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A new kind of double Chebyshev polynomial approximation on unbounded domains

Ayşe Betül Koç* and Aydın Kurnaz

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Department of Mathematics, Faculty of Science, Selcuk University, Konya, Turkey

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Citation and License

Boundary Value Problems 2013, 2013:10  doi:10.1186/1687-2770-2013-10


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/10


Received:2 October 2012
Accepted:4 January 2013
Published:22 January 2013

© 2013 Koç and Kurnaz; 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

In this study, a new solution scheme for the partial differential equations with variable coefficients defined on a large domain, especially including infinities, has been investigated. For this purpose, a spectral basis, called exponential Chebyshev (EC) polynomials, has been extended to a new kind of double Chebyshev polynomials. Many outstanding properties of those polynomials have been shown. The applicability and efficiency have been verified on an illustrative example.

MSC: 35A25.

Keywords:
partial differential equations; pseudospectral-collocation method; matrix method; unbounded domains

1 Introduction

The importance of special functions and orthogonal polynomials occupies a central position in the numerical analysis. Most common solution techniques of differential equations with these polynomials can be seen in [1-12]. One of the most important of those special functions is Chebyshev polynomials. The well-known first kind Chebyshev polynomials [1] are orthogonal with respect to the weight-function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M1">View MathML</a> on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M2">View MathML</a>. These polynomials have many applications in different areas of interest, and a lot of studies are devoted to show the merits of them in various ways. One of the application fields of Chebyshev polynomials can appear in the solution of differential equations. For example, Chebyshev polynomial approximations have been used to solve ordinary differential equations with boundary conditions in [1], with collocation points in [13], the general class of linear differential equations in [14,15], linear-integro differential equations with collocation points in [16], the system of high-order linear differential and integral equations with variable coefficients in [17,18], and the Sturm-Liouville problems in [19].

Some of the fundamental ideas of Chebyshev polynomials in one-variable techniques have been extended and developed to multi-variable cases by the studies of Fox et al.[1], Basu [20], Doha [21] and Mason et al.[5]. In recent years, the Chebyshev matrix method for the solution of partial differential equations (PDEs) has been proposed by Kesan [22] and Akyuz-Dascioglu [23] as well.

On the other hand, all of the above studies are considered on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M3">View MathML</a> in which Chebyshev polynomials are defined. Therefore, this limitation causes a failure of the Chebyshev approach in the problems that are naturally defined on larger domains, especially including infinity. Then, Guo et al.[24] has proposed a modified type of Chebyshev polynomials as an alternative to the solutions of the problems given in a nonnegative real domain. In his study, the basis functions called rational Chebyshev polynomials are orthogonal in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M4">View MathML</a> and are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M5">View MathML</a>

Parand et al. and Sezer et al. successfully applied spectral methods to solve problems on semi-infinite intervals [25,26]. These approaches can be identified as the methods of rational Chebyshev Tau and rational Chebyshev collocation, respectively. However, this kind of extension also fails to solve all of the problems over the whole real domain. More recently, we have introduced a new modified type of Chebyshev polynomials that is developed to handle the problems in the whole real range called exponential Chebyshev (EC) polynomials [27].

In this study, we have shown the extension of the EC polynomial method to multi-variable case, especially, to two-variable problems.

2 Properties of double EC polynomials

The well-known first kind Chebyshev polynomials are orthogonal in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M3">View MathML</a> with respect to the weight-function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M7">View MathML</a> and can be simply determined with the help of the recurrence formula [1]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M8">View MathML</a>

(2.1)

Therefore, the exponential Chebyshev (EC) functions are recently defined in a similar fashion as follows [27].

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M9">View MathML</a>

be a function space with the weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M10">View MathML</a>. We also assume that, for a nonnegative integer n, the nth derivative of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M11">View MathML</a> is also in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M12">View MathML</a>. Then an EC polynomial can be given by

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M14">View MathML</a> .

This definition leads to the three-term recurrence equation for EC polynomials

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M15">View MathML</a>

(2.2)

This definition also satisfies the orthogonality condition [27]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M16">View MathML</a>

(2.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M18">View MathML</a> is the Kronecker function.

Double EC functions

Basu [20] has given the product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M19">View MathML</a> which is a form of bivariate Chebyshev polynomials. Mason et al.[5] and Doha [11] have also mentioned a Chebyshev polynomial expression for an infinitely differentiable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M20">View MathML</a> defined on the square <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M21">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M22">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M24">View MathML</a> are Chebyshev polynomials of the first kind, and the double primes indicate that the first term is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M25">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M27">View MathML</a> are to be taken as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M29">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M30">View MathML</a>, respectively.

