Received 12 April 2016; accepted 12 June 2016; published 15 June 2016
In this paper, we consider the standard form of the Goursat problem   as provided below
This equation has been examined by several numerical methods such as Runge-Kutta method, finite difference method, finite elements method and Adomian Decomposition Method (ADM).
We will prove the applicability and effectiveness of RDTM on solving linear and non-linear Goursat problems. The main advantage of RDTM is that it can be applied directly to the problems without requiring linearization, discretization or perturbation.
2. One Dimensional Differential Transform Method
The differential transform of the function is defined as follows
where is the original function and is the transformed function. Here means the kth deriv-
ative with respect to x.
The differential inverse transform of is defined as
Combining (3) and (4) yields
From (3) and (4) it is easy to see that the concept of the differential transform is derived from Taylor series expansion [see Table 1].
3. Analysis of the Reduced Differential Transform Method
The basic definitions of reduced differential transform method are introduced below.
Assume that the function is analytic and continuously differentiable with respect to time t and space x in the domain of interest, then let
Where the t-dimensional spectrum function is the transformed function. The differential inverse of is defined as follows
Then combining Equation (6) and (7) we can write
Table 1. One-dimensional differential transformation  .
Now, we express the Goursat problem in the standard operator form.
With the initial conditions
We applying RDTM of Equations (1) and (2) giving
where is transformed value of and is transformed value of.
By iterative calculations we obtain the following values of as:
From (7) we have.
One can get the exact solution of (1) by substituting (14) and (15) in (16).
With reference to the articles   . We easy prove the transformation in the following Table 2.
Table 2. Reduced differential transformation.
4. Application and Results
In this section, we apply the method to some linear and non linear Goursat problem in order to demonstrate its efficiency.
A. The linear homogeneous Goursat problem
We first consider the linear homogeneous Goursat problem defined below
Where is a linear function of u.
Example 1: Consider the homogeneous Goursat problem
Taking RDTM of (19) and (20), we obtain
Substituting (22) into (21) and using the recurrence relation, we will reach to the results listed below.
And so on. In general, we have for substituting all into (7) yields the solution
. This result is in full agreement with the one obtained in  by VIM.
Example 2: Now consider the homogeneous Goursat problem
Applying RDT to (23) and (24) we obtain
Substituting (26) into (25) and using the recurrence relation we have
And so on. By substituting all into (7), the solution becomes. This results perfectly matches the results obtained in  by VIM.
B. The linear inhomogeneous Goursat problem:
We now consider inhomogeneous Goursat problem
where is linear function of u.
Example 3: We first consider the linear in homogeneous Goursat problem.
Taking RDM of (27) and (28) will lead to
Substituting (30) into (29) and using the recurrence relation we have
And so on. In general, we have for substituting all in (7) yields the solution
, this result is again identical to the one obtained in  by VIM .
Example 4: Consider the linear in homogeneous Goursat problem
Taking RDM of (31) and (32) gives rise to
Substituting (34) into (33) and using the recurrence relation we have
And so on. In general, we have except substituting all in (7) yields the solu-
tion, This result is again in full agreement with one obtained in  by VIM .
C. The non-linear Goursat problem:
is non-linear term.
Here, we apply RDTM to non-linear Goursat problem.
Example 5: We first consider the non-linear Goursat problem .
Taking RDM of (37), (38) yields
Substituting (41) into (40) and (39) and using the recurrence relation we have
yields the solution which is in full agreement with one obtained in  by VIM.
Example 6: We finally consider the non-linear Goursat problem
Taking RDM of (42) and (43) we obtain
where is the reduced transform of such that
And so on,
Substituting (45) into (44) and using the recurrence relation we have and so on.
Therefore, the solution is which is the exact solution.
In this section, we use the Adomian decomposition method to obtain the solution of (1) and (2). and discuss the comparison between the reduced differential transform method and the Adomian decomposition method.
With reference to the article  -  , (1) can be rewritten In an operator form as:
The inverse operators and can defined as:
By applying and respectively to both sides of (46) and substituting gives
Adomian method admits the use of recursive relation
We applying adomian decomposition method to examples (3) and (6) to illustrate the comparison between the two method.
Following the pervious discussion and using (49) Equations (27) and (28) gives
This result is again identical to the one obtained by the RDTM example (3).
Again applying pervious discussion and using (49) Equations (42) and (43) gives
where are adomian polynomials for the nonlinear term giving by
Then the closed form solution is giving by
This result is again identical to the one obtained by RDTM in example (6).
We have carried out the comparative study between the reduced differential transform method and the Adomian decomposition method by handling the Goursat problem, Two numerical examples have shown that the reduced differential transform method is a very simple technique to handle linear and nonlinear Goursat problem than the Adomian decomposition method, and also, it is demonstrated that the reduced differential transform method solves linear and nonlinear Goursat problem without using any complicated polynomials like as the Adomian polynomials.
In addition, the obtained series solution by the reduced differential transform method converges faster than those obtained by the Adomian decomposition method. It is concluded that this simple reduced differential transform method is a powerful technique to handle linear and nonlinear initial value problems.
The Goursat problem has been analyzed using reduced differential transform method. All the illustrative examples have shown that the reduced differential transform method is powerful mathematical tool to solving Goursat problem. It is also a promising method to solve other nonlinear equations, the presented method reduces the computational difficulties existing in the other traditional methods and all the calculations can be done by simple manipulations.
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