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Fuzzy approximate solutions of second-order fuzzy linear boundary value problems

Xiaobin Guo13*, Dequan Shang2 and Xiaoquan Lu3

Author Affiliations

1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China

2 Department of Public Courses, Gansu College of Traditional Chinese Medicine, Lanzhou, 730000, China

3 College of Chemistry and Chemical Engineering, Northwest Normal University, Lanzhou, 730070, China

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Boundary Value Problems 2013, 2013:212  doi:10.1186/1687-2770-2013-212


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/212


Received:21 January 2013
Accepted:2 August 2013
Published:30 September 2013

© 2013 Guo et al.; 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 paper, approximate solutions of second-order linear differential equations with fuzzy boundary conditions, in which coefficient functions maintain the sign, are investigated. The fuzzy linear boundary value problem is converted to a crisp function system of linear equations by the undetermined fuzzy coefficients method. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method.

Keywords:
fuzzy numbers; matrix analysis; fuzzy boundary value problems; fuzzy approximate solutions

1 Introduction

Nowadays, fuzzy differential equations (FDEs) is a popular topic studied by many researchers since it is utilized widely for the purpose of modeling problems in science and engineering. Most of the practical problems require the solution of a fuzzy differential equation (FDE) which satisfies fuzzy initial or boundary conditions, therefore a fuzzy initial or boundary problem should be solved. However, many fuzzy initial or boundary value problems could not be solved exactly, sometimes it is even impossible to find their analytical solutions. Thus, considering their approximate solutions is becoming more important.

Prior to discussing fuzzy differential equations and their associated numerical algorithms, it is necessary to present an appropriate brief introduction to derivative of the fuzzy-valued function. The concept of a fuzzy derivative was first introduced by Chang and Zadeh [1], followed up by Dubois and Prade [2] who used the extension principle in their approach. Other fuzzy derivative concepts were proposed by Puri and Ralescu [3] and Goetschel and Vaxman [4] as an extension of the Hukuhara derivative of multivalued functions. Kandel and Byatt [5,6] applied the concept of fuzzy differential equation to the analysis of fuzzy dynamical problems.

The numerical methods for solving fuzzy differential equation are introduced in [7-9]. In 2001, Buckley and Feuring [10] presented two analytical methods for solving an nth-order fuzzy linear differential equation with fuzzy initial conditions. Their first method of solution was to fuzzify the crisp solution and then check to see if it satisfies the fuzzy differential equations with fuzzy initial conditions. The second method was the reverse of the first method; in that they firstly solved the fuzzy initial value problem and then checked to see if it defined a fuzzy function. In 2008, Allahviranllo et al.[11] obtained the approximate solution of nth-order linear differential equations with fuzzy initial conditions by using the collocation method. In 2003, O’Regan et al.[12] proved a super-linear result for fuzzy boundary value problems relying on a general Schauder theorem in the metric space. Meanwhile Lakshmikantham et al.[13] investigated the solution of two-point boundary value problems associated with nonlinear fuzzy differential equations by using the extension principle. In 2008, Chen Minghao et al.[14] obtained the conclusion: two-point boundary value problems have the analytic solution only on condition that the new structure and properties to the fuzzy number are given. But for second-order fuzzy linear boundary value problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M1">View MathML</a>

(1.1)

associated with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M2">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M3">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M4">View MathML</a>

(1.4)

it is not the case. Once the coefficient functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M7">View MathML</a> are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> maintain the sign on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M8">View MathML</a>, a unique solution must exist.

In this paper, we consider the approximate solution of a class of second-order linear differential Eq. (1.1) under fuzzy boundary value conditions (1.2), (1.3) and (1.4). Based on the undetermined fuzzy coefficients method, we convert a second-order linear differential equation to the crisp system of linear equations. Secondly, we investigate their cases according to indifferent cases of coefficient functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> from the original systems and build the corresponding crisp systems of linear equations. Then we derive a fuzzy approximate solution of the fuzzy linear differential equation from solving crisp function systems of linear equations. Finally, some examples are given to illustrate the proposed method. The structure of this paper is organized as follows.

