Geodesic
On solutions of the traveling wave type for the nonlinear heat equation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 196 (2021), pp. 36-43

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In this paper, we consider the problem of finding solutions to a nonlinear heat equation with a power-law nonlinearity, which have the form of a traveling wave and simulate the propagation of disturbances along a cold background with a finite speed. We show that the construction can be reduced to the Cauchy problem for a second-order ordinary differential equation with a singular coefficient of the highest derivative. For this Cauchy problem, the theorem on the existence and uniqueness of a smooth solution is proved. We develop an algorithm for constructing an approximate solution based on the boundary-element method and also present the results of computational experiments with numerical estimates of the parameters of the solution.
Keywords: nonlinear heat equation, existence theorem, uniqueness theorem, series, boundary-element method.
Mots-clés : exact solution, convergence 
@article{INTO_2021_196_a3,
     author = {A. L. Kazakov and P. A. Kuznetsov and L. F. Spevak},
     title = {On solutions of the traveling wave type for the nonlinear heat equation},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {36--43},
     publisher = {mathdoc},
     volume = {196},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_196_a3/}
}
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A. L. Kazakov; P. A. Kuznetsov; L. F. Spevak. On solutions of the traveling wave type for the nonlinear heat equation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 196 (2021), pp. 36-43. http://geodesic.mathdoc.fr/item/INTO_2021_196_a3/