Convolution Type Integral Formula For High Oscillation Integral And Its Applicatio

Highly oscillatory integrals and integral equations play an important role in various fields,such as electromagnetics,acoustic scattering,fluid dynamics and so on.Generally,it is difficult to obtain accurate numerical results using traditional numerical quadrature methods.Therefore,numerical solutions of highly oscillatory integrals and integral equations are hot topics in computational mathematics.This thesis constructs a novel convolution quadrature for convolution integrals using Hermite collocation methods and its correction formula.The main research results are listed as follows:(1)A convolution quadrature based on the Hermite collocation method is developed for general convolution integrals,which improves convergence rates with help of derivatives and is able to provide accurate approximations to convolution integrals with singular,multiple scale or highly oscillatory kernels.Through analysis of the initial value problem,the error analysis with respect to the step size has been established.Furthermore,a correction formula has been constructed to promote the convergence rate.(2)The convolution quadrature based on Hermite collocation method and its correction formula are applied to the computation of convolution integrals with highly oscillatory Bessel kernels,and the convergence analysis with respect to the oscillation is carried out.The sharp convergence rate is obtained.Compared with the classical convolution quadrature,the proposed method enjoys a higher convergence rate.(3)The convolution quadrature based on the Hermite collocation method and its correction formula are applied to numerical solutions of the first-kind Volterra integral equations with Bessel kernel.The numerical results indicate that the proposed method is able to efficiently solve the first-kind Volterra integral equation,and has the property that the higher the oscillation,the better the approximation.