A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations

This article adapts an operational matrix formulation of the collocation method for the one- and two-dimensional nonlinear fractional sub-diffusion equations (FSDEs). In the proposed collocation approach, the double and triple shifted Jacobi polynomials are used as base functions for approximate solutions of the one- and two-dimensional cases. The space and time fractional derivatives given in the underline problems are expressed by means of Jacobi operational matrices. This investigates spectral collocation schemes for both temporal and spatial discretizations. Thereby, the expansion coefficients are then determined by reducing the FSDEs, with their initial and boundary conditions, into systems of nonlinear algebraic equations which are far easier to be solved. Furthermore, the error of the approximate solution is estimated theoretically along with graphical analysis to confirm the exponential convergence rate of the proposed method in both spatial and temporal discretizations. In order to show the high accuracy of our algorithms, we report the numerical results of some numerical examples and compare our numerical results with those reported in the literature.

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