Fast numerical study on spatial nonuniform grids for two-dimensional fractional coupled equations with fractional Neumann boundary conditions

In this paper, a study on the fast numerical analysis based on spatial nonuniform grids and inverse problem for the two-dimensional space–time fractional coupled equations with fractional Neumann boundary conditions are conducted. The second order L1${}^{+}$ method combined with the Crank–Nicolson (CN) method in time and the fractional block-centered finite difference (BCFD) method based on spatial nonuniform grids in space are employed. To improve computational efficiency, a fast version fractional BCFD algorithm based on the Krylov subspace iterative methods and the spatial sum-of-exponentials (SOE) technology is also constructed. Besides, to conduct the fractional parameter identification problem for the coupled model, an efficient hybrid Black Widow Optimization and Cuckoo Search (BWOCS) algorithm is applied. Numerical example is given to verify the correctness and efficiency of the proposed methods.