Finite Element Methods For Some Types Of Distributed-order Partial Differential Equations

The distributed-order partial differential equations can accurately describe some physical processes that integer and fractional partial differential equations cannot de-scribe.Specifically,the time distributed-order partial differential equations have obvious advantages over other mathematical models in describing anomalous diffusion processes with memory and genetic characteristics.In this thesis,we focus on constructing finite ele-ment methods for solving several types of distributed-order partial differential equations.For every numerical scheme in this thesis,the corresponding stability and convergence analysis are given,and some representative numerical examples are presented to verify the correctness of our theoretical results.The main contributions of our work include the following three aspects:?1?In Chapter 3,the unstructured mesh Galerkin finite element method with a weighted and shifted Gr¨unwald difference approximation and Composite Trapezoid for-mula is presented to solve the nonhomogeneous two-dimensional distributed-order time fractional Cable equation.The Crank-Nicolson type discretization of the finite element scheme is implemented to obtain the numerical solution.The stability and convergence of the numerical scheme are discussed and derived.Finally,some numerical examples on convex domains are given to confirm our theoretical results.?2?In Chapter 4,the time two-mesh?TT-M?finite element method is considered to calculate the two-dimensional nonlinear time distributed-order and space fractional diffu-sion equation.In time,the TT-M algorithm combined with both the implicit second-order?backward difference scheme and Crank-Nicolson scheme for computing the numerical solution at time t₁is used to speed up the calculation.At the same time,the spatial direction is approximated by the finite element method.The detailed analyses of sta-bility and error are also given,and the second-order time convergence accuracy can be arrived at.Finally,some numerical examples are shown to illustrate the effectiveness of our numerical method.?3?In Chapter 5,the error of the finite element solution for a distribut-ed-order diffusion-wave equation is considered.The time fractional derivatives are defined in the Caputo sense,and their orders?,?belong to the intervals?0,1?and?1,2?,respec-tively.Some computationally effective numerical methods are proposed to simulate the distributed-order time-fractional diffusion-wave equation.In the time direction,the mid-point quadrature rule is used to transform the distributed-order term into the multi-term time fractional terms,then L1 formula and L2 formula are chosen to approximate the Caputo fractional derivatives.Further,the spatial direction is discretized by the Galerkin finite element method.The stability and error estimation of the fully discrete scheme based on the H¹norm are proved.Finally,a numerical example is given to illustrate the correctness and effectiveness of the theoretical analysis.