In Section 2, we recall some basic definitions and results about fuzzy numbers as well as fuzzy derivative of the fuzzy-valued function. In Sections 3, 4 and 5, we build crisp function systems of linear equations via analyzing different cases based on the coefficient functions of the fuzzy linear differential equation in detail. The proposed algorithms are illustrated by solving some examples in Section 6 and the conclusion is drawn in Section 7.

2 Preliminaries

2.1 Fuzzy numbers

Definition 2.1[1]

A fuzzy number is a fuzzy set like <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M14">View MathML</a> which satisfies:

(1) u is upper semi-continuous,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M15">View MathML</a> outside some interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M16">View MathML</a>,

(3) there are real numbers a, b such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M17">View MathML</a> and

(3.1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M18">View MathML</a> is monotonic increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M19">View MathML</a>,

(3.2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M18">View MathML</a> is monotonic decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M21">View MathML</a>,

(3.3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M23">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M24">View MathML</a> be the set of all real fuzzy numbers which are normal, upper semi-continuous, convex and compactly supported fuzzy sets.

Definition 2.2[2]

A fuzzy number u in a parametric form is a pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M25">View MathML</a> of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M28">View MathML</a>, which satisfies the following requirements:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M26">View MathML</a> is a bounded monotonic increasing left continuous function,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M27">View MathML</a> is a bounded monotonic decreasing left continuous function,

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M28">View MathML</a>.

A crisp number x is simply represented by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M28">View MathML</a>. By appropriate definitions, the fuzzy number space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M35">View MathML</a> becomes a convex cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M24">View MathML</a> which could be embedded isomorphically and isometrically into a Banach space [15,16].

Definition 2.3[2]

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M39">View MathML</a> and arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M40">View MathML</a>. Then

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M41">View MathML</a> iff <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M43">View MathML</a>,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M44">View MathML</a>,

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M45">View MathML</a>,

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M46">View MathML</a>

Definition 2.4[17]

For arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M47">View MathML</a>, the quantity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M48">View MathML</a>

is the distance between fuzzy numbers u and v.

2.2 Second-order fuzzy boundary value problem

Definition 2.5[3]

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M49">View MathML</a>. If there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M50">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M51">View MathML</a>, then z is called the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M52">View MathML</a>-difference of fuzzy numbers x and y, and it is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M53">View MathML</a>.

In this paper the ⊖ sign stands always for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M52">View MathML</a>-difference, and let us remark that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M55">View MathML</a>.

Definition 2.6[18]

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M57">View MathML</a>. We say that f is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M52">View MathML</a> differential at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M59">View MathML</a>, if there exists an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M60">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M61">View MathML</a> sufficiently small, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M63">View MathML</a> and the limits (in the metric D)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M64">View MathML</a>

Lemma 2.1[19]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M65">View MathML</a>is differential on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M66">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M68">View MathML</a>are nonnegative and monotonic increasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M66">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M71">View MathML</a>is differential on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M66">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M73">View MathML</a>

Definition 2.7 The second-order differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M74">View MathML</a>

(2.1)

with the boundary value conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M75">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M76">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M77">View MathML</a>

(2.4)

are called the second-order fuzzy boundary value problems (FBVPs). The differential Eq. (2.1) along with fuzzy boundary value conditions (2.2), (2.3) and (2.4) are said to be second-order fuzzy differential equation No. 1, No. 2 and No. 3 boundary value problems, respectively.

In particular, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M78">View MathML</a> is a linear function with respect to y and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M79">View MathML</a>, Eq. (2.1) is reduced to Eq. (1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M80">View MathML</a>

and it is a linear differential equation. In this paper, we discuss the approximate solution of the second-order fuzzy linear differential function boundary value problem. For simplicity, we only discuss the second-order fuzzy linear differential function with fuzzy boundary value conditions (2.3) and (2.4).

