The Research On Spectral-collocation Methods And Improved Convergence For Several Volterra Kinds Of Integral Differential Equations

In this thesis, a spectral collocation method for solve nonlinear Volterra and weakly singular Volterra differential integral equations are introduced, and we construct the high-accuracy algorithm to analyze the estimate of error and the global convergence of these methods. Numerical experiment results are provided to prove the effectiveness of the proposed method.in chapter 2, a Legendre collocation method is employed to solve the nonlinear Volterra-Fredholm-Hammerstein integral equations, and a rigorous error analysis is also provided for the proposed method, which indicate that the numerical errors in L2-norm and L?-norm will decay exponentially given that the kernel function is sufficiently smooth. To illustrate the efficiency of the method some numerical experiment results are presented.In chapter 3, a Chebyshev spectral-collocation method is proposed to solve the Volterra integral equations of second kind with weakly singular kernel. Because of the speciality of the weakly singular kernel, the integral term in the resulting VIE is approximated by Gauss quadrature formulas using the Chebyshev colloca-tion points with weight of ?-1/2,-1/2, hence, a highly accurate numerical solution is obtained. finally, through the error analysis of the numerical format and the nu-merical solutions from Chebyshev spectral-collocation methods, the convergence of spectral rate based on L? and L2 is obtained, and the effectiveness of the proposed methods is proved through numerical experiments.The fourth chapter is the extension of chapter 3, we extend the weakly sin-gular kernel (t-s)-1/2 to (t-s)-? of Volterra integral equation. Using the Jacobi spectral collocation methods with weight of w-?,-? to approximate the Volterra integral equations,we acquire the numerical format of Jacobi spectral collocation methods. In order to obtain the error analysis and the convergence analysis of these methods, we introduce the definition of fractional order integral differential equations, and use some properties of fractional order integral differential equa-tions to prove the error analysis and the convergence analysis of Jacobi spectral collocation methods. the convergence of spectral rate based on L? and L2 is ob-tained, and the effectiveness of the Jacobi spectral collocation methods is proved through numerical experiments.Moreover, the fifth chapter was extended on the basis of the fourth chapter, we extend Volterra differential integral equation with smooth weakly singular kernels to fractional order Volterra differential integral equation. And because of the relationship between the fractional order Volterra equations and Volterra type differential integral equations with smooth weakly singular kernels, then we can use Jacobi spectral-collocation method to solve the fractional order Volterra integral differential equations. The convergence of spectral rate for the proposed method is established by the L?-norm and the weighted L2-norm. Numerical experiments results are provided to prove the effectiveness of the given methods.