An efficient meshless computational technique to simulate nonlinear anomalous reaction–diffusion process in two-dimensional space

In this paper, a numerical simulation of an anomalous reaction–diffusion process in two-dimensional space with a nonlinear source term is presented. An efficient and powerful computational technique based on a combination of a time integration scheme and meshless local Petrov–Galerkin method is implemented for solving the problem. An implicit time stepping scheme with second-order accuracy is used to discretize the problem in the time direction. The unconditional stability of the proposed time discretization scheme is proved in an appropriate Sobolev space. In order to obtain a fully discrete model, the primary spatial domain is represented by a set of regularly distributed nodes and spatial shape functions are constructed on the distributed field nodes using the radial point interpolation method. Then, a local weak form meshless method based on the radial point interpolation shape functions is used to discretize the problem in the spatial direction. The efficiency and accuracy of the proposed computational procedure are demonstrated by numerical examples. Numerical results show that the method approximates solutions of the model in excellent agreement with analytical results. Moreover, it can be observed that the method gives the expected second-order temporal convergence rate independent of the fractional order which confirms the theoretical predictions.