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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Analysis of the new homotopy perturbation method for linear and nonlinear problems

Ali Demir1*, Sertaç Erman1, Berrak Özgür1 and Esra Korkmaz2

Author Affiliations

1 Department of Mathematics, Kocaeli University Umuttepe, Izmit, Kocaeli, 41380, Turkey

2 Ardahan University, Merkez, Ardahan, 75000, Turkey

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Boundary Value Problems 2013, 2013:61  doi:10.1186/1687-2770-2013-61

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/61


Received:14 December 2012
Accepted:6 March 2013
Published:26 March 2013

© 2013 Demir 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 article, a new homotopy technique is presented for the mathematical analysis of finding the solution of a first-order inhomogeneous partial differential equation (PDE) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M1">View MathML</a>. The homotopy perturbation method (HPM) and the decomposition of a source function are used together to develop this new technique. The homotopy constructed in this technique is based on the decomposition of a source function. Various decompositions of source functions lead to various homotopies. Using the fact that the decomposition of a source function affects the convergence of a solution leads us to development of a new method for the decomposition of a source function to accelerate the convergence of a solution. The purpose of this study is to show that constructing the proper homotopy by decomposing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> in a correct way determines the solution with less computational work than using the existing approach while supplying quantitatively reliable results. Moreover, this method can be generalized to all inhomogeneous PDE problems.

1 Introduction

Many problems in the fields of physics, engineering and biology are modeled by linear and nonlinear PDEs. In recent years, the homotopy perturbation method (HPM) has been employed to solve these PDEs [1-6]. HPM has been used extensively to solve nonlinear boundary and initial value problems. Therefore HPM is of great interest to many researchers and scientists. HPM, first presented by Ji Huan He [7-9], is a powerful mathematical tool to investigate a wide variety of problems arising in different fields. It is obtained by successfully coupling homotopy theory in topology with perturbation theory. In HPM, a complicated problem under study is continuously deformed into a simple problem which is easy to solve to obtain an analytic or approximate solution [10].

In this work, a new homotopy perturbation technique is proposed to find the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M3">View MathML</a> based on the decomposition of a right-hand side function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> which leads to the construction of new homotopies. It is shown that decomposing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> in a correct way, requiring less computational work than the existing approach, helps us to determine the solution. The results reveal that the proposed method is very effective and simple.

The paper is organized as follows. In Section 2, an analysis of the new homotopy perturbation method is given. The construction of a new homotopy based on the decomposition of a source function for a first-order inhomogeneous PDE is covered in Section 3. Some examples are given to illustrate the construction of a new homotopy in Section 4. Finally, some concluding remarks are given in Section 5.

2 New homotopy perturbation method

To illustrate the basic ideas of the homotopy analysis method, we consider the following nonlinear differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M6">View MathML</a>

(1)

with the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M7">View MathML</a>

(2)

where A is a general differential operator, B is a boundary operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M8">View MathML</a> is a known analytical function, and Γ is the boundary of domain Ω.

The operator A in (1) can be rewritten as a sum of L and N, where L and N are linear and nonlinear parts of A, respectively, as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M9">View MathML</a>

By the homotopy technique, we construct the homotopy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M10">View MathML</a>

(3)

which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M11">View MathML</a>

(4)

where

p is an embedding parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M13">View MathML</a> is an initial approximation of (1), which satisfies the boundary conditions. As p changes from zero to unity, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M14">View MathML</a> changes from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M13">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M16">View MathML</a>. In this technique, the convergence of a solution depends on the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M13">View MathML</a>, that is, we can have different approximate solutions for different <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M13">View MathML</a>.

Let us decompose the source function as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M19">View MathML</a>. If we take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M20">View MathML</a>

in (3), we obtain the following homotopy:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M21">View MathML</a>

(5)

which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M22">View MathML</a>

(6)

Obviously, from (6) we have

As the embedding parameter p changes from zero to unity, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M14">View MathML</a> changes from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M25">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M16">View MathML</a>.

According to He’s homotopy perturbation method, we can first use the embedding parameter p as a small parameter and assume that the solution of (6) can be written as a power series in p as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M27">View MathML</a>

(7)

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M28">View MathML</a>, we get the approximate solution of (1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M29">View MathML</a>

In the next section, it is shown that the decomposition of the right-hand side function has a great effect on the amount of calculation and the speed of convergence of the solution.

3 Construction of a new homotopy based on the decomposition of a source function

Let us consider the following boundary value problem with the following inhomogeneous PDE:

(8)

(9)

where a, b, g and f are continuous functions in some region of the plane and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M32">View MathML</a>. By solving this boundary value problem by the homotopy perturbation method, we obtain an approximate or exact solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M3">View MathML</a>. Before proceeding further, let us introduce the integral operator S defined in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M34">View MathML</a>

(10)

Then the derivative of operator S with respect to y is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M35">View MathML</a>

Rewriting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> as a sum of two functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M37">View MathML</a> and then constructing a homotopy in equation (6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M38">View MathML</a>

(11)

which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M39">View MathML</a>

(12)

Substituting (7) into (12), and equating the coefficient of the terms by the same power in p, we have

(13)

If there exists the relationship

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M41">View MathML</a>

(14)

between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M43">View MathML</a>, then we have from (13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M44">View MathML</a>

and

Hence the approximate or exact solution of problem (8)-(9) is obtained as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M46">View MathML</a>

and the source function is obtained from (14) in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M47">View MathML</a>

(15)

Therefore equation (15) is a necessary condition to accelerate the convergence. If equation (15) has a solution, then we obtain the approximate or exact solution of problem (8)-(9) in two steps.

