A. L. Kazakov, L. F. Spevak
ANALYTICAL AND NUMERICAL RADIALLY SYMMETRIC SOLUTIONS
TO A HEAT EQUATION WITH ARBITRARY NONLINEARITY
DOI: 10.17804/24109908.2023.2.049064 The paper deals with the construction of radially symmetric heat waves, which are solutions to the heat conduction equation with an arbitrary form of nonlinearity under nonzero boundary condition specified on a moving manifold. The boundary value problem under study is a generalization of those solved by us earlier. Firstly, the class of the considered parabolic equations is extended; secondly, the boundary condition generating a heat wave in a space of arbitrary dimensionality has a more general form. A new theorem of the existence and uniqueness of the heatwavetype analytical solution is proved for this problem. An approximate method of constructing solutions of the required form is proposed, which is based on expansion in radial basis functions combined with the collocation method. In each time step, the solution is constructed in two stages. The first stage is solving a problem in the region bounded by a specified moving manifold and a heat wave front, which is a priori unknown and evaluated during solving. Herewith, a special substitution is used, i.e. the required function and the spatial variable change their roles. In the second stage, the solution is completed in the region bounded by the positions of the specified moving manifold on a current step and at the initial time. The boundary conditions are defined from the firststep solution. In the test example, the solutions constructed by the developed algorithm are compared with the known exact solution. Calculations show a good accuracy of the numerical solutions at various values of the numerical parameters, including space dimensionality. The observed numerical convergence with respect to the time step shows the correctness of the proposed computational procedure.
Acknowledgments: The study was carried out in accordance with state assignment No. AAAAA181180207901405 for the IES UB RAS. Keywords: nonlinear heat equation, radially symmetric solution, heat wave, power series, collocation method, radial basis functions References:
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Article reference
Kazakov A. L., Spevak L. F. Analytical and Numerical Radially Symmetric Solutions to a Heat Equation with Arbitrary Nonlinearity // Diagnostics, Resource and Mechanics of materials and structures. 
2023.  Iss. 2.  P. 4964.  DOI: 10.17804/24109908.2023.2.049064. 
URL: http://eng.dreamjournal.org/issues/20232/20232_400.html (accessed: 06/22/2024).
