Numerical Method for Solving an Inverse Problem for Non-linear Parabolic Equation with Unknown Initial Conditions

Abstract

The paper is devoted to the inverse problem for a nonlinear parabolic equation with unknown initial conditions. A computational scheme for solving this problem is proposed. This approach allows  to obtain the numerical solution in internal points of domain  and the unknown boundary function. The proposed scheme is based on the using of finite-difference equations and regularization technique. We investigate the local stability of computational method and stability of the numerical solutions. We obtained the dependence of stability on the discretization steps and level error of the initial data The proposed scheme proved the basis for development of numerical method and for the computational experiment. The experimental results are also presented in this paper, and confirm the effectiveness of the method.