Definition

Based on Basu’s study, now we introduce double EC polynomials in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M31">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M33">View MathML</a> are EC polynomials defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M34">View MathML</a>

Recurrence relation The polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M35">View MathML</a> satisfies the recurrence relations

(2.5)

(2.6)

If the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M38">View MathML</a> is continuous throughout the whole infinite domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M39">View MathML</a>, then the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M35">View MathML</a>’s are biorthogonal with respect to the weight function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M41">View MathML</a>

(2.7)

and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M42">View MathML</a>

(2.8)

Multiplication<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M43">View MathML</a> is said to be of higher order than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M44">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M45">View MathML</a>. Then the following result holds:

(2.9)

Function approximation

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M20">View MathML</a> be an infinitely differentiable function defined on the square <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M48">View MathML</a>. Then it may be expressed in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M49">View MathML</a>

(2.10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M50">View MathML</a>

(2.11)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M20">View MathML</a> in Eq. (2.10) is truncated up to the mth and nth terms, then it can be written in the matrix form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M52">View MathML</a>

(2.12)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M53">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M54">View MathML</a> EC polynomial matrix with entries <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M35">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M56">View MathML</a>

(2.13)

and A is an unknown coefficient vector,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M57">View MathML</a>

(2.14)

Matrix relations of the derivatives of a function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M58">View MathML</a>th-order partial derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M20">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M60">View MathML</a>

(2.15)

and its matrix form is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M61">View MathML</a>

(2.16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M63">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M64">View MathML</a>

Proposition 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M20">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M66">View MathML</a>th-order derivative be given by (2.12) and (2.16), respectively. Then there exists a relation between the double EC coefficient row vector<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M67">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M58">View MathML</a>th-order partial derivatives of the vector<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M69">View MathML</a>of size<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M54">View MathML</a>as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M71">View MathML</a>

(2.17)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M72">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M73">View MathML</a>are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M74">View MathML</a>operational matrices for partial derivatives given in the following forms:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M75">View MathML</a>

and

Here, IandOare<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M77">View MathML</a>identity and zero matrices, respectively, andTdenotes the usual matrix transpose.

Proof Taking the partial derivatives of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M79">View MathML</a> and both sides of the recurrence relation (2.5) with respect to x, we get

(2.18)

(2.19)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M82">View MathML</a>

(2.20)

By using the relations (2.18)-(2.20) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M83">View MathML</a> the elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M84">View MathML</a> of the matrix of partial derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M72">View MathML</a> can be obtained from the following equalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M86">View MathML</a>

(2.21)

Similarly, taking the partial derivatives of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M88">View MathML</a> and both sides of the recurrence relation (2.6) with respect to y, respectively, we write

(2.22)

(2.23)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M91">View MathML</a>

(2.24)

Then with the help of the relations (2.22)-(2.24), the elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M92">View MathML</a> of the matrices of partial derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M73">View MathML</a> can be obtained from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M94">View MathML</a>

(2.25)

We have noted here that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M95">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M96">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M97">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M98">View MathML</a>.

From (2.21) and (2.25), the following equalities hold for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M99">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M100">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M101">View MathML</a>

(2.26)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M102">View MathML</a>

(2.27)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M104">View MathML</a> and I denotes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M77">View MathML</a> identity matrix.

Then utilizing the equalities in (2.26) and (2.27), the explicit relation between the double EC polynomial row vector and those of its derivatives has been proved as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M106">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M107">View MathML</a>

 □

Remark<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M108">View MathML</a>.

CorollaryFrom Eqs. (2.16) and (2.17), it is clear that the derivatives of the function are expressed in terms of double EC coefficients as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M109">View MathML</a>

(2.28)

3 Collocation method with double EC polynomials

In the process of obtaining the numerical solutions of partial differential equations with the double EC method, the main idea or major step is to evaluate the necessary Chebyshev coefficients of the unknown function. So, in Section 2, we give the explicit relations between the polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M35">View MathML</a> of an unknown function and those of its derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M111">View MathML</a> for different nonnegative integer values of i and j.

In this section, we consider the higher-order linear PDE with variable coefficients of a general form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M112">View MathML</a>

(3.1)

with the conditions mentioned in [23] as three possible cases:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M113">View MathML</a>

(3.2)

and/or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M114">View MathML</a>

(3.3)

and/or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M115">View MathML</a>

(3.4)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M117">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M123">View MathML</a> are known functions on the square <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M48">View MathML</a>. We now describe an approximate solution of this problem by means of double EC series as defined in (2.10). Our aim is to find the EC coefficients in the vector A. For this reason, we can represent the given problem and its conditions by a system of linear algebraic equations by using collocation points.

