Highly Oscillatory Integral Equations And Their Numerical Solutions

Highly oscillatory problems have already extensively been appeared in many scientific and engineering applications,for example scattering problems in electromagnetics and acoustics with live vitally related while the study of highly oscillatory integral equations is a vital part in highly oscillatory problems.However because of high oscillation of the highly oscillatory integral equations,traditional numerical methods for integral equations meet an extremely difficult numerical challenge such that numerical solution of highly oscillatory integral equations still remains a challenging numerical problem.In this thesis,we study extensively and deeply on highly oscillatory integral equations and their effective numerical solution.The main work and innovation of this thesis are embodied as follows.(1)New oscillation notion and new oscillatory spaces are proposed.A new oscillation notion is proposed from the point of the effect which the oscillation has on numerical analysis.It can describe the degree of oscillation effect on approximation accuracy.New oscillatory function spaces are also constructed based on the new notion of oscillation which includes oscillatory spaces with different oscillatory order and structured oscillatory space with an oscillatory structure.These spaces play a vital role in analyzing the solution of highly oscillatory integral equations.(2)Oscillation of the solution of highly oscillatory integral equations is studied.Based on the new notion of oscillation and oscillatory spaces,two kinds of highly oscillatory Fredholm and Volterra integral equations of the second kind are carefully analyzed.The research indicates that the solution of these two kinds oscillatory integral equations has certain oscillatory structure and can be represented by the summation of some products of non-oscillatory functions and known elemental oscillatory functions in terms of the new oscillation and is non-oscillatory in the new structured oscillatory space.(3)Oscillation preserving Galerkin methods(OPGM)and oscillation preserving collocation methods(OPCM)are proposed for highly oscillatory integral equations.The oscillation preserving methods adopt some simple oscillatory functions which capture the oscillation of the solution of the integral equations such that the oscillatory structure of the solution can be preserved and then make the approximation accuracy independent of the high oscillation.Numerical results show that the proposed methods possess the optimal convergence order uniformly with respect to the oscillation and they are numerically stable when the oscillation is rapid enough.(4)Multi-frequencies oscillatory interpolation is proposed which is the base for oscillation preserving collocation methods.The fundamental interpolation functions include not only the classical polynomials but also a set of oscillatory functions with different oscillatory frequencies such that the approximation error will not increase as the frequency increases in estimating the oscillatory functions.(5)Highly oscillatory Volterra integral equations with Bessel kernel which has practical applications are solved by OPCM based on a non-uniform mesh.Numerical results illustrate that OPCM based on a non-uniform mesh is effective for the solution of this kind highly oscillatory equations which is not influenced by the high oscillation.In the end of this thesis three possible research directions are proposed: oscillation analysis of solution of more complicated highly oscillatory integral equations,fast algorithms and parallel computing of oscillation preserving methods and more applications of oscillation preserving methods.