Numerical solution of a modified anomalous diffusion equation with nonlinear source term through meshless singular boundary method

In this study, singular boundary method is employed for solving a modified anomalous diffusion process in two dimensional space with nonlinear source term with initial and Dirichlet-type boundary conditions. The process is modeled as a two dimensional nonlinear time-fractional sub-diffusion equation in sense of Riemann-Liouville fractional derivatives. A splitting scheme is applied to split the solution of the inhomogeneous governing equation into homogeneous solution and particular solution. We present the numerical operation for calculating the particular solution and homogeneous solution. For gaining approximation particular solution and homogeneous solution we employed MPS method and SBM method, respectively. We use θ-weighted finite difference method as time discretization for time derivatives. We employ Predictor-Corrector Algorithm for the nonlinear source term. A comparison check between SBM and other methods is given to show the accuracy of SBM applying on the equation. Consequently, some numerical examples with different domains is tested and compared with the exact analytical solutions to display the validity and accuracy of the numerical method and compared with other methods.

Anomalous nonlinear diffusion processSingular boundary method(SBM)Time-fractional derivativesMethod of Particular solutionFundamental solution