Wavelet Galerkin Method For Nonlinear Integral Equations Of The Second Kind

This thesis continues the theme of the papers [1, 2] and develops the wavelet Galerkin method for the nonlinear Fredholm and Volterra integral equations of the second kind. These equations are used as mathematical models for many problems in physics and engineering. This type of equations covers many important applications including boundary integral equations, which occurs as reformulations of other mathematical problems. Recently, many authors present efficient numerical solutions for the linear Fredholm and Volterra integral equations of the second kind by Galerkin and collocation methods. When it comes to the nonlinear Fredholm and Volterra integral equation, attention should be paid to [2]. In [2], Y. Mahmoudi used so called Legendre wavelets to solve the nonlinear Fredholm and Volterra integral equations and got a good result. However, in strictly sense, the basis functions used in [2] is not wavelet functions but scaling functions.Firstly, in the present thesis a kind of orthonormal Legendre wavelets bases of space S kn [a , b ] (the space of piecewise polynomials of degree less than or equal to k on the interval [a , b ] with n uniform knots) is constructed. And the wavelets constructed are discontinuous and compactly supported. And wavelet Galerkin schemes based on discontinuous compactly supported orthogonal Legendre wavelets to solve the nonlinear Fredholm and Volterra integral equations of the second kind are presented, respectively. The nonlinear part of the integral equation is approximated by the Legendre wavelets constructed on the interval [a , b], and then the nonlinear integral equation is reduced to a system of nonlinear equations which can be solved with Newton iterative method. In additional, a complete analysis for the convergence and the error estimate is given. Three illustrative examples are included to demonstrate that our wavelets provide is stable. And it is shown that our algorithm yields very accurate results by less computational cost.On one hand, our methods can be extended and applied to a wide class of nonlinear Fredholm integral equation of the second kind and the system of nonlinear integral equations, linear and nonlinear integro-differential equations. On the other hand, combining the theory of finite element method with wavelet can improve the accuracy of the numerical solution by finite element method.