Unstructured mesh finite volume methods for fractional-in-space nonlinear reaction–diffusion equations on irregular domains

Abstract

Fractional models have been used as a powerful tool to study anomalous diffusion in human brains or heart issues with structural heterogeneity. However, the biological issues are always with irregular shapes. Many published methods for solving the fractional models are no longer effective for irregular domains. In this paper, using a finite volume method based on an unstructured mesh, we numerically solve a space fractional nonlinear reaction–diffusion model based on a fractional Laplacian operator (−∇2)α∕2 and nonlinear source term. The highlight of this method is that the considered solution domains can be arbitrarily shaped due to the fact that the meshes are unstructured, which is important in real applications. Here, we consider two dimensions fractional-in-space nonlinear reaction–diffusion equations (FISNRDE-2D) with homogeneous Neumann boundary conditions. The efficiency of our method is showcased by solving the fractional-in-space Allen–Cahn model, the Fractional FitzHugh–Nagumo model and the fractional Gray–Scott model.