Comparing dierent degrees of nonlinearity for inverse problem for parabolic equation

Abstract

In this work we consider one dimensional nonlinear parabolic equation with unknown function on the right side of space variable. As an additional information we are given a function which describes a solution on the left side and thus the problem is overdened on the left side. The problem is solved by gradient method. The main target is to understand an inuence of the nonlinearity degree of the equation on convergence of the numerical algorithm. For that we take dierent degrees of nonlinear term in the equation, construct a numerical solution and give the results in graphical form. Also we enlarge a time interval and consider a convergence of the algorithm. Some negative eects can be avoided by enlarging the time interval. We give all formulae to solve a direct problem and adjoint problem, give references where to nd how to obtain a gradient for the functional given on nonlinear parabolic equation. We also describe the step-by-step algorithm of the solution of the problem. Higher degrees of the nonlinearity make the numerical solution less accurate, but at the same time it makes the functional properties of the equation much better. Inuence of these two aspects is considered in the work. Also some comments are given on some moments for the numerical algorithm, such as choosing a constant coecient in gradient method.