New Approach for Solving Partial Differential Equations Based on Collocation Method

In this paper, a new approach for solving partial differential equations was introduced. The collocation method based on LA-transform and proposed the solution as a power series that conforming Taylor series. The method attacks the problem in a direct way and in a straightforward fashion without using linearization, or any other restrictive assumption that may change the behavior of the equation under discussion. 
Five illustrated examples are introduced to clarifying the accuracy, ease implementation and efficiency of suggested method. The LA-transform was used to eliminate the linear differential operator in the differential equation.


Introduction
Differential equations can be used to describe physical, engineering, biological and chemical phenomena as a mathematical manner, as well as their use in economic, sciences and engineering. Differential equations have developed and become increasingly important in all fields of science and their applications. Therefore, getting the solution of the differential equation is very important in mathematics and these fields. Only the simplest differential equations can be solved to obtain the exact solution. Many methods have been proposed to obtain approximate or analytic solutions to solve it, such as, homotopy analysis method (HAM) [1 -5], homotopy perturbation method (HPM) [6 -11], Admoain decomposition method (ADM) [12 -17], variational iteration method (VIM) [18 -20], artificial neural network (Ann) [21 -25], Laplace decomposition method [26,27], Sumud u decomposition method [28][29][30] and Collocation Method [31][32][33]. All decomposition methods are proposed the solution as a series form and then the solution obtained iteratively.
In this paper, we present a new approach for solving PDEs based on suggested the solution as a series form that actually matches with Taylor series.
In the next section, we will introduce the definition of the LA-transform and main of its properties that are used in suggested approch to eliminate the linear differential operator in the differential equation.

LA-Transform and its Inverse
LA-transform is integral transform suggested by Luma and Alaa in [35] which is defined as follows: Where v is a real number, which is improper integral converges. Table (1) gives the main properties of this transform.  We noted that The inverse transform has a linear combination property, i.e., For more details and the advantages of this transform see [35].
In this method the unknown function ( , ) can be expressed as infinite series of the form: The next step is to determine the terms (n= 0, 1, 2 …).
Taking the LA-transform (with respect to the variable t) for the equation (2) to get: From the properties in the Table (1), equation (6) becomes: From equation (3) we have: So: Taking the inverse of the LA-transform for both sides of equation (9), to get: Now in equation (10) we can get (depending on equation (4) and since the operator R is independent of Also, the nonlinear part ( ) of equation (10), can be written as follows: Then the nonlinear part of equation (10) can be written as: Finally, we can write the inhomogeneous term as follows: Substituting equations (11), (14) and (15) Then substituting equation (19) in (4) to get ( , ) .