Modified Green's functions and the third boundary value problem for the Helmholtz equation
The authors consider the method of boundary integral equations for the solution of the third boundary value problem for Helmholtz' equation in an exterior domain. It is well known that in the simplest formulation the integral equation may not be uniquely solvable although the boundary value problem is. Various other formulations have been devised to get rid of this difficulty. The authors propose another one which consists in adding a series of ''spherical wave functions'' to the usual fundamental solution. (The same method has been applied by R. E. Kleinman and G. F. Roach to the corresponding Dirichlet and Neumann Problems.) They succeed in exhibiting a range for the intervening coefficients which guarantees unique solvability for the integral equation. Two optimality criteria for choosing the coefficients are investigated.
scientific article; zbMATH DE number 3888136