Quasi-wavelet And Compact Difference Methods For Partial Integro-differential Equation

Along with the development of science and technology, some people found a equation from live:the partial integro-differential equation. This equation is widely applied in heat conduction with memory, compression of porous vis-coelastic, population and dynamics, and so on. Because the kernel term in this equation has the character of memory, solving this equation is a difficult problem, which also become one of popular problems in recently.The main of this paper is by using quasi-wavelet method, compact dif-ference method and finite difference method to study three kinds of different partial integro-differential equations. The first equation is a integro-differential equation with a positive memory term which based on unbounded domain. The second equation and the third equation are both integro-differential equa-tion with a singular kernel term. This paper are separated by six parts. The background, the studying development of integro-differential equation, and the frame of this paper is introduced in part 1. The main idea of part 2 and part 3 is study the first integro-differential equation. The main idea of part 4 is study the second integro-differential equation. The main idea of part 5 is study the third integro-differential equation, this paper is end up with a conclusion and the future work.It is the first time that the quasi-wavelet method was used to solve a integro-differential equation on an unbounded domain in part 2. We first choose an arbitrary point xO(xO ?R) and an arbitrary number p(p ?(0,+∞)), thus we get a bounded interval [xO-p,xO+p] in R, then we will use quasi-wavelet method to handle the problem on bounded domain [xO- p,xO+p]. On one side, because this domain is chosen by random, and the quasi-wavelet have the character of localization, the computed value and the solution are different. In order to get more accurate computed value, we must remove some values. On the other hand, because the points of xO,p are chosen by random, we can get the numerical solution on the any domain which is on R. When we calculate the numerical solution, we use the Forward Euler method in time and quasi-wavelet method in space, and we use the triangular rule using the values of the integrand at the right-hand end-points of the domain to handle the convolution term. We not only study the one-dimensional problem, but also study the two-dimensional problem. At last we give some numerical example to demonstrate the solution.In part 3, we study the integro-differential equation on an unbounded domain by Crank-Nicolson method and quasi-wavelet method. The way we try to handle the unbounded domain is same as above. We use Crank-Nicolson method in time and quasi-wavelet method also in space, but the convolution term was handled by a continuous piecewise linear interpolant. We also study the one-dimensional problem and the two-dimensional problem, and we use the numerical example as part 2 for a better comparison.It is not only in the engineering but also in mathematics area, the finite difference method was used by most engineer and scholar. This is because the finite difference method is an easy and useful method, so it is popular. In recently many scholar solved the integro-differential problem by finite differ-ence method, but the compact difference method haven’t been used to solve this problem. Because the order of compact difference is 4 and the order of finite difference method is 2, we handle the integro-differential equation with a weakly singular kernel term by a compact difference scheme. After given the discrete scheme, we demonstrate the stability and the convergence strictly, and we use the numerical example to demonstrate the theoretical analysis.On one hand, after realizing the fact that fully order can not be demon-strated by handling the integro-differential equation with a weakly singular kernel term, we are interested in a new method to deprive the domain(graded mesh). If the graded mesh was used to deprive the domain, the mesh was dense near the singular point and was sparse far from the singular point which compensate the character of singular of solution. On the other hand, after realizing that we need plenty of memory space to store these data, we consider to use the alternating direction method to handle this problem to reduce the time. Therefore, we use the alternating direction implicit method non-uniform meshes to study the problem with a singular term. The implicit-Euler scheme for temporal direction, and the second-order difference method for spatial di-rection are proposed and analyzed. The convergence order is O(k+h2x+h2y). We also give the demonstration of stability and convergence. At last we use two numerical example to coincide with the theoretical analysis.Based on the analysis in section 5, the graded mesh was used to solve the integral-differential equation in deep in section 6. In section 6 we used the implicit Crank-Nicolson scheme for solving this problem. The alternat-ing direction implicit Crank-Nicolson scheme was considered for temporal dis-cretization, instead of implicit Euler scheme. The second order difference scheme also be considered for spatial discretization. The convergence order is O(k2+h2x+h2y). The stability and the convergence are proven. Two numer-ical examples which are the same as section 5 are presented to demonstrate the convergence behavior.