Antireduction and exact solutions of nonlinear heat equations. (English) Zbl 0954.35037

Fushchych, Wilhelm; Zhdanov, Renat

J. Nonlinear Math. Phys. 1, No. 1, 60-64 (1994).

The paper is dealing with nonlinear heat-conductivity equations of a general form \[ u_t = \left(a(u)u_x\right)_x + F(u) \] for a real variable \(u(x,t)\). Trying substitutions of various types, the authors find special forms of the functions \(a(u)\) and \(F(u)\) that admit an exact reduction of the underlying nonlinear parabolic PDE to a (set of) ODE(s), which the authors call “antireduction”. A list of the corresponding functions, substitutions, and special exact solutions to the resultant ODE systems is given. In some cases, the obtained solutions are solitary waves. It is stressed that the results presented in this paper cannot be obtained by means of the traditional technique based on the Lie-group theory.

Reviewer: Boris A.Malomed (Tel Aviv)

MSC:

35C05	Solutions to PDEs in closed form
35K55	Nonlinear parabolic equations
35A25	Other special methods applied to PDEs

Keywords:

ansatz; solitary wave; substitutions of various types