In this paper a new method for solving Goursat problem is introduced using Reduced Differential Transform Method (RDTM). The approximate analytical solution of the problem is calculated in the form of series with easily computable components. The comparison of the methodology presented in this paper with some other well known techniques demonstrates the effectiveness and power of the newly proposed methodology.
In this paper, we consider the standard form of the Goursat problem [
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.
The differential transform of the function
where
ative with respect to x.
The differential inverse transform of
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
The basic definitions of reduced differential transform method are introduced below.
Definition
Assume that the function
Where the t-dimensional spectrum function
Then combining Equation (6) and (7) we can write
| Function form | Transformed form |
|---|---|
Now, we express the Goursat problem in the standard operator form.
With the initial conditions
where
We applying RDTM of Equations (1) and (2) giving
where
By iterative calculations we obtain the following values of
From (7) we have.
One can get the exact solution of (1) by substituting (14) and (15) in (16).
With reference to the articles [
| Functional form | Transformed form |
|---|---|
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
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
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
B. The linear inhomogeneous Goursat problem:
We now consider inhomogeneous Goursat problem
where
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
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
tion
C. The non-linear Goursat problem:
where
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
And
Substituting (41) into (40) and (39) and using the recurrence relation we have
Example 6: We finally consider the non-linear Goursat problem
Taking RDM of (42) and (43) we obtain
where
And so on,
Substituting (45) into (44) and using the recurrence relation we have
Therefore, the solution is
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 [
where
The inverse operators
By applying
Adomian method admits the use of recursive relation
where
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 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
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.
Sharaf Mohmoud,Mohamed Gubara, (2016) Reduced Differential Transform Method for Solving Linear and Nonlinear Goursat Problem. Applied Mathematics,07,1049-1056. doi: 10.4236/am.2016.710092