Abstract
In this article, a new homotopy technique is presented for the mathematical analysis of finding the solution of a firstorder inhomogeneous partial differential equation (PDE) . 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 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 [16]. 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 [79], 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 based on the decomposition of a righthand side function which leads to the construction of new homotopies. It is shown that decomposing 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 firstorder 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:
with the boundary conditions
where A is a general differential operator, B is a boundary operator, 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:
By the homotopy technique, we construct the homotopy
which is equivalent to
where
p is an embedding parameter, is an initial approximation of (1), which satisfies the boundary conditions. As p changes from zero to unity, changes from to . In this technique, the convergence of a solution depends on the choice of , that is, we can have different approximate solutions for different .
Let us decompose the source function as . If we take
in (3), we obtain the following homotopy:
which is equivalent to
Obviously, from (6) we have
As the embedding parameter p changes from zero to unity, changes from to .
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:
Setting , we get the approximate solution of (1)
In the next section, it is shown that the decomposition of the righthand 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:
where a, b, g and f are continuous functions in some region of the plane and . By solving this boundary value problem by the homotopy perturbation method, we obtain an approximate or exact solution . Before proceeding further, let us introduce the integral operator S defined in the following form:
Then the derivative of operator S with respect to y is defined as
Rewriting as a sum of two functions and then constructing a homotopy in equation (6), we have
which is equivalent to
Substituting (7) into (12), and equating the coefficient of the terms by the same power in p, we have
If there exists the relationship
between and , then we have from (13)
and
Hence the approximate or exact solution of problem (8)(9) is obtained as
and the source function is obtained from (14) in the following form:
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 in such a way that the functions and have the relationship in (14). If we have such a case, then we are looking for an arbitrary such that the functions and have the following relationship:
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 in the following linear problem:
where a and b are constants.
4.1 Case 1: the source function is a polynomial
We consider that and is an nthorder polynomial in problem (17)(18). Hence can be written in the following form:
where are constants and α, β are natural numbers.
If the polynomial is decomposed as
from (15), we can get the solution of the problem in two steps. We take as an nthorder polynomial given as
where are constants. Then becomes
In order to determine the unknown coefficients in terms of , we substitute (21), (22), and (23) into (20)
and then using (19), we get the following equations:
Then we obtain
As a result, constructing the following homotopy:
leads us to reaching the solution in two steps.
4.2 Case 2: the source function is a sum of two functions and
Let us take in problem (17)(18). Since is a continuous function, and are also continuous functions. Based on (15), is decomposed as follows:
Then we can get the solution of the problem in two steps. Hence the function can be rewritten in the following form:
The solutions of the above ordinary differential equation are
and
On the other hand, we have
and
If either (28) or (29) satisfy equation (30), and equation (31) has a solution, then it is possible to decompose the function to accelerate the convergence of the solution. As a result, by constructing the following homotopy:
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
By constructing the following homotopy:
which is equivalent to
we get the functions , as it is explained in Section 4.2. It is obvious that satisfies equation (30). Moreover, from equation (31), can be found as follows:
If we put the function and in the homotopy (33) or (34), we get
and we get the following:
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:
In this problem, the righthand side function is a thirdorder polynomial. From (15), we have
Based on the source function , if we take as
and substitute (40) into (39), we obtain
Based on these results, we have , and the homotopy is constructed in the following form:
and we get the following:
Hence, we obtain the exact solution in the following form:
with shortlength calculation.
5.3 Example 3
Let us find the solution of following BVP:
As in the previous examples, using (15) we get
Based on the source function , we conclude that is in the following form:
Substituting (45) into (44), we obtain
Now, these results let us take , and construct the homotopy in the following form:
which leads to the following equations:
Hence, the exact solution becomes
which is obtained by minimum amount of calculations.
5.4 Example 4
Let us find the solution of the following firstorder nonlinear BVP:
Using (15) we get
Based on the source function , we find is in the following form:
Substituting (50) into (49), we obtain
Now we can take , and construct the homotopy in the following form:
which leads to the following equations:
Hence, we obtain the exact solution
with shortlength calculation.
6 Conclusion
In this paper, we employ a new homotopy perturbation method to obtain the solution of a firstorder inhomogeneous PDE. In this method, each decomposition of the source function leads to a new homotopy. However, we develop a method to obtain the proper decomposition of 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 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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