Solving this ODE and applying inverse transform they obtain the exact solution of the problem. Ranjit Dhunde and G. L. Waghmare [2] used double Laplace transform and Mohand M. Abdelrahim Mehgob [4] applied double Elzaki transform to solve PIDE. They converted PIDE to an algebraic equation and Appling inverse transform they obtained exact solution. Differential Transform Method is a semi-analytical numerical technique which depends on Taylor's series for the solution of differential and integral equations. The concept of differential transform method was first introduced by Zhou [9] who solved linear and non-linear initial value problems in electric circuit analysis. Recently, The various types of differential equations, integro differential equations, and Volterra integral equations solved by using two-dimensional DTM [12-19]. The modified differential transform method is a modified version of two-dimensional DTM, and it will take less computational time and effort to solve PIDEs. In this paper, we will find an analytical-numerical solution of linear partial integro differential equations (PIDE) with convolution kernel using a modified differential transform method.
Partial Integro Differential Equations (PIDEs) occur naturally in various fields of science and technology. The main purpose of this paper is to study how to solve linear partial integro differential equations with convolution kernel by using the Laplace-Differential Transform Method (LDTM). This method is a simple and reliable technique for solving such equations. The efficiency and reliability of this method is also illustrated with some examples. The result obtained by this method is compared with the result obtained by Double Laplace Transform and Double Elzaki Transform method.
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