The Numercial Solutions For Three Evolution Partial Differential Equations

In this dissertation, We will study the numerical solutions of three evolution partial differential equations by finite clement method,finite difference method and spectral method. The first evolution partial differential equation is a partial integro-differential equation with a weakly singular kernel, which the form is along with the initial conditions and the boundarv conditions Here,ut=(?)u/(?)t, the kernelβ(t)=tα-1/τ(α),0<α<1,is a singular kernel at t=0, andτdenotes the Gamma function. We construct the stable and second order scheme to efficiently solve the above integro-differential equation,The equation is discretized in time by the finite central difference and in space by the finite element method, obtained a semi-discrete scheme and a fully discrete scheme;then proved the stability and convergence of two schemes. A numerical example demonstrates the theoretical results.The second evolution partial differential equation is a fourth-order partial integro-differential equation with a weakly singular kernel, which the form isalong with the initial conditions and the boundary conditionsu(t,-1)=0, u(t,1)=0, ux(t,-1)=0, ux(t,1)=0, t∈(0,T], (0.0.15) Here,let T=(-1,1), K(t-s)=(t-s)α-1/τ(α). For the above integro-differential equation, the equation is discretized in space by the spectral method and in time by the finite central difference method and the Lagrange interpolated method, obtained a semi-discrete scheme and a fully discrete scheme; then proved the stability and convergence of the semi-discrete scheme.The third evolution partial differential equation is the singular perturbation convection diffusion equation, consider the model problem along with the initial conditions and the boundary conditions here,α≠0,ε>0. For the singular perturbation convection diffusion equation, using finite difference method, propose and analyze a series of the simplest difference schemes and parallel scheme with the idea of combinational difference quotient, the stability is proven strictly, which has better stabilty, wider application scope and calculates simply, need less time and have higher efficiency. Error estimate are derived for all schemes and numerical experiments verify closely the results of the theoretical analysis. In order to calculates simply,need less time and more convenience, in addition, construct a class of parallel scheme which parallel calculate in time direction, prove the stability of the parallel scheme, derive some good numerical results by parallel iteration algorithm.