In this paper, we consider a fully discrete finite element method (FEM) to solve the two-dimensional nonlinear Fisher' equation with Riesz fractional derivatives in space. This method is chiefly performed by using Crank–Nicolson discretization in conjunction with a linearized approach in time and FEM in space. The existence, uniqueness of the weak solution, and the numerical stability of the scheme are proved in great detail. The optimal error estimate computed by $L^2$-norm showed both in time and space is derived by introducing a fractional orthogonal projection. Moreover, several numerical examples are conducted on unstructured triangular meshes by a properly designed algorithm.

In this article, we propose an exponential B-spline approach to obtain approximate solutions for the fractional sub-diffusion equation of Caputo type. The presented method is established via a uniform nodal collocation strategy by using an exponential B-spline based interpolation in conjunction with an effective finite difference scheme in time. The unique solvability is rigorously proved. The unconditional stability is well illustrated via a procedure closely resembling the classic von Neumann technique. A series of numerical examples are carried out, and by contrast to other algorithms available in the open literature, numerical results confirm the validity and superiority of our method.