A numerical solution to the Gel'fand-Levitan-Marchenko equation

摘要

In this paper a numerical method is developed to solve the inverse scattering problem associated with two Gel'fand-Levitan-Marchenko integral equations. This inverse problem is represented by two coupled partial differential equations. The solution of these partial differential equations is two kernel functions. These kernels are also related to each other through the Gel'fand-Levitan-Marchenko integral equations. Legendre-Gauss-Lobatto nodes are used to construct the Nth polynominal interpolation to approximate the solution of the kernel functions which are related to the scattering potential. An example is given to demonstrate the accuracy of the proposed method.