Numerical solution of the continuation problem for a hyperbolic differential equation

Hongliang Ren, Purdue University

Abstract

This thesis discusses the solution of a wave equation in a region above the $x\sb3=0$ plane with given Cauchy-type boundary conditions in the $x\sb3$ direction. Two problems are treated. In the first, where the given Dirichlet and Neumann data on the plane $x\sb3=0$ are independent, while in the second, where they are linearly related in terms of the Neumann operator (corresponding to a down going wave). When c is a constant, Fourier transformations are used to reduce the three-dimensional wave equation to a one-dimensional Klein-Gordon equation. By Riemann's method, the solution of Klein-Gordon equation is constructed with the given Dirichlet and Neumann boundary conditions in $x\sb3$ direction. A regularization procedure is developed for transforming the boundary data into a form for which the problem is well-posed. A finite difference formula is obtained which can be used in numerical implementation. These results are applicable to the layer-stripping approach to the inverse problem. When c is a non-constant, a coordinate transformation is used to reduce the coefficient of $u\sb{x\sb3x\sb3}$ to a unit constant. Using Romanov's method, differential integral equations are set up with respect to u and $u\sb{x\sb3}.$ It is shown that, if the boundary conditions belong to the analytic Banach space $A\sb{s},$ then the solution of the wave equation exists locally and is unique globally. A finite difference scheme is proposed for solving the wave equation numerically. Computational results demonstrate the usefulness of the method.

Degree

Ph.D.

Advisors

Weston, Purdue University.

Subject Area

Mathematics

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