A High Order Difference Scheme For Fractional Sub-diffusion Equations

Fractional order partial differential equations are widely used in biology, chemistry,finance, fluid mechanics, mechanics of materials and so on. At present, having someresearch about the analytical solution of fractional order partial differential equations, butthe analytical solution of a lot of fractional order partial differential equations is still hardto come by, especially for non-linear equations. Therefore, people pay more attention tothe numerical algorithms for solving fractional order partial differential equations.We first briefly review the history of fractional calculus, fractional diffusion waveequations and their applications, and give some commonly used definitions of fractionalderivative. We sketch several discrete methods to approximate the Caputo definition andthe Riemann-Liouville definition, and we summarize and analyze the research status ofnumerical algorithm for fractional sub-diffusion equations.In this paper, a high order difference scheme is derived for factional sub-diffusionequations by using L12formula to approximate Caputo fractional derivative and usingthe compact difference scheme of4orders to approximate the second spatial partialderivatives. The truncation error of this scheme is O(τ3α+h4), where h denotes spaceinterval,τ denotes time interval,α is the order of fractional derivative. Then we researchthe existence and uniqueness of the solution of this difference equation. And undercertain conditions, we obtain the stability and convergence of this scheme in infinitenorm by using energy method.Finally, some numerical experiments are given to verify the theoretical results thatare obtained in the front of this paper. And we use Thomas algorithm to solve thetridiagonal linear equations that appear in the scheme. L1compact finite differencescheme and high order difference scheme are used respectively to solve fractionalsub-diffusion equations. Numerical results show that the accuracy of high orderdifference scheme is better. This scheme is suitable to solve fractional sub-diffusionequations.