Self-similar solutions to a multidimensional singular heat equation with power nonlinearity

Abstract

The paper deals with the construction of exact solutions to a singular heat equation with power nonlinearity in the case of numerous independent variables with spatial (e.g. axial or central) symmetry. A new class of self-similar solutions is proposed, which reduce to solving the Cauchy problem for a second-order nonlinear ordinary differential equation having singularities at the higher derivative with respect to the required function and/or the independent variable. The ordinary differential equation is studied in two ways: analytically and numerically. The analytical study uses a truncated Taylor series with recurrently computed coefficients, for which explicit formulas are obtained. The numerical solution to the problem uses an iteration algorithm based on the collocation method and radial basis functions. The numerical analysis shows the convergence of the proposed numerical algorithm and its sufficient accuracy enabling one to use the found self-similar solutions to verify approximate solutions to the original heat equation. Besides, the numerical analysis has allowed the radius of convergence of the constructed Taylor series to be evaluated. The form of the constructed self-similar solutions, namely their unboundedness near the symmetry center (axis), enables us to study the behavior and exactness of the numerical solutions to the nonlinear singular parabolic-type equation that have been obtained by the stepwise solution method proposed by us earlier and possess the same property.