Now, the collocation points can be determined in the inner domain as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M125">View MathML</a>

(3.5)

and at the boundaries

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M127">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M129">View MathML</a>.

Since EC polynomials are convergent at both boundaries, namely their values are either 1 or −1, the appearance of infinity in the collocation points does not cause a loss in the method.

Therefore, when we substitute the collocation points into the problem (3.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M130">View MathML</a>

(3.6)

The system (3.6) can be written in the matrix form as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M131">View MathML</a>

(3.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M132">View MathML</a> denotes the diagonal matrix with the elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M133">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M134">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M135">View MathML</a>) and F denotes the column matrix with the elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M136">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M134">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M138">View MathML</a>).

Putting the collocation points into derivatives of the unknown function as in Eq. (2.28) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M139">View MathML</a>

(3.8)

where E is the block matrix given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M140">View MathML</a>

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M141">View MathML</a>, we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M142">View MathML</a>

(3.9)

Therefore, from Eq. (3.7), we get a system of the matrix equation for the PDE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M143">View MathML</a>

(3.10)

which corresponds to a system of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M144">View MathML</a> linear algebraic equations with unknown double EC coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M145">View MathML</a>.

It is also noted that the structures of matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M132">View MathML</a> and F vary according to the number of collocation points and the structure of the problem. However, E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M73">View MathML</a> do not change their nature for fixed values of m and n which are truncation limits of the EC series. In other words, the changes in E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M73">View MathML</a> are just dependent on the number of collocation points.

Briefly, we can denote the expression in the parenthesis of (3.10) by W and write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M151">View MathML</a>

(3.11)

Then the augmented matrix of Eq. (3.11) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M152">View MathML</a>

(3.12)

Applying the same procedure for the given conditions (3.2)-(3.4), we have

(3.13)

(3.14)

(3.15)

Then these can be written in a compact form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M156">View MathML</a>

(3.16)

where V is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M157">View MathML</a> matrix and R is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M158">View MathML</a> matrix, so that h is the rank of all row matrices belonging to the given condition. The augmented matrices of the conditions become

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M159">View MathML</a>

(3.17)

Consequently, (3.12) together with (3.17) can be written in a new augmented matrix form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M160">View MathML</a>

(3.18)

This form can be achieved by replacing some rows of (3.12) by the rows of (3.17) accordingly, or adding those rows to the matrix (3.12) provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M161">View MathML</a>. Then it can be written in the following compact form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M162">View MathML</a>

(3.19)

Finally, the vector A (thereby the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M145">View MathML</a>) is determined by applying some numerical methods (e.g., Gauss elimination) designed especially to solve the system of linear equations. Therefore, the approximate solution can be obtained. In other words, it gives the double EC series solution of the problem (3.1) with given conditions.

4 Illustration

Now, we give an example to show the ability and efficiency of the double EC polynomial approximation method.

Example

Let us consider the linear partial differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M164">View MathML</a>

with the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M165">View MathML</a>

It is known that the exact solution of the problem is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M166">View MathML</a>.

Absolute errors of the proposed procedure at the grid points are tabulated for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M167">View MathML</a> in Table 1.

Table 1. Absolute errors of Example at different points

Contour plots of the exact solutions and the approximate solutions are given for the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M169">View MathML</a> in (a) and (b) and for the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M171">View MathML</a> in (c) and (d) of Figure 1, respectively. Figure 2 shows a graphical representation of the exact solution and, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/10/mathml/M167">View MathML</a>, the approximate solution of the example.

thumbnailFigure 1. Contour plots of exact and approximate solutions.

thumbnailFigure 2. Exact and approximate solution of the example.

5 Conclusion

In this article, a new solution scheme for the partial differential equation with variable coefficients defined on unbounded domains has been investigated and EC polynomials have been extended to double EC polynomials to solve multi-variable problems. It is also noted that the double EC-collocation method is very effective and has a direct ability to solve multi-variable (especially two-variable) problems in the infinite domain. For computational purposes, this approach also avoids more computations by using sparse operational matrices and saves much memory. On the other hand, the double EC polynomial approach deals directly with infinite boundaries, and their operational matrices are of few non-zero entries lain along two subdiagonals.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ABK wrote the first draft and AK corrected and improved the final version. All authors read and approved the final draft.

Acknowledgements

This study was supported by the Research Projects Center (BAP) of Selcuk University. The authors would like to thank the editor and referees for their valuable comments and remarks which led to a great improvement of the article. Also, ABK and AK would like to thank the Selcuk University and TUBITAK for their support. We note here that this study was presented orally at the International Conference on Applied Analysis and Algebra (ICAAA 2012), Istanbul, 20-24 June, (2012).

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