Principle of structural analogy of solutions and its application to nonlinear PDEs and delay PDEs

TL;DR

This paper introduces a principle of structural analogy for solutions, enabling the construction of exact solutions to complex nonlinear PDEs, including those with delay and time-dependent coefficients, by relating them to simpler auxiliary equations.

Contribution

The paper develops a novel approach based on structural analogy to find exact solutions of nonlinear PDEs with delays and time-dependent coefficients, expanding solution methods for complex equations.

Findings

Exact solutions for nonlinear PDEs with delay were constructed.

Generalized separable solutions can handle variable delay PDEs.

The approach simplifies solving complex nonlinear PDEs by relating them to auxiliary equations.

Abstract

Using the principle of structural analogy of solutions, approaches have been developed for constructing exact solutions of complex nonlinear PDEs, including PDEs with delay, based on the use of special solutions to auxiliary simpler related equations. It is shown that to obtain exact solutions of nonlinear non-autonomous PDEs, the coefficients of which depend on time, it is possible to use generalized and functional separable solutions of simpler autonomous PDEs, the coefficients of which do not depend on time. Specific examples of constructing exact solutions to nonlinear PDEs, the coefficients of which depend arbitrarily on time, are considered. It has been discovered that generalized and functional separable solutions of nonlinear PDEs with constant delay can be used to construct exact solutions of more complex nonlinear PDEs with variable delay of general form. A number of nonlinear reaction-diffusion type PDEs with variable delay are described, which allow exact solutions with generalized separation of variables.