Local radial point interpolation method with high order time discretization for time fractional one-dimensional (1D) and two-dimensional (2D) Burgers equation with low and high Reynolds numbers

We solve time fractional one-dimensional (1D) and two-dimensional (2D) Burgers equation numerically, in this study. The Burgers equation is an important nonlinear partial differential equation which has many applications. Also its time fractional version is used for modelling some physical phenomena. The time fractional derivative in this study is in Caputo sense and we approximate it with a difference scheme whose order of accuracy is $3-\alpha$, $\alpha\in(0,1)$ where $\alpha$ is fractional order. On the other hand, space derivatives are approximated via meshless local radial point interpolation method (LRPIM). In this way, a numerical method is developed for the time fractional Burgers equation. 1D and 2D problems are solved via the proposed method to assess its accuracy. Also, some 2D problems defined on irregular domains are solved. Obtained results are compared with exact solutions (if available) and some available numerical methods in literature such as finite element, finite difference and wavelet collocation methods. The obtained results and comparisons prove accuracy and efficacy of the proposed method.