Numerical Methods For Solving Fractional Differential Equation With Non-smooth Solutions

In last few decades,fractional calculus is widely used to describe the anomalous diffusion phe-nomena of physical systems.Fractional differential equations have also been important mathematical models in application,such as,the spread of pollutants in soil layer,viscoelastic mechanics,seawater intrusion in the freshwater layer,etc.In this paper,we consider numerical methods for solving fractional differential equations with non-smooth solution.Firstly,we adopt the Lubich’s approximation to discrete the Caputo fractional derivative and adding correction terms to deal with the non-smooth solution,which allows us to solve the non-smooth part of the solution exactly,so that we can achieve the expected accuracy and convergence order.Secondly,we utilize the extrapolation method and Taylor expansion to deal with the right hand nonlinear term,and obtain two kinds of implicit-explicit schemes.Moreover,we study the convergence and stability of the proposed schemes and present the stability domain.We present plenty of numerical examples to show the efficiency and flexibility of the proposed schemes.It is shown that for nonlinear fractional differential equations with or without analytical solution,the schemes can be of second-order convergence.Furthermore,numerical schemes for solving nonlinear multi-term fractional differential equations are also given.Numerical experiment shows that the proposed schemes can work efficiently for the problem.