Review of methods for constructing exact solutions of equations of mathematical physics based on simpler solutions

A. V. Aksenov   ; A. D. Polyanin

Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 2, pp. 149-180

Résumé

We present a brief review of methods based on the use of simpler solutions for constructing exact solutions of both nonlinear equations of mathematical physics and partial functional-differential equations. These methods are underlain by two general ideas: 1) simple exact solutions of the considered equations can be used to find more complicated solutions of the same equations; 2) exact solutions can be used as a basis for constructing solutions of either more complicated related equations or other classes of equations having similar nonlinear terms. In particular, we show how more complicated exact solutions can be found starting with simple solutions and using the shift and scale transformations; we show that sufficiently complicated solutions can in some cases be obtained by adding terms to simpler solutions; we consider situations where simple solutions of the same type can be used for constructing more complicated compound solutions; we describe the method for constructing exact solutions with several spatial variables starting with solutions of related equations with a single spatial variable. Most of the proposed methods lead to a small amount of intermediate calculations. Their efficiency is illustrated with particular examples. We consider the nonlinear heat conduction equations, reaction–diffusion equations, nonlinear wave equations, hydrodynamic, and gas dynamics equations. We show that some solutions of partial differential equations can be used to construct exact solutions of more complicated equations with delay. We describe the method that allows constructing exact solutions of partial functional-differential equations that have the desired functions with either stretched or contracted arguments.

Détail

Mots-clés : exact solutions, reaction–diffusion equations
Keywords: nonlinear partial differential equations, nonlinear wave equations, functional-differential equation with constant and variable delay, solutions with generalized separation of variables.

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