A. L. Kazakov, L. F. Spevak, O. A. Nefedova

ON THE NUMERICAL-ANALYTICAL APPROACHES TO SOLVING A NONLINEAR HEAT CONDUCTION EQUATION WITH A SINGULARITY

The paper deals with the construction and study of solutions to a nonlinear heat conduction equation in the case of the power-law dependence of the thermal conductivity coefficient on temperature. The parabolic type of the equation degenerates in the case of the zero value of the required function. It acquires some properties typical of first-order equations; particularly, it has solutions with a free boundary in the form of a heat wave propagating at a finite velocity over the cold front. Two types of the boundary conditions are discussed: the specified law of motion of the heat front and the boundary condition specified on a static manifold, initiating a heat wave. A comparative analysis of our developed approaches to solving the boundary value problems is made; namely, local analytical solutions are constructed by the power series method, and numerical-analytical solutions are constructed based on the boundary element method on a specified finite time interval. For some particular cases, the construction reduces to the Cauchy problem for a second-order ordinary nonlinear differential equation with a singularity before the higher derivative. The solution of this equation enables a partially self-similar solution to be constructed for the initial problem. The advantages and applicability of each approach are described. Examples are considered.

Acknowledgments: The work was partially supported by the Complex Program of UB RAS (project No. 18-1-1-5) and the RFBR, project No. 16-01-00608.

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