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)
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
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,
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,
Let us decompose the source function as
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,
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
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
Then the derivative of operator S with respect to y is defined as
Rewriting
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
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 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
where a and b are constants.
4.1 Case 1: the source function
f
(
x
,
y
)
is a polynomial
We consider that
where
If the polynomial
from (15), we can get the solution of the problem in two steps. We take
where
and
In order to determine the unknown coefficients
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
f
(
x
,
y
)
is a sum of two functions
r
(
x
)
and
t
(
y
)
Let us take
Then we can get the solution of the problem in two steps. Hence the function
Since
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
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
If we put the function
and we get the following:
Hence, the solution
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
Based on the source function
and substitute (40) into (39), we obtain
Based on these results, we have
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
Substituting (45) into (44), we obtain
Now, these results let us take
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
Substituting (50) into (49), we obtain
Now we can take
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
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).
References

Biazar, J, Azimi, F: He’s homotopy perturbation method for solving Helmoltz equation. Int. J. Contemp. Math. Sci.. 3, 739–744 (2008)

Biazar, J, Ghazvini, H: Convergence of the homotopy perturbation method for partial differential equations. Nonlinear Anal., Real World Appl.. 10, 2633–2640 (2009). Publisher Full Text

Sweilam, NH, Khader, MM: Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method. Comput. Math. Appl.. 58, 2134–2141 (2009). Publisher Full Text

Biazar, J, Ghazvini, H: Homotopy perturbation method for solving hyperbolic partial differential equations. Comput. Math. Appl.. 56, 453–458 (2008). Publisher Full Text

Junfeng, L: An analytical approach to the sineGordon equation using the modified homotopy perturbation method. Comput. Math. Appl.. 58, 2313–2319 (2009). Publisher Full Text

Neamaty, A, Darzi, R: Comparison between the variational iteration method and the homotopy perturbation method for the SturmLiouville differential equation. Bound. Value Probl.. 2010, Article ID 317369 (2010)

He, JH: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng.. 178, 257–262 (1999). Publisher Full Text

He, JH: A coupling method of homotopy technique and a perturbation technique for nonlinear problems. Int. J. NonLinear Mech.. 35, 37–43 (2000). PubMed Abstract  Publisher Full Text

He, JH: Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput.. 135, 73–79 (2003). Publisher Full Text

He, JH: Comparison of homotopy method and homotopy analysis method. Appl. Math. Comput.. 156, 527–539 (2004). Publisher Full Text