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A. L. Kazakov, L. F. Spevak, E. L. Spevak


The paper deals with the construction of exact solutions to a nonlinear heat equation with degeneration in the case of the zero value of the required function. Generically self-similar solutions and traveling wave solutions are considered, the construction of which reduces to solving Cauchy problems for a nonlinear second-order ordinary differential equation with a singularity before the higher derivative. Two approaches are proposed to solve the Cauchy problems: the analytical solution by the power series method and the numerical solution by the boundary element method on a specified segment. A complex computational experiment is carried out to compare the above two methods with each other and with the finite difference methods, namely the Euler method and the fourth-order Runge-Kutta method. Power series segments are used on the first step of the finite difference solutions in order to resolve the singularity. The comparison of the application domains, the accuracy of the solutions and their dependence on the parameters of a certain problem shows that the boundary element method is the most universal, although not the most accurate for some particular examples. The conclusions drawn allow us to construct benchmark solutions to verify the approximate solutions of the nonlinear heat equation by various methods in a wide range of parameter values.

Acknowledgements: The work was supported by the RFBR, project No. 20-07-00407.

Keywords: nonlinear heat equation, exact solution, generically self-similar solution, traveling wave, power se-ries, boundary element method


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