Several Novel Wavelet-based Numerical Methods For Solving Integral And Differential Equations

Large number of practical problems as well as physical phenomena can be modeled as integral equations or partial differential equations. Because the majority of nonlinear integral and partial differential equations have no closed-form solutions, it is of both theoretical and practical importance to develop efficient numerical algorithms for such equations.This thesis is focused on the numerical solution of such equations by use of the Legendre wavelet (LW). The main contributions achieved are listed below.(1) The Sobolev space is characterized with the Legendre multiwavelet (LMW), and the approximation error is estimated for two common norms. These results lay a theoretical foundation for the subsequent work.(2) The extended Legendre wavelet (ELW) is constructed, and its properties are investigated in detail. Also, the Legendre wavelet neural network (LWNN) is constructed, which is suited for the approximation of nonlinear functions due to striking advantages such as simple structure, optimal approximation order, fast convergence speed, and lower computational cost.(3) A refined LMW nonstandard representation of the integral operator and the corresponding fast algorithm are presented. The resulting computational matrix is sparse, lower dimension and block diagonal. Another outstanding advantage is that the integral operator matrices for all the subintervals are identical. As a result, the time cost needed for calculating the integral operator significantly reduces.(4) Based on the LWNN and refined LMW representation of integral operator, a novel method for solving integral equations, known as the refined LW method, is suggested, which is then applied to the solution of the Lane-Emden equation and the Fredholm equation, respectively. For the Lane-Emden equation, the method is performed according to the following steps: (a) convert the Lane-Emden equation to an integral equation; (b) approximate the resulting nonlinear function by the LWNN; (c) by calling the refined Legendre wavelet method, compute the integral operator, product operator and integer power operator for each subinterval, leading to a linear system; (d) get the numerical solution for each subinterval by solving the linear system; and (f) obtain the solution of the original equation by combining these solutions. The proposed method can also be applied to finding the optimal solution of an integral equation.(5) A weak LW variational form is constructed by replacing the bases of the variational form with LW. On this basis, a novel method for solving partial differential equations, known as the weak LW-Galerkin method, is devised by incorporating the weak LW variational form into the classical Wavelet-Galerkin method. This method possesses two remarkable advantages: (a) the boundary conditions are easier to deal with, and (b) the time cost acquired for the calculation of the connection coefficients of the integral operator is decreased. For the Poisson equation, the weak LW-Galerkin form is constructed, and the approximation order of the numerical solution is estimated for two different norms. The theoretical results justify the weak LW-Galerkin method.(6) A novel method for solving partial differential equations, known as the mixed discontinuous Legendre Wavelet-Galerkin (MDLWG) method, is proposed. This method has advantages such as sparse representation of the operator, consistency, high order accuracy and adaptive algorithm. Moreover, the proposed method can significantly lower the computational complexity because (a) the elementwise computation involved in the boundary condition can be carried out efficiently with the aid of the piecewise Legendre polynomials, and (b) the connection coefficients involve only lower order derivatives of the basis functions, and the corresponding matrix is block diagonal. This method is applied to solving the second elliptic partial differential equations with Dirichlet boundary condition, and the error of the numerical fluxes involved in the computation of the boundary values is estimated. The result obtained justifies our method.(7) A new method for solving the advection equation, known as the DLMWD method is designed by incorporating the LMW into the classical discontinuous Galerkin (DG) method; the adaptive algorithm is described, and its approximating error of the upwind numerical flux is estimated. Specifically, the elements are arranged in a suitable sequence, and needn't to be assembled as a full system. So, the solution can be calculated in an element-by-element fashion. What is more important, the upwind flux, differential operator and boundary condition can be evaluated with lower complexity.Numerical experiments demonstrate that all the proposed methods are valid. The MDLWG and DLWG methods significantly improve on the standard DG method and are applicable to various partial differential equations.