Natural Boundary Element Method For The Problems In An Exterior Elliptic Domain

In this paper, we investigate the natural boundary element method for the boundary value problem of Helmholtz equation in an exterior elliptic domain. ?The main work of the dissertation can be summarized as follows:In Part I, Mathieu functions (a kind of important spacial function) used are introduced, the definitions of Mathieu functions and modified Mathieu functions with physical background are given, the relationships of them are presented when parameter is q and—q .In Part II, by the principle of the natural boundary reduction, we obtain the Poisson integral formula and the natural integral equation of this problem, develop a numerical method to solve the natural integral equation. For computation, we discuss the numerical methods of the Mathieu functions when 0 < q < 20 in details. Then, the variationsl formulation of the problem is derived. Existence and uniqueness of solution of the variational problem are established. Finally, we present some numerical examples to demonstrate the performance of our method.