A LOCAL MESHLESS RBF METHOD FOR SOLVING FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH OPTIMAL SHAPE PARAMETERS

Abstract

This paper investigates the application of the meshless local radial basis functions collocation method (LRBFCM) for the numerical solution of fractional integro-differential equation and two-dimensional fractional Volterra integral equation.  Unlike the traditional global RBF collocation method, dividing the collocation of the problem in the global domain into many local regions, and therefore, the ill-conditioning of the problem is reduced and becomes highly stable.  Here, we use the multiquadric (MQ) radial basis function that includes a shape parameter, which plays an important role in the accuracy of method.  Scaling of the shape parameter to make local RBF approximation insensitive is performed by particle swarm optimization (PSO) algorithm.  Some test problems are studied and the numerical results shows the efficiency of the method.