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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 [

The Sumudu transform is defined over the set of the functions

The existence and the uniqueness were discussed in [

In [

Further in [

A very interesting fact about Sumudu transform is that the original function and its Sumudu transform have the same Taylor coefficients except a factor

Similarly, the Sumudu transform sends combinations,

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

Let

We consider the general inhomogeneous nonlinear equation with initial conditions given below:

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:

Substitution of (

On comparing both sides of (

In more general, we have

On applying the inverse Sumudu transform to (

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

By taking Sumudu transform for (

By applying the inverse Sumudu transform for (

Using (

In (

The few components of the Adomian polynomials are given as follows:

From the above equations we obtain

Then the first few terms of

Therefore the solution obtained by LDM is given as follows:

Consider the system of nonlinear coupled partial differential equation

Applying the Sumudu transform (denoted by

On using inverse Sumudu transform in (

The recursive relations are

By this recursive relation we can find other components of the solution

The solution of above system is given by

Consider the following homogeneous linear system of PDEs:

Taking the Sumudu transform on both sides of (

Using the decomposition series (

The SADM presents the recursive relations

Taking the inverse Sumudu transform of both sides of (

Then the solutions follows

Consider the system of nonlinear partial differential equations

On using Sumudu transform on both sides of (

Similar to the previous example, we rewrite

By taking the inverse Sumudu transform we have

Using the inverse Sumudu transform on (

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.