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Hassan Eltayeb, Adem Kılıçman, "Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations", Abstract and Applied Analysis, vol. 2012, Article ID 412948, 13 pages, 2012. https://doi.org/10.1155/2012/412948
Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations
We develop a method to obtain approximate solutions of nonlinear system of partial differential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled partial differential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples, and results of the present technique have close agreement with approximate solutions obtained with the help of Adomian decomposition method (ADM).
Most of phenomena in nature are described by nonlinear differential equations. So scientists in different branches of science try to solve them. But because of nonlinear part of these groups of equations, finding an exact solution is not easy. Different analytical methods have been applied to find a solution to them. For example, Adomian has presented and developed a so-called decomposition method for solving algebraic, differential, integrodifferential, differential-delay and partial differential equations. In the nonlinear case for ordinary differential equations and partial differential equations, the method has the advantage of dealing directly with the problem [1, 2]. These equations are solved without transforming them to more simple ones. The method avoids linearization, perturbation, discretization, or any unrealistic assumptions [3, 4]. It was suggested in  that the noise terms appears always for inhomogeneous equations. Most recently, Wazwaz  established a necessary condition that is essentially needed to ensure the appearance of “noise terms” in the inhomogeneous equations. In the present paper, the intimate connection between the Sumudu transform theory and decomposition method arises in the solution of nonlinear partial differential equations is demonstrated.
The Sumudu transform is defined over the set of the functions by the following formula:
The existence and the uniqueness were discussed in , for further details and properties of the Sumudu transform and its derivatives we refer to . In , some fundamental properties of the Sumudu transform were established.
In , this new transform was applied to the one-dimensional neutron transport equation. In fact one can easily show that there is a strong relationship between double Sumudu and double Laplace transforms, see .
Further in , the Sumudu transform was extended to the distributions and some of their properties were also studied in . Recently Kılıçman et al. applied this transform to solve the system of differential equations, see .
A very interesting fact about Sumudu transform is that the original function and its Sumudu transform have the same Taylor coefficients except a factor . Thus if then see .
Similarly, the Sumudu transform sends combinations, , into permutations, and hence it will be useful in the discrete systems. Further
Thus we further note that since many practical engineering problems involve mechanical or electrical systems where action is defined by discontinuous or impulsive forcing terms. Then the Sumudu transform can be effectively used to solve ordinary differential equations as well as partial differential equations and engineering problems. Recently, the Sumudu transform was introduced as a new integral transform on a time scale to solve a system of dynamic equations, see . Then the results were applied on ordinary differential equations when , difference equations when , but also, for -difference equations when , where or for which has important applications in quantum theory and on different types of time scales like , , and the space of the harmonic numbers. During this study we use the following Sumudu transform of derivatives.
Theorem 1.1. Let be in , and let denote the Sumudu transform of the th derivative, of , then for For more details, see .
We consider the general inhomogeneous nonlinear equation with initial conditions given below: where is the highest order derivative which is assumed to be easily invertible, is a linear differential operator of order less than , represents the nonlinear terms and is the source term. First we explain the main idea of SDM: the method consists of applying Sumudu transform
Using the differential property of Laplace transform and initial conditions we get
By arrangement we have
The second step in Sumudu decomposition method is that we represent solution as an infinite series: and the nonlinear term can be decomposed as where are Adomian polynomials  of , , , , and it can be calculated by formula
On comparing both sides of (1.14) and by using standard ADM we have: then it follows that
In more general, we have
On applying the inverse Sumudu transform to (1.15) and (1.17), we get where represents the term that is arising from source term and prescribed initial conditions. On using the inverse Sumudu transform to and using the given condition we get where the function , obtained from a term by using the initial condition is given by the terms appears while applying the inverse Sumudu transform on the source term and using the given conditions. We define where , . Then we verify that satisfies the original equation (1.7). We now consider the particular form of inhomogeneous nonlinear partial differential equations: with the initial condition where is second-order differential operator, represents a general non-linear differential operator where as is source term. The methodology consists of applying Sumudu transform first on both sides of (1.10) and (1.23),
Then by the second step in Sumudu decomposition method and inverse transform as in the previous we have
Now in order to illustrate STDM we consider some examples. Consider a nonlinear partial differential equation with initial conditions
By applying the inverse Sumudu transform for (2.3), we get which assumes a series solution of the function and is given by
In (2.6) and are Adomian polynomials that represents nonlinear terms. So Adomian polynomials are given as follows:
The few components of the Adomian polynomials are given as follows:
From the above equations we obtain
Then the first few terms of follow immediately upon setting
Therefore the solution obtained by LDM is given as follows:
Example 2.1. Consider the system of nonlinear coupled partial differential equation
with initial conditions
Applying the Sumudu transform (denoted by ) we have
On using inverse Sumudu transform in (2.14), our required recursive relation is given by
The recursive relations are where , , and are Adomian polynomials representing the nonlinear terms  in above equations. The few components of Adomian polynomials are given as follows
By this recursive relation we can find other components of the solution
The solution of above system is given by
Example 2.2. Consider the following homogeneous linear system of PDEs:
with initial conditions
Taking the Sumudu transform on both sides of (2.20), then by using the differentiation property of Sumudu transform and initial conditions, (2.21) gives
Using the decomposition series (2.23) for the linear terms , and , , we obtain
The SADM presents the recursive relations
Taking the inverse Sumudu transform of both sides of (2.25) we have and so on for other components. Using (1.11), the series solutions are given by
Then the solutions follows
Example 2.3. Consider the system of nonlinear partial differential equations
with initial conditions
On using Sumudu transform on both sides of (2.29), and by taking Sumudu transform for the initial conditions of (2.30) we get
Similar to the previous example, we rewrite and by the infinite series (1.11), then inserting these series into both sides of (2.31) yields where the terms and are handled with the help of Adomian polynomials by (1.12) that represent the nonlinear terms and , respectively. We have a few terms of the Adomian polynomials for and which are given by
By taking the inverse Sumudu transform we have
Using the inverse Sumudu transform on (2.35) we have
The rest terms can be determined in the same way. Therefore, the series solutions are given by
Then the solution for the above system is as follows:
The Sumudu transform-Adomian decomposition method has been applied to linear and nonlinear systems of partial differential equations. Three examples have been presented, this method shows that it is very useful and reliable for any nonlinear partial differential equation systems. Therefore, this method can be applied to many complicated linear and nonlinear PDEs.
The authors would like to express their sincere thanks and gratitude to the reviewers for their valuable comments and suggestions for the improvement of this paper. The first author acknowledges the support by the Research Center, College of Science, King Saud University. The second author also gratefully acknowledges the partial support by the University Putra Malaysia under the Research University Grant Scheme (RUGS) 05-01-09-0720RU and FRGS 01-11-09-723FR.
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