Abstract
The paper deals with two-dimensional boundary-value problems for the degenerate nonlinear parabolic equation with a source term, which describes the process of heat conduction in the case of the power-law temperature dependence of the heat conductivity coefficient. We consider a heat wave propagation problem with a specified zero front in the case of two spatial variables. The solution existence and uniqueness theorem is proved in the class of analytic functions. The solution is constructed as a power series with coefficients to be calculated by a proposed constructive recurrent procedure. An algorithm based on the boundary element method using the dual reciprocity method is developed to solve the problem numerically. The efficiency of the application of the dual reciprocity method for various systems of radial basis functions is analyzed. An approach to constructing invariant solutions of the problem in the case of central symmetry is proposed. The constructed solutions are used to verify the developed numerical algorithm. The test calculations have shown the high efficiency of the algorithm.
Highlights
IntroductionSymmetry 2020, 12, 921 and Tσ = 0, which is always true when σ > 0, there is one-to-one correspondence between the values of the functions u and T; the properties of the derivatives need to be further investigated
We consider a quasilinear parabolic equation in the case of power-law nonlinearity, which models the heat conduction process: Tt = αdiv(Tσ ∇T ) + H(T ). (1)Here, t is time, T is the required function, α, σ are positive constants, H(T ) is a specified source function, and H(0) = 0, div, ∇ are spatial divergence and spatial gradient.Equation (1) is the complete form of the porous medium equation [1]
The boundary element method (BEM) solutions obtained with the use of all the selected radial basis functions (RBF) have small relative errors, so that the algorithm is stable with respect to the choice of RBFs
Summary
Symmetry 2020, 12, 921 and Tσ = 0, which is always true when σ > 0, there is one-to-one correspondence between the values of the functions u and T; the properties of the derivatives need to be further investigated It is Equation (2) that will be studied in what follows. For approximate methods of constructing solutions to boundary value problems for nonlinear equations of mathematical physics, it is very seldom possible to prove any rigorous statements of convergence, excluding statements of the local convergence of power series [21]. New classes of wave-type exact solutions to the nonlinear heat conduction equation without sources (sinks) were obtained in [26], their construction being reducible to the integration of the Cauchy problems for the generalized Liénard equation [27,28]
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