Open Access Research

Chebyshev wavelets method for solving Bratu’s problem

Changqing Yang* and Jianhua Hou

Author Affiliations

Department of Science, Huaihai Institute of Technology, Lianyungang, Jiangsu, 222005, China

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Boundary Value Problems 2013, 2013:142  doi:10.1186/1687-2770-2013-142


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/142


Received:14 May 2013
Accepted:16 May 2013
Published:2 June 2013

© 2013 Yang and Hou; 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

A numerical method for one-dimensional Bratu’s problem is presented in this work. The method is based on Chebyshev wavelets approximates. The operational matrix of derivative of Chebyshev wavelets is introduced. The matrix together with the collocation method are then utilized to transform the differential equation into a system of algebraic equations. Numerical examples are presented to verify the efficiency and accuracy of the proposed algorithm. The results reveal that the method is accurate and easy to implement.

1 Introduction

In this paper, we consider the boundary-value problem and initial value problem of Bratu’s problem. It is well known that Bratu’s boundary value problem in one-dimensional planar coordinates is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M1">View MathML</a>

(1)

with the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M2">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M3">View MathML</a> is a constant, the exact solution of equation (1) is given by [1]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M4">View MathML</a>

(2)

where θ satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M5">View MathML</a>

(3)

The problem has zero, one or two solutions when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M8">View MathML</a>, respectively, where the critical value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M9">View MathML</a> satisfies the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M10">View MathML</a>

It was evaluated in [1-3] that the critical value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M9">View MathML</a> is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M12">View MathML</a>.

In addition, an initial value problem of Bratu’s problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M13">View MathML</a>

(4)

with the initial conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M14">View MathML</a> will be investigated.

Bratu’s problem is also used in a large variety of applications such as the fuel ignition model of the thermal combustion theory, the model of thermal reaction process, the Chandrasekhar model of the expansion of the universe, questions in geometry and relativity about the Chandrasekhar model, chemical reaction theory, radiative heat transfer and nanotechnology [4-11].

A substantial amount of research work has been done for the study of Bratu’s problem. Boyd [2,12] employed Chebyshev polynomial expansions and the Gegenbauer as base functions. Syam and Hamdan [8] presented the Laplace decomposition method for solving Bratu’s problem. Also, Aksoy and Pakdemirli [13] developed a perturbation solution to Bratu-type equations. Wazwaz [10] presented the Adomian decomposition method for solving Bratu’s problem. In addition, the applications of spline method, wavelet method and Sinc-Galerkin method for solution of Bratu’s problem have been used by [14-17].

In recent years, the wavelet applications in dealing with dynamic system problems, especially in solving differential equations with two-point boundary value constraints have been discussed in many papers [4,16,18]. By transforming differential equations into algebraic equations, the solution may be found by determining the corresponding coefficients that satisfy the algebraic equations. Some efforts have been made to solve Bratu’s problem by using the wavelet collocation method [16].

In the present article, we apply the Chebyshev wavelets method to find the approximate solution of Bratu’s problem. The method is based on expanding the solution by Chebyshev wavelets with unknown coefficients. The properties of Chebyshev wavelets together with the collocation method are utilized to evaluate the unknown coefficients and then an approximate solution to (1) is identified.

2 Chebyshev wavelets and their properties

2.1 Wavelets and Chebyshev wavelets

In recent years, wavelets have been very successful in many science and engineering fields. They constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M15">View MathML</a>. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets [19]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M16">View MathML</a>

Chebyshev wavelets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M17">View MathML</a> have four arguments, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M18">View MathML</a>, k can assume any positive integer, m is the degree of Chebyshev polynomials of first kind and x denotes the time.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M19">View MathML</a>

(5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M20">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M18">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M23">View MathML</a> are the well-known Chebyshev polynomials of order m, which are orthogonal with respect to the weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M24">View MathML</a> and satisfy the following recursive formula:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M25">View MathML</a>

We should note that the set of Chebyshev wavelets is orthogonal with respect to the weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M26">View MathML</a>.

The derivative of Chebyshev polynomials is a linear combination of lower-order Chebyshev polynomials, in fact [20],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M27">View MathML</a>

(6)

2.2 Function approximation

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28">View MathML</a> defined over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M29">View MathML</a> may be expanded as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M30">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M31">View MathML</a>, in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M32">View MathML</a> denotes the inner product with the weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M33">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28">View MathML</a> in (7) is truncated, then (7) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M35">View MathML</a>

(8)

where C and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M36">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M37">View MathML</a> matrices given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M38">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M39">View MathML</a>

3 Chebyshev wavelets operational matrix of derivative

In this section we first derive the operational matrix D of derivative which plays a great role in dealing with Bratu’s problem.

