Constructing Exact and Approximate Diffusion Wave Solutions for a Quasilinear Parabolic Equation with Power Nonlinearities

The paper studies a degenerate nonlinear parabolic equation containing a convective term and a source (reaction) term. It considers the construction of approximate solutions to this equation with a specified law of diffusion wave motion, the existence of these solutions being proved in our previous studies. A stepwise algorithm of the numerical solution with a time-difference scheme is proposed, the second-order difference scheme being used in such problems for the first time. At each step the problem is solved iteratively on the basis of a radial basis function (RBF) collocation method. In order to verify the numerical solution algorithm, two classes of exact generalized traveling wave solutions are proposed, whose construction is reduced to solving a Cauchy problem for second order ordinary differential equations (ODEs) with a singularity at the higher derivative. The theorem of the existence and uniqueness of the analytical solution in the form of a power series is proved for it, and the estimates of the radius of convergence are obtained. The Euler method is used to prove a similar statement concerning the existence of a continuous solution in the non-analytical case. The RBF collocation method is also applied for the approximate solution of the Cauchy problem. The solutions to the Cauchy problem are numerically analyzed, and this has enabled us to reveal and describe some of their properties, including those not previously observed, and to assess the accuracy of the method.