A. L. Kazakov and L. F. Spevak

ANALYTICAL AND NUMERICAL SOLUTIONS TO A NONLINEAR HEAT EQUATION WITH A FRACTIONAL TIME DERIVATIVE

The paper presents an analytical and numerical study of a degenerate nonlinear heat equation containing a conformable fractional time derivative. It briefly reviews the use of fractional calculus in mathematical models of natural-science processes and phenomena, as well as methods for solving problems containing fractional derivatives. The theorem of the existence and uniqueness of the thermal wave analytical solution at a specified zero front is proved for the equation under study. The analytical solution is constructed in the form of a power series. An incremental algorithm for constructing a numerical solution with difference approximation in time is proposed. The iteration approach based on the collocation method and approximation by radial basis functions is applied at each step. Test calculations are made in order to verify the numerical algorithm, where the numerical solutions are compared with the truncated series representing the analytical solutions.

Acknowledgment: The study was financially supported by the Russian Ministry of Science and Higher Education under the project on Studying and Modeling of the Rheology and Heat-and-Mass Transfer Phenomena in Structured Environments at Variable Initial and Boundary Conditions, No. 124020600042-9.

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