3 Method for solving No. 2 FBVPs

3.1 The undetermined fuzzy coefficients method

The undetermined fuzzy coefficients method is to seek an approximate solution as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M81">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M83">View MathML</a>, are positive basic functions whose all differentiations are positive. We compute the fuzzy coefficients in (3.1) by setting the error to zero as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M84">View MathML</a>

(3.2)

Then we substitute (3.1) in (3.2) and represent them in parametric forms, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M85">View MathML</a>

(3.3)

Lemma 3.1[11]

Let basic functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M83">View MathML</a>, and all of their differentiations be positive, without loss of generality. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M88">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M90">View MathML</a>.

In order to solve Eq. (1.1) with condition (1.3), we need to investigate the system of Eq. (3.3). In this section we consider two cases.

3.2 Case 1 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M91">View MathML</a>

Suppose that coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> are nonnegative. From (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M94">View MathML</a>

(3.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M95">View MathML</a>

(3.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M96">View MathML</a>

(3.6)

And when coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> are negative, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M99">View MathML</a>

(3.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M100">View MathML</a>

(3.8)

If (3.1) is substituted in (3.4) and (3.5), respectively, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M101">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M102">View MathML</a>

also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M103">View MathML</a>

By setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M104">View MathML</a>

the following system is obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M105">View MathML</a>

(3.9)

Equation (3.9) is a system of linear equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M106">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M107">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M108">View MathML</a>

And

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M109">View MathML</a>

The variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M111">View MathML</a> are obtained by solving (3.9) by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M113">View MathML</a>. These variables yield the fuzzy approximate solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M114">View MathML</a>.

In the same way, when coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> are negative, we build the corresponding system of linear equations as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M117">View MathML</a>

(3.10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M118">View MathML</a>

3.3 Case 2 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M119">View MathML</a>

Suppose that coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a> is nonnegative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> is negative. From (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M122">View MathML</a>

(3.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M123">View MathML</a>

(3.12)

When coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a> is negative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> is nonnegative, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M126">View MathML</a>

(3.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M127">View MathML</a>

(3.14)

If (3.1) is substituted in (3.11) and (3.12), respectively, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M128">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M129">View MathML</a>

also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M130">View MathML</a>

By setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M131">View MathML</a>

the following system is obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M132">View MathML</a>

(3.15)

Equation (3.15) is a system of linear equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M106">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M134">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M135">View MathML</a>

And

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M136">View MathML</a>

The variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M111">View MathML</a> are obtained by solving (3.15) by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M113">View MathML</a>. These variables yield the fuzzy approximate solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M114">View MathML</a>.

In the same way, when coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a> is negative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> is nonnegative, we set up the corresponding system of linear equations as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M144">View MathML</a>

(3.16)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M145">View MathML</a>

4 Method for solving No. 3 FBVPs

4.1 The undetermined fuzzy coefficients method

Similarly, we compute the fuzzy coefficients in (3.1) by setting the error to zero as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M146">View MathML</a>

(4.1)

Then we substitute (3.1) in (4.1) and represent them in parametric forms:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M147">View MathML</a>

(4.2)

In order to solve Eq. (1.1) with Eq. (1.4), we need to investigate the system of Eq. (4.3). In this section we also consider two cases.

4.2 Case 1 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M91">View MathML</a>

Suppose that coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> are nonnegative. From (4.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M151">View MathML</a>

(4.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M152">View MathML</a>

(4.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M153">View MathML</a>

(4.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M154">View MathML</a>

(4.6)

And when coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> are negative, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M157">View MathML</a>

(4.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M158">View MathML</a>

(4.8)

If (3.1) is substituted in (4.3) and (4.4), respectively, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M159">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M160">View MathML</a>

also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M161">View MathML</a>

By setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M162">View MathML</a>

the following system is obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M163">View MathML</a>

(4.9)

Equation (4.9) is a system of linear equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M106">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M165">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M166">View MathML</a>

And

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M167">View MathML</a>

The variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M111">View MathML</a> are obtained by solving (4.9) by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M113">View MathML</a>. These variables yield the fuzzy approximate solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M114">View MathML</a>.

