a>

Then it follows that

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M156">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M157">View MathML</a> is given and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M158">View MathML</a> has computable upper bound, we can define an enclosure for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a> by solving the following system of inequalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M160">View MathML</a>

(2.23)

Its solution is an interval containing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a>, and over which the graph

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M162">View MathML</a>

is squeezed between the graphs

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M163">View MathML</a>

(2.24)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M164">View MathML</a>

(2.25)

Using the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M165">View MathML</a>

uniformly over any compact set, and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a> is a simple root, we obtain, for large N and sufficiently small ε,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M167">View MathML</a>

in a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a>. Hence the graph of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M169">View MathML</a> intersects the graphs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M171">View MathML</a> at two points with abscissae <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M172">View MathML</a> and the solution of the system of inequalities (2.23) is the interval

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M173">View MathML</a>

and in particular <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M174">View MathML</a>. Summarizing the above discussion, we arrive at the following lemma which is similar to that of [27] for Sturm-Liouville problems.

Lemma 2.1For any eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M152">View MathML</a>, we can find<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M176">View MathML</a>and sufficiently smallεsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M174">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M178">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M179">View MathML</a>

(2.26)

Proof Since all eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58">View MathML</a> are simple, then for large N and sufficiently small ε, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M181">View MathML</a> in a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a>. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M183">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M184">View MathML</a>

has two distinct solutions which we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M185">View MathML</a>. The decay of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M186">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M188">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M189">View MathML</a> will ensure the existence of the solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M190">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M191">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M192">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M193">View MathML</a>. For the second point, we recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M194">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M192">View MathML</a> and as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M189">View MathML</a>. Hence, by taking the limit, we obtain

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M198">View MathML</a>. This leads us to conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M199">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a> is a simple root.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M201">View MathML</a>. Then (2.17) and (2.21) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M202">View MathML</a>

(2.27)

Therefore θ, m must be chosen so that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M129">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M204">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a> be an eigenvalue and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M206">View MathML</a> be its approximation. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M207">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M208">View MathML</a>. From (2.27) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M209">View MathML</a>. Now we estimate the error <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M210">View MathML</a> for an eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a>. □

Theorem 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a>be an eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58">View MathML</a>. For sufficient largeN, we have the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M214">View MathML</a>

(2.28)

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M215">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M192">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M189">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M218">View MathML</a>, then from (2.27) and after replacing λ by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M206">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M220">View MathML</a>

(2.29)

Using the mean value theorem yields that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M221">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M222">View MathML</a>

(2.30)

Since the eigenvalues are simple, then for sufficiently large N<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M223">View MathML</a> and we get (2.28). The rest of the proof follows from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M224">View MathML</a> converges uniformly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M93">View MathML</a> in ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M188">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M189">View MathML</a>. □

3 The case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68">View MathML</a>

This section includes briefly a treatment similar to that of the previous section for the eigenvalue problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68">View MathML</a> introduced in Section 1 above. Notice that the condition (1.18) implies that the analysis of problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68">View MathML</a> is not included in that of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M232">View MathML</a> be a solution of (1.15) satisfying the following initial:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M233">View MathML</a>

(3.1)

Therefore, the eigenvalues of the problem in question are the zeros of the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M234">View MathML</a>

(3.2)

Similarly to [[22], p.220], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M235">View MathML</a> satisfies the system of integral equations

(3.3)

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M87">View MathML</a>, are the Volterra operators defined in (2.5) above. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M241">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M242">View MathML</a> to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M243">View MathML</a>

(3.5)

As in [12] we split <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M244">View MathML</a> into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M245">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M246">View MathML</a> is the known part

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M247">View MathML</a>

(3.7)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M248">View MathML</a> is the unknown one

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M249">View MathML</a>

(3.8)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M248">View MathML</a> is entire in λ for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M251">View MathML</a> for which, see [12],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M252">View MathML</a>

(3.9)

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M253">View MathML</a> to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M254">View MathML</a>

(3.10)

where θ is sufficiently small, for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M110">View MathML</a> and m are as in the previous section, but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M256">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M257">View MathML</a>

(3.11)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M258">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M259">View MathML</a>

