Differential equations theory is an important Mathematical branch which is used to describe practical problems in physics, chemistry, and biology and so on . It is well known that many phenomena in scientific fields such as reaction-diffusion process, population growth, solid physics, fluid dynamics, Mathematical biology and chemical kinetics, can be modelled by systems of linear or nonlinear PDEs. In order to understand and analyze these phenomena well, it needs to know solution of systems of these linear or nonlinear PDEs. So, it is a crucial work to obtain solutions of systems of linear or nonlinear PDEs in the science. With this idea; scientists and mathematicians have developed and searched some methods such as Hirota bilinear method , Exp-function method , tanh method  , sine-cosine method , Galerkin method  and Differential transform method (DTM) . It is more difficult to obtain solutions of nonlinear PDEs than those of linear differential equations. Therefore, it may not always be possible to obtain analytical solutions of these equations  . In this case, it is used analytical methods giving series solutions. In these kinds of methods, the solutions are sought in the form of series    . Analytical methods are based on finding the other terms of the series from given initial conditions for the problem being considered. At this point, it is encountered the concept of convergence of the series. So, it is necessary to perform convergence analysis of these methods. As this convergence analysis can be carried out theoretically, one can gain information about the convergence of the series solution by looking at the absolute error between the numerical solution and the analytical solution. In some Analytic methods, a very good convergence can be achieved with only a few terms of the series, but more terms can be needed in some problems. That is, if the terms of the series increase, this provides better convergence to the analytical solution.
In this study, I have used LDM to solve a system of nonlinear partial differential equations for two different initial conditions. Later, we compared the obtained results with exact solutions and solution obtained by Method of Differential Quadrature . In this paper, we are not going to explain the LDM. For that, I have referred papers    to illustrate this method for a nonlinear system of PDE’s.
Consider a system of nonlinear partial differential equations on our interest of region given by:
with initial condition
here, we have consider the general form of boundary conditions. Taking Laplace transform of Equations (2.1) and (2.2) with respect to t, we get
Taking inverse Laplace transform of above system with respect to “t”, we get
Let us suppose that,
be the solution of given system of Equations (2.1), (2.2) in series form. Also we can decompose the nonlinear terms appeared in given system by using adomian polynomials, namely
where , , and are adomian polynomials . From the Equations (2.5), (2.6), (2.7) and (2.8), we get
Comparing the both sides of above system of equations, we get the following recursive relations
Note that the solution of (2.1), (2.2) can exhibit a shock phenomenon for finite t; we select f(x, y) and g(x, y) such that the shock occurs for a value of t far from our region of interest. Let
Therefore from the recursive relation (2.9) and (2.10), we get
then , can be calculate as
Also, and are calculated as
Substitute all the values of and in the Equation (2.7), we get
This is an exact solution of the given system of nonlinear partial differential Equations (2.1) and (2.2). We have verified this through the substitution, which is identical to the solution obtained by R. E. Bellman using the method of differential quadrature . Let we change the initial conditions to
From the recursive relation (2.9), (2.10) and above initial conditions, we get
Also, and are calculated as
and so on. Substitute all the values of and in Equation (2.7), we get
(The shock occurs at ). This is an approximate solution of given system of equations.
From the examples above, we can clearly say that we can calculate and when explicitly solutions exist for given initial functions. More importantly, the methodology    does have potential application to the system of nonlinear partial differential equations and clearly in the case of stochastic parameters as well. The given system of equation has a unique solution for the given boundary conditions.
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