A local mesh-less collocation method for solving a class of time-dependent fractional integral equations: 2D fractional evolution equation

The local radial basis function (LRBF) method is an excellent tool for solving variable-order time-fractional evolution equations. This method reduces the expensive computational cost of the conventional global RBF collocation (GRBFC) approach. The present paper represents an efficient mesh-less LRBF collocation approach for solving the two-dimensional (2D) fractional evolution equation for the arbitrary fractional order in complex-shaped domains. Having been used an appropriate finite difference (FD) technique to discrete the time variable, the LRBF method is applied in order to solve the equation. Unlike the conventional GRBFC method, consideration of the local nodes in a subdomain surrounding the given local point is needed for the proposed LRBF method. This method can reduce the ill-conditioning of the global resultant RBF interpolation matrix while it is also well efficient for large scale problems. Some illustrative examples are given to demonstrate the efficiency and proficiency of the method.