(3.12)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M260">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M253">View MathML</a> belongs to the Paley-Wiener space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M117">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M118">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M264">View MathML</a>, then we can reconstruct the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M265">View MathML</a> via the following sampling formula:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M266">View MathML</a>

(3.13)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M123">View MathML</a>, and approximate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M265">View MathML</a> by its truncated series <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M270">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M271">View MathML</a>

(3.14)

Since all eigenvalues are real, then from now on we restrict ourselves to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M127">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M273">View MathML</a>, the truncation error, cf. (1.5), is given for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M129">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M275">View MathML</a>

(3.15)

where

(3.16)

The samples <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M277">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M278">View MathML</a>, in general, are not known explicitly. So, we approximate them by solving numerically <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M134">View MathML</a> initial value problems at the nodes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M135">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M281">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M282">View MathML</a> be the approximations of the samples of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M283">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M284">View MathML</a>, respectively. Now we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M285">View MathML</a>, which approximates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M270">View MathML</a>

(3.17)

Using standard methods for solving initial problems, we may assume that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M143">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M289">View MathML</a>

(3.18)

for a sufficiently small ε. From (2.13) we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M265">View MathML</a> satisfies the condition (1.9) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M256">View MathML</a> and therefore whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M147">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M293">View MathML</a>

(3.19)

where there is a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M294">View MathML</a> for which, cf. (1.10),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M295">View MathML</a>

(3.20)

Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M296">View MathML</a>

As in the above section, we have the following lemma.

Lemma 3.1For any eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M152">View MathML</a>of the problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68">View MathML</a>, we can find<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M176">View MathML</a>and sufficiently smallεsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M300">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M178">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M302">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M303">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M304">View MathML</a>are the solutions of the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M305">View MathML</a>

(3.21)

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M306">View MathML</a>

(3.22)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M307">View MathML</a>. Then (3.15) and (3.19) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M308">View MathML</a>

(3.23)

Therefore, θ, m must be chosen so that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M129">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M310">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a> be an eigenvalue and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M206">View MathML</a> be its approximation. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M313">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M314">View MathML</a>. From (3.23) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M315">View MathML</a>. Now we estimate the error <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M210">View MathML</a> for an eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a>. Finally, we have the following estimate.

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M159">View MathML</a>be an eigenvalue of the problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68">View MathML</a>. For sufficient largeN, we have the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M320">View MathML</a>

(3.24)

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M215">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M192">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M189">View MathML</a>.

In the following section, we have taken <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M324">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M118">View MathML</a>, in order to avoid the first singularity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M326">View MathML</a>.

4 Examples

This section includes three detailed worked examples illustrating the above technique accompanied by comparison with the sinc-method derived in [12]. It is clearly seen that the Hermite interpolations method gives remarkably better results. The first two examples are computed in [12] with the classical sinc-method where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M327">View MathML</a>. But in the last example, where eigenvalues cannot be computed concretely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M328">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M329">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M330">View MathML</a> we mean the absolute errors associated with the results of the classical sinc-method and our new method (Hermite interpolations), respectively. We indicate in these examples the effect of the amplitude error in the method by determining enclosure intervals for different values of ε. We also indicate the effect of the parameters m and θ by several choices. Each example is exhibited via figures that accurately illustrate the procedure near to some of the approximated eigenvalues. More explanations are given below. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M331">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M332">View MathML</a> are defined by

(4.1)

(4.2)

respectively. Recall also that the enclosure intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M335">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M336">View MathML</a> are determined by solving

(4.3)

(4.4)

respectively. We would like to mention that MATHEMATICA has been used to obtain the exact values for the three examples where eigenvalues cannot be computed concretely. MATHEMATICA is also used in rounding the exact eigenvalues, which are square roots.

Example 1

The boundary value problem

(4.5)

(4.6)

is a special case of the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M58">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M343">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M344">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M345">View MathML</a>. Here the characteristic function is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M346">View MathML</a>

(4.7)

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M95">View MathML</a> will be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M348">View MathML</a>

(4.8)

As is clearly seen, eigenvalues cannot be computed explicitly. Five tables indicate the application of our technique to this problem and the effect of ε, θ and m (Tables 1, 2, 3, 4 and 5). By exact, we mean the zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M93">View MathML</a> computed by Mathematica.

