Numerical Methods For Highly Oscillatory Integrals And Integral Equations

Highly oscillatory integrals and integral equations arise from harmonic analysis,fluid mechanics,electromagnetic wave scattering problems,image analysis,epidemic models,re-sponse of feedback systems to periodic input signals and so on.Applying the classical methods to the oscillatory problems is always inefficient.So it is necessary to develop special numerical methods for highly oscillatory problems.This doctoral dissertation is concerned with numer-ical methods for highly oscillatory integrals and the second kind Volterra integral equations.For the oscillatory integrals of the Fourier type,we focus on the construction of the adaptive Filon type method.For highly oscillatory integral equations,we mainly study the numerical scheme based on collocation.The whole dissertation contains the following several parts:In Chapter 1,the background on highly oscillatory problems is shown at first.Then the present state about the efficient numerical methods for highly oscillatory integrals and integral equations is presented.Also,the main works of this dissertation are listed.Chapter 2 is devoted to studying the numerical method for highly oscillatory integrals,i.e.,the adaptive Filon type method with high asymptotic order.By analyzing the error of the Filon type method in detail,we obtain some special points.Fion type method based on these points possesses a high asymptotic order.By some special functions,we propose the so called adaptive Filon type method.We further decrease the error by adding the interior nodes(Chebyshev nodes)and prove its convergence.Chapter 3 to Chapter 5,efficient numerical methods for highly oscillatory Volterra inte-gral equations are considered.Chapter 3 studys the exponential fitting collocation method for a class of Volterra integral equations.This method is based on the exponential fitting interpola-tion and the collocation method.The convergence of the method is proved and the comparison with the polynomial collocation is given at last.In Chapter 4,we get our numerical scheme by discretizing the highly oscillatory integral with the Filon method in a class of Volterra integral equations whose solutions contain os-cillatory functions.Then,the convergence results are obtained.Compared with the classical numerical methods,such methods have an asymptotic order which means that the numerical results produced by these methods will become more accurate when the frequency ? becomes larger.Several numerical experiments are provided to verify the theoretical results in the end.Chapter 5 is about the numerical methods for Volterra integral equations with highly oscillatory kernels.The semi-discretized scheme is got by discretizing the integral with Filon method.Furthermore,highly oscillatory integrals in the matrices are discretized to obtain the fully discretized scheme.The convergence results of the semi and fully discretized schemes are proved.The error analysis indicates that they all have an asymp-totic order.At last,the numerical examples are given to illustrate the efficiency of our methods.