Zhanbin Yuan

Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains

In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme. A finite element method is developed to solve 2D-space fractional diffusion equations with nonlinear source term.We develop a finite element method for FPDEs on irregular domain.We obtain explicit expressions for fractional derivatives of shape functions.We give the details on how to compute fractional stiffness matrix.Fractional FitzHugh-Nagumo model is solved on circular domain.

Numerical algorithm for three-dimensional space fractional advection diffusion equation

Space fractional advection diffusion equations are better to describe anomalous diffusion phenomena because of non-locality of fractional derivatives, which causes people to confront great trouble in problem solving while enjoying the convenience from mathematical modelling, especially in high dimensional cases. In this paper, we solve the three-dimensional problem by the process of dimension by dimension, which can be achieved through a predictor-corrector algorithm. In time discretization, Crank-Nicolson scheme is adopted to match second-order difference operator of the space direction. Then, the efficiency of this method is demonstrated by some numerical examples finally.

An exponential B-spline collocation method for the fractional sub-diffusion equation

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.