In the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M40">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M41">View MathML</a>

Applying (6) the derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M42">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M43">View MathML</a>

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M44">View MathML</a> is zero outside the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M45">View MathML</a>, so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M46">View MathML</a>

(9)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M47">View MathML</a>

In fact we have shown that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M48">View MathML</a>

(10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M49">View MathML</a>

From (10), it can be generalized for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M50">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M51">View MathML</a>

(11)

4 Solution of Bratu’s problem

Consider Bratu’s problem given in (1). In order to use Chebyshev wavelets, we first approximate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M53">View MathML</a>

Applying (11) we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M54">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M55">View MathML</a>

(12)

We now collocate (12) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M56">View MathML</a> points at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M57">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M58">View MathML</a>

(13)

Suitable collocation points are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M59">View MathML</a>

Thus with the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M2">View MathML</a>, we have

(14)

(15)

Equations (13), (14) and (15) generate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M63">View MathML</a> set of nonlinear equations. The approximate solution of the vector C is obtained by solving the nonlinear system using the Gauss-Newton method.

5 Error analysis

Theorem 5.1A function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M64">View MathML</a>, with bounded second derivative, say<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M65">View MathML</a>, can be expanded as an infinite sum of Chebyshev wavelets, and the series converges uniformly to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28">View MathML</a>, that is[21],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M67">View MathML</a>

Since the truncated Chebyshev wavelets series is an approximate solution of Bratu’s problem, so one has an error function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M68">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M70">View MathML</a>

The error bound of the approximate solution by using Chebyshev wavelets series is given by the following theorem.

Theorem 5.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M71">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M72">View MathML</a>is the approximate solution using the Chebyshev wavelets method. Then the error bound would be obtained as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M73">View MathML</a>

Proof Applying the definition of norm in the inner product space, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M74">View MathML</a>

Because the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M75">View MathML</a> is divided into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M76">View MathML</a> subintervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M18">View MathML</a>, then we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M79">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M80">View MathML</a> is the interpolating polynomial of degree m which agrees with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28">View MathML</a> at the Chebyshev nodes on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M82">View MathML</a> with the following error bound for interpolating [22,23]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M83">View MathML</a>

Therefore, using the above equation, we would get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M84">View MathML</a>

 □

6 Numerical examples

To illustrate the ability and reliability of the method for Bratu’s problem, some examples are provided. The results reveal that the method is very effective and simple.

Example 6.1 Consider the first case for Bratu’s equation as follows, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M85">View MathML</a>[14,15]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M86">View MathML</a>

(16)

We solve the equation by using the Chebyshev wavelets method with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M88">View MathML</a>. The numerical results obtained are presented in Table 1. Table 1 shows the comparison between the absolute error of exact and approximate solutions for various values of M (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M87">View MathML</a>). Moreover, higher accuracy can be achieved by taking higher order approximations.

Table 1. Computed absolute errors for Example 6.1

Example 6.2 Consider the initial value problem [10,14-16,24]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M90">View MathML</a>

(17)

The exact solution is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M91">View MathML</a>. Here we solve it using Chebyshev wavelets, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M93">View MathML</a>. First we assume that the unknown function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M28">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M95">View MathML</a>

Applying (12) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M96">View MathML</a>

(18)

Using the initial condition, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/142/mathml/M97">View MathML</a>

(19)

Equations (18) and (19) generate a system of nonlinear equations. These equations can be solved for unknown coefficients of the vector C. A comparison between the exact and the approximate solutions is demonstrated in Figure 1. From Figure 1, it can be found that the obtained approximate solutions are very close to the exact solution. In addition, Table 2 shows the exact and approximate solutions using the method presented in Section 3 and compares the results with the method presented in [16]. Also, by comparing the results of the table, we see that the results of the proposed method are more accurate.

thumbnailFigure 1. Comparison of solutions for Example 6.2.

Table 2. Comparison of the results of the Chebyshev and Legendre wavelets method for Example 6.2

7 Conclusions

The aim of present work is to develop an efficient and accurate method for solving Bratu’s problems. The Chebyshev wavelet operational matrix of derivative together with the collocation method are used to reduce the problem to the solution of nonlinear algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

CY completed the main study, carried out the results of this article and drafted the manuscript. JH checked the proofs and verified the calculation. All the authors read and approved the final manuscript.

Acknowledgements

Project is supported by the Huaihai Institute of Technology (No. Z2001151).

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