Lectures on Cauchy’s Problem in Linear Partial Differential Equations. By J. Hadamard. Pp. viii+316. 15s.net. 1923. (Per Oxford University Press.)

The method of fundamental solutions (MFS) is a relatively new method for the numerical solution of boundary value problems and initial/boundary value problems governed by certain partial differential equations. The ease with which it can be implemented and its effectiveness have made it a very popular tool for the solution of a large variety of problems arising in science and engineering. In recent years, it has been used extensively for a particular class of such problems, namely inverse problems. In this study, in view of the growing interest in this area, we review the applications of the MFS to inverse and related problems, over the last decade.

A survey of applications of the MFS to inverse problems

Identifying an unknown source in time-fractional diffusion equation by a truncation method

In this paper, we consider the inverse scattering problem of recovering the shape of a perfectly conducting cavity from one source and several measurements placed on a curve inside the cavity. Under restrictive assumptions on the size of the cavity, a uniqueness theorem for finitely many excitations is given. Based on a system of nonlinear and ill-posed integral equations for the unknown boundary, we apply a regularized Newton iterative approach to find the boundary. We present the mathematical foundation of the method and give several numerical examples to show the viability of the method.

Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem

Calculation of the heat flux near the liquid–gas–solid contact line

The inverse scattering problem for cavities

Two regularization methods for the Cauchy problems of the Helmholtz equation

We consider the inverse scattering problem of determining the shape of a cavity with impedance boundary condition from sources and measurements placed on a curve inside the cavity. It is shown that both the shape $\partial D$ of the cavity and the surface impedance ? are uniquely determined by the measured data and numerical methods are given for determining both $\partial D$ and ? where neither one is known a priori. Numerical examples are given showing the viability of our method.

Hai-Hua Qin

Papers

Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem

The inverse scattering problem for cavities

Two regularization methods for the Cauchy problems of the Helmholtz equation

The inverse scattering problem for cavities with impedance boundary condition

Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation