On The Asymptotic Orders And Efficient Numercial Methods For Volterra Integral Equations With Highly Oscillatory Kernels

Abstract:Integral equations with highly oscillatory kernels widely appear in the area of electromagnetic scattering and quantum mechanics, etc. As an important part of highly oscillatory problems, numerical solutions of these equations are a hot topic in computational mathematics. When the kernel function highly oscillates, classical methods become inefficiency. Therefore, it is necessary to develop helpful methods for calculating these equations. Since computation of integral equations can be changed into the numerical integration by discreteness, numerical methods for highly oscillatory integrals play a critical role in these problems. Based on the research on numerical approaches for highly oscillatory integrals, we, in Chapter2, consider a class of algorithms for numerical solutions of Volterra integral equations of the second kind with highly oscillatory trigonometric kernels, containing the direct-Filon method based on linear interpolation, the high-order Filon method, the piecewise constant collocation method, and the continuous piecewise linear collocation method. By analyzing the asymptotic properties of the solution and its derivatives, we present the estimate on the asymptotic order of the direct-Filon method, which shows that the higher the oscillation of the kernel function, the better the method. Numerical experiments also demonstrate the effectiveness of the given order and the efficiency of these methods for solving Volterra integral equations with highly oscillatory kernels. In Chapter3, we employ these methods to solving Volterra integral equations of the first kind with highly oscillatory trigonometric kernels. Numerical tests show that all these methods, except for the piecewise constant method, behave well when the kernel function becomes highly oscillatory. The innovations of this thesis are developing the asymptotic property of solutions of a class of Volterra integral equations and introducing the asymptotic order of the direct-Filon method.The paper contains13pictures,11charts and65...