However, it is not always possible to decompose the source function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> in such a way that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M43">View MathML</a> have the relationship in (14). If we have such a case, then we are looking for an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M51">View MathML</a> such that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M43">View MathML</a> have the following relationship:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M54">View MathML</a>

(16)

In this case, we get the approximate or exact solution of the problem in more than two steps. Moreover, we can get the solution in the form of series.

4 Analysis of the new homotopy perturbation method for linear problems

In this section we illustrate the theory of the new homotopy perturbation method given in Sections 2 and 3 for some special <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> in the following linear problem:

(17)

(18)

where a and b are constants.

4.1 Case 1: the source function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> is a polynomial

We consider that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> is an nth-order polynomial in problem (17)-(18). Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> can be written in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M62">View MathML</a>

(19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M63">View MathML</a> are constants and α, β are natural numbers.

If the polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> is decomposed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M65">View MathML</a>

(20)

from (15), we can get the solution of the problem in two steps. We take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42">View MathML</a> as an nth-order polynomial given as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M67">View MathML</a>

(21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M68">View MathML</a> are constants. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M69">View MathML</a> becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M70">View MathML</a>

(22)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M71">View MathML</a> becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M72">View MathML</a>

(23)

In order to determine the unknown coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M68">View MathML</a> in terms of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M63">View MathML</a>, we substitute (21), (22), and (23) into (20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M75">View MathML</a>

(24)

and then using (19), we get the following equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M76">View MathML</a>

Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M77">View MathML</a>

As a result, constructing the following homotopy:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M78">View MathML</a>

leads us to reaching the solution in two steps.

4.2 Case 2: the source function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> is a sum of two functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M81">View MathML</a>

Let us take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M82">View MathML</a> in problem (17)-(18). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> is a continuous function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M81">View MathML</a> are also continuous functions. Based on (15), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M82">View MathML</a> is decomposed as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M87">View MathML</a>

(25)

Then we can get the solution of the problem in two steps. Hence the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> can be rewritten in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M89">View MathML</a>

(26)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M82">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M91">View MathML</a>

(27)

The solutions of the above ordinary differential equation are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M92">View MathML</a>

(28)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M93">View MathML</a>

(29)

On the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M94">View MathML</a>

(30)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M95">View MathML</a>

(31)

If either (28) or (29) satisfy equation (30), and equation (31) has a solution, then it is possible to decompose the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> to accelerate the convergence of the solution. As a result, by constructing the following homotopy:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M97">View MathML</a>

we reach the solution in two steps.

5 Numerical examples

5.1 Example 1

Consider the inhomogeneous linear boundary value problem (BVP) with constant coefficients

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M98">View MathML</a>

(32)

By constructing the following homotopy:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M99">View MathML</a>

(33)

which is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M100">View MathML</a>

(34)

we get the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M102">View MathML</a> as it is explained in Section 4.2. It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M103">View MathML</a> satisfies equation (30). Moreover, from equation (31), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M101">View MathML</a> can be found as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M105">View MathML</a>

(35)

If we put the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M102">View MathML</a> in the homotopy (33) or (34), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M108">View MathML</a>

(36)

and we get the following:

Hence, the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M3">View MathML</a> becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M111">View MathML</a>

(37)

which is the exact solution of the problem with minimum amount of calculation.

5.2 Example 2

Consider the following inhomogeneous linear BVP with variable coefficients:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M112">View MathML</a>

(38)

In this problem, the right-hand side function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> is a third-order polynomial. From (15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M114">View MathML</a>

(39)

Based on the source function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M115">View MathML</a>, if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M117">View MathML</a>

(40)

and substitute (40) into (39), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M118">View MathML</a>

Based on these results, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M120">View MathML</a> and the homotopy is constructed in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M121">View MathML</a>

(41)

and we get the following:

Hence, we obtain the exact solution in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M123">View MathML</a>

(42)

with short-length calculation.

5.3 Example 3

Let us find the solution of following BVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M124">View MathML</a>

(43)

As in the previous examples, using (15) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M125">View MathML</a>

(44)

Based on the source function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M126">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42">View MathML</a> is in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M128">View MathML</a>

(45)

Substituting (45) into (44), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M129">View MathML</a>

Now, these results let us take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M131">View MathML</a> and construct the homotopy in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M132">View MathML</a>

(46)

which leads to the following equations:

Hence, the exact solution becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M134">View MathML</a>

(47)

which is obtained by minimum amount of calculations.

5.4 Example 4

Let us find the solution of the following first-order non-linear BVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M135">View MathML</a>

(48)

Using (15) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M136">View MathML</a>

(49)

Based on the source function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M137">View MathML</a>, we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M42">View MathML</a> is in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M139">View MathML</a>

(50)

Substituting (50) into (49), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M140">View MathML</a>

Now we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M142">View MathML</a> and construct the homotopy in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M143">View MathML</a>

(51)

which leads to the following equations:

Hence, we obtain the exact solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M145">View MathML</a>

(52)

with short-length calculation.

6 Conclusion

In this paper, we employ a new homotopy perturbation method to obtain the solution of a first-order inhomogeneous PDE. In this method, each decomposition of the source function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> leads to a new homotopy. However, we develop a method to obtain the proper decomposition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> which lets us obtain the solution with minimum computation and accelerate the convergence of the solution. This study shows that the decomposition of the source function has a great effect on the amount of computations and the acceleration of the convergence of the solution. Comparing to the standard one, decomposing the source function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/61/mathml/M2">View MathML</a> properly is a simple and very effective tool for calculating the exact or approximate solutions with less computational work.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read, checked and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The research was supported by parts by the Scientific and Technical Research Council of Turkey (TUBITAK).

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