Similarly, when coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> are negative, we build the corresponding system of linear equations as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M175">View MathML</a>

(4.10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M176">View MathML</a>

4.3 Case 2 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M119">View MathML</a>

Suppose that coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a> is nonnegative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> is negative. From (4.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M180">View MathML</a>

(4.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M181">View MathML</a>

(4.12)

When coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a> is negative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> is nonnegative, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M184">View MathML</a>

(4.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M185">View MathML</a>

(4.14)

If (3.1) is substituted in (4.11) and (4.12), respectively, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M186">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M187">View MathML</a>

also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M188">View MathML</a>

By setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M189">View MathML</a>

the following system is obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M190">View MathML</a>

(4.15)

Equation (4.15) is a system of linear equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M191">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M192">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M193">View MathML</a>

And

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M194">View MathML</a>

The variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M111">View MathML</a> are obtained by solving (4.14) by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M113">View MathML</a>. These variables yield the fuzzy approximate solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M114">View MathML</a>.

Similarly, when coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M5">View MathML</a> is negative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M6">View MathML</a> is nonnegative, we extend the corresponding system of linear equations as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M202">View MathML</a>

(4.16)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M203">View MathML</a>

Likewise, for Eq. (1.1) with fuzzy boundary conditions (1.2), the following results are obvious.

5 Approximate solutions of second-order FLBVPs

The above model linear Eqs. (3.9), (3.10), (3.15), (3.16), (4.9), (4.10), (4.15) and (4.16) are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M204">View MathML</a> function systems of linear equations and they have the same form as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M205">View MathML</a>

(5.1)

In the process of solving Eq. (5.1) by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M207">View MathML</a>, no matter whether it is consistent or inconsistent, we obtain the minimal norm least squares solution [20] by using the generalized inverse of the coefficient matrix S, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M208">View MathML</a>

(5.2)

Thus we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M209">View MathML</a>

Therefore, we obtain the fuzzy approximate solution of the original fuzzy linear differential equation as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M210">View MathML</a>

(5.3)

6 Numerical examples

Example 6.1 Consider the following second-order fuzzy linear differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M211">View MathML</a>

The exact solution is as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M212">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M214">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M215">View MathML</a>

From (3.11), we build the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M216">View MathML</a>

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M217">View MathML</a>, the parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M224">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M225">View MathML</a> are obtained, and by putting them into (5.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M226">View MathML</a>

Tables 1, 2, 3 and 4 show the comparisons between the exact solution and the approximate solution at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M227">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M228">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M229">View MathML</a>; all data are computed by Matlab7.x.

Table 1. Comparisons between the exact solution and the approximate solution

Table 2. Comparisons between the exact solution and the approximate solution

Table 3. Comparisons between the exact solution and the approximate solution

Table 4. Comparisons between the exact solution and the approximate solution

Example 6.2 Consider the following second-order fuzzy linear differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M238">View MathML</a>

The exact solution of the equation is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M239">View MathML</a>

The extended linear equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M106">View MathML</a> is as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M241">View MathML</a>

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M242">View MathML</a>, the parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M250">View MathML</a> are obtained. Tables 5 and 6 show comparisons between the exact solution and the approximate solution at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M228">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/212/mathml/M229">View MathML</a>; all data were calculated by Matlab7.x.

Table 5. Comparisons between the exact solution and the approximate solution

Table 6. Comparisons between the exact solution and the approximate solution

Form Tables 1, 2, 3, 4, 5 and 6, we know that the approximate solutions obtained from the proposed method are best close to the exact solutions of original linear deferential equations with fuzzy boundary value conditions.

7 Conclusion

In this paper the approximate method similar to the undetermined coefficients method, based on a positive basis for solving second-order fuzzy linear boundary value problems, was discussed. Three classes of boundary conditions and the general case were considered. According to the sign of coefficient functions of the fuzzy linear differential equation, the corresponding function systems of linear equations were set up. Following each other, fuzzy approximate solutions were obtained by solving a crisp function extended system of linear equations. Numerical examples show that our methods are practical and efficient.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

The work is supported by the Natural Scientific Funds of PR China (71061013, 21175108) and the Youth Research Ability Project of Northwest Normal University (NWNU-LKQN-1120).

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