The solution of the inverse problem for the diffusion equation based on Laguerre transformation

In this work a method of solving the inverse problem for the diffusion equations, based on Laguerre spectral transformation, is suggested. The problem is considered in 1D space. Diffusion equation is obtained from Maxwell's equations in the low-frequency limit. By the given solution in a certain point of space a distribution of the conductivity in the media is found. The optimization method of solution is used. The Laguerre's harmonics function is minimized. The minimization is made using the conjugate gradient method or the Newton method. The results of defining the conductivity in the horizontally layered media are presented. An influence of the accuracy of the edge problem approximation upon that of the inverse problem solution, is analyzed. The accuracies of the inverse problem solution method, based on Laguerre transformation, and the method using Fourier transformation, are compared.