Table 3. Absolute error<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M374">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M364">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M365">View MathML</a>

Table 4. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M351">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M352">View MathML</a>, the exact solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M388">View MathML</a>are all inside the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M389">View MathML</a>for different values ofε

Table 5. With<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M364">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M365">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M388">View MathML</a>are all inside the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M389">View MathML</a>for different values ofε

Figures 1 and 2 illustrate the comparison between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M422">View MathML</a> for different values of m and θ. Figures 3 and 4, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M351">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M352">View MathML</a>, illustrate the enclosure intervals for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M426">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M427">View MathML</a>, respectively. Also, Figures 5 and 6 illustrate the enclosure intervals for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M426">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M427">View MathML</a>, respectively, but for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M364">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M365">View MathML</a>.

thumbnailFigure 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M433">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M435">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M436">View MathML</a>.

thumbnailFigure 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M433">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M440">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M441">View MathML</a>.

thumbnailFigure 3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M442">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M444">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M435">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M436">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M448">View MathML</a>.

thumbnailFigure 4. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M442">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M444">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M435">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M436">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M455">View MathML</a>.

thumbnailFigure 5. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M442">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M444">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M440">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M441">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M448">View MathML</a>.

thumbnailFigure 6. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M442">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M432">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M444">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M440">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M441">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M455">View MathML</a>.

Example 2

The Dirac system

(4.9)

(4.10)

is a special case of the problem treated in the previous section with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M473">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M474">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M475">View MathML</a>. The characteristic function is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M476">View MathML</a>

(4.11)

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M246">View MathML</a> will be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M478">View MathML</a>

(4.12)

As in the previous example, Figures 7, 8, 9, 10, 11 and 12 illustrate the results of Tables 6, 7, 8, 9 and 10. Figures 7 and 8 illustrate the comparison between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M244">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M480">View MathML</a> for different values of m and θ. Figures 9 and 10, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M482">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M483">View MathML</a>, illustrate the enclosure intervals for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M426">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M427">View MathML</a>, respectively. Also, Figures 11 and 12 illustrate the enclosure intervals for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M426">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M427">View MathML</a>, respectively, but for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M488">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M489">View MathML</a>.

thumbnailFigure 7. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M491">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M493">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M494">View MathML</a>.

thumbnailFigure 8. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M491">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M498">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M499">View MathML</a>.

thumbnailFigure 9. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M493">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M494">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M448">View MathML</a>.

thumbnailFigure 10. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M493">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M494">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M455">View MathML</a>.

thumbnailFigure 11. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M498">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M499">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M448">View MathML</a>.

thumbnailFigure 12. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M498">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M499">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M455">View MathML</a>.

Table 8. Absolute error<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M374">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M488">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M489">View MathML</a>

Table 9. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M482">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M483">View MathML</a>, the exact solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M388">View MathML</a>are all inside the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M567">View MathML</a>for different values ofε

Table 10. With<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M488">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M489">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M388">View MathML</a>are all inside the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M567">View MathML</a>for different values ofε

Example 3

The boundary value problem

(4.13)

(4.14)

is a special case of the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M68">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M602">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M603">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M604">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M605">View MathML</a>. Here the characteristic function is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M606">View MathML</a>

(4.15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M607">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M608">View MathML</a> are Airy functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M609">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M610">View MathML</a>, respectively, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M611">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M612">View MathML</a> are derivatives of Airy functions. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M246">View MathML</a> will be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M614">View MathML</a>

(4.16)

Figures 13, 14 and Tables 11, 12 illustrate the applications of the method to this problem.

thumbnailFigure 13. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M619">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M620">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M621">View MathML</a>.

thumbnailFigure 14. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M500">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M490">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M502">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M434">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M619">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M620">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M455">View MathML</a>.

Table 12. With<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M350">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M630">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M631">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M388">View MathML</a>are all inside the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/36/mathml/M567">View MathML</a>for different values ofε

Competing interests

The author declares that he has no competing interests.

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.

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