Analytical and numerical solutions to the problem of diffusion wave initiation for a quasilinear parabolic system

Solutions of the diffusion wave type are constructed and analyzed for a system of two degenerate nonlinear parabolic equations. The problem of initiating a diffusion wave is considered for an arbitrary form of nonlinearity in the system and for arbitrary directions of motion of the zero fronts of the two target functions. A theorem is proved on the existence of four different analytical solutions depending on the propagation directions of the zero fronts. A new numerical method is proposed, which for the first time enables the solution of the problem for the case of oppositely directed motion of the two zero fronts. A new exact solution is explicitly constructed and used to verify the computational results. A numerical experiment is performed, demonstrating the convergence of the numerical method and its effectiveness across various problem parameters.