Highly Accurate Fast Difference Schemes For Solving Several Fourth-order Time Multi-term Fractional Sub-diffusion Equations

Fractional partial differential equations(PDEs)have been shown to provide some more powerful means for modeling challenging phenomena such as anomalously diffusive transport,compared to their integer-order analogues and attracted extensive research.Due to the memory effect of the fractional differential operator,the numerical discretizations of time-fractional PDEs generate numerical schemes that involve the numerical solutions at all previous time steps.Direct time-stepping methods usually require large amount of storage and expensive computational cost.It is significant to develop fast and memory-saving methods for numerically solving time-fractional differential equations.This thesis mainly focuses on the fast finite difference methods for solving several fourth-order time multi-term fractional sub-diffusion equations with the Dirichlet boundary conditions and the highly accurate finite difference methods for solving fourth-order time multi-term variable-order fractional sub-diffusion equations with the first Dirichlet boundary conditions.At first,for the fourth-order time multi-term fractional sub-diffusion equations with the first Dirichlet boundary condition,the method of order reduction is applied to treat the fourth-order derivative in space.At some super-convergence points,the multi-term Caputo derivatives are fast evaluated based on the sum-of-exponentials(SOE)approximation for the kernel functions appeared in Caputo fractional derivatives.The unconditional stability and convergence of the presented scheme are illustrated by the discrete energy method.Finally,some numerical examples are given to verify the convergence accuracy of the proposed scheme.Compared with the direct scheme without the acceleration in time direction,the CPU time of the current fast scheme is largely reduced.Secondly,a fast difference scheme for solving the fourth-order time multi-term fractional sub-diffusion equations with the second Dirichlet boundary condition is disccused.The time fractional derivatives are discretized by the fast L2-1_?formula and a highly accurate fast finite difference scheme is established.Using the discrete energy method,the convergence of the presented difference scheme can reachO(?~2+h~4+?)in L2 norm,with?a small positive constant,?and h the temporal and spatial step size,respectively.Numerical experiments are carried out to confirm the accuracy and effectiveness of the proposed fast difference scheme.Finally,the numerical method is developed to solve the fourth-order time multi-term variable-order fractional sub-diffusion equations with the first Dirichlet boundary condition.At some paticular points,we develop a numerical differentiation formula to approximate the linear combination of variable-order Caputo fractional derivatives and reveal its numerical accuracy.The convergence rate of the numerical differentiation formula is of order no less than two in time at some special points,which are determined by finding the root of a nonlinear equation.In order to solve the time multi-term variable-order fractional sub-diffusion equations,the proposed numerical formula is applied to develop the highly accurate finite difference scheme.Both the theoretical analysis and numerical experiments show the validity of the present finite difference scheme.