Approximation of partial integro differential equations with a weakly singular kernel using local meshless method

In this work, a meshless method in local setting and Laplace transform are coupled to approximate partial integro-differential equations (PIDEs). The typical method of obtaining the approximate solution of such type of equations is to use the meshless time stepping process. The major drawback with meshless time stepping process is the time instability. In this work the time instability is avoided using Laplace transform, and the issue of shape parameter sensitivity, and ill-conditioning of differentiation matrix which is commonly encountered in global meshless methods is resolved using local meshless method. To check the efficiency and accuracy of the proposed method, numerical approximation of different PIDEs are obtained and validated against the exact solutions.