Solution of the problem of initiating the heat wave for a nonlinear heat conduction equation using the boundary element method

The paper is devoted to constructing approximate heat wave solutions propagating along the cold front at a finite speed for a nonlinear (quasi-linear) heat conduction equation with a power nonlinearity. The coefficient of the higher derivatives vanishes on the front of the heat wave, i.e., the equation degenerates. One- and two-dimensional problems about the initiation of a heat wave by the boundary mode specified on a given fixed manifold are studied. Algorithms for solving this problem based on the boundary element method and a special change of variables as a result of which the unknown function and the independent spatial variable exchange their roles are proposed. The solution of the transformed problem in the form of a converging power series is constructed. These algorithms are implemented in computer programs, and test computations are performed. Their results are compared with truncated power series mentioned above and with the known exact solutions; the results are in good agreement.