The Analytical and a Higher-Accuracy Numerical Solution of a Free Boundary Problem in a Class of Discontinuous Functions

A new method is suggested for obtaining the exact and numerical solutions of the initial-boundary value problem for a nonlinear parabolic type equation in the domain with the free boundary. With this aim, a special auxiliary problem having some advantages over the main problem and being equivalent to the main problem in a definite sense is introduced. The auxiliary problem allows us to obtain the weak solution in a class of discontinuous functions. Moreover, on the basis of the auxiliary problem a higher-resolution numerical method is developed so that the solution accurately describes all physical properties of the problem. In order to extract the significance of the numerical solutions obtained by using the suggested auxiliary problem, some computer experiments are carried out.