| Abstract: | ![]()
In this paper, we establish a novel fractional model arising in the chemical reaction and develop an efficient spectral method for the three-dimensionalRiesz-like space fractional nonlinear coupled reaction-diffusion equations. Based onthe backward difference method for time stepping and the Legendre-Galerkin spectral method for space discretization, we construct a fully discrete numerical schemewhich leads to a linear algebraic system. Then a direct method based on the matrixdiagonalization approach is proposed to solve the linear algebraic system, where thecost of the algorithm is of a small multiple of $N^4$ ($N$ is the polynomial degree ineach spatial coordinate) flops for each time level. In addition, the stability and convergence analysis are rigorously established. We obtain the optimal error estimatein space, and the results also show that the fully discrete scheme is unconditionallystable and convergent of order one in time. Furthermore, numerical experimentsare presented to confirm the theoretical claims. As the applications of the proposedmethod, the fractional Gray-Scott model is solved to capture the pattern formationwith an analysis of the properties of the fractional powers. |
