Application Of Meshfree Collocation Method For The Numerical Integration Of Some Fractional Order Partial Differential Equations

Meshfree collocation method based on Gaussian radial basis functions is proposed for the numerical integration of some higher-order partial differential equations(PDEs)which arise in regularized solitary waves,diffusion wave equation in the reaction process,and advection-dispersion mobile-immobile equation in porous media.The main motivation for this choice is the application of proposed method in higher dimensions.The meshfree approach eliminates the requirement of meshing and approximates numerical solutions using a set of uniform or random points.On the other hand,finite-difference,finite element,and finite volume are mesh-grid methods.The main issue of these methods is the requirement of the grid structure,which can be complicated and time consuming,especially in higher dimensions.To avoid these problems,some meshfree methods have been developed,such as Adomain decomposition method,Galerkin method and radial basis function method.The aim of this research is the application of meshfree collocation method to some challenging PDE models including higher-order time-fractional nonlinear PDEs,time-space fractional-order PDEs and time-space dependent fractional-order PDEs.Time derivatives are dealt with Coimbra fractional derivative operator or Caputo fractional derivative operator and finite difference method whereas space fractional derivatives over finite domain are dealt with Riesz fractional derivative operator,Riemann-Liouville fractional derivative operator and Gr(?)nwald-Letnikov fractional derivative operator.The proposed method is successfully tested on various examples and the results are compared with exact solutions and reported in the recent literature.The error between numerical and exact solutions is analyzed by maximum error norms.Furthermore,the time and space rates of convergence are also computed.