Approximate solution of stochastic Allen–Cahn equation of fractional order using finite difference and RBF-based meshfree method

The main goal of this research work is to show that the combined method based on time discrete scheme and radial basis function (RBF) based meshfree method is appropriate to approximate the solution of stochastic Allen–Cahn equation of fractional order. In this method, we start with the definition of Caputo fractional derivative and use Lagrange interpolation technique to obtain a representation of time fractional derivative. Then the finite difference method is used to discretize the stochastic Allen–Cahn equation in time direction and substitute the obtained formulas for Caputo fractional derivative into the resulting finite difference equations. Finally, a meshfree method based on RBF is used to estimate the unknown function in spatial direction. This method transforms the problem under consideration into a nonlinear system in each time step, which is solved via fixed point technique and LU decomposition method. Acceptable accuracy and efficiency of the presented approach are investigated by some mathematical criterions such as RMS error, infinity error, 2-norm error, and experimental convergence order.