Nonlinear functional substitutions and transformations for nonlinear diffusion and wave equations

Abstract

Background. The research considers the problem of constructing exact solutions of nonlinear wave equations and diffusion type using the method of nonlinear functional substitutions. Materials and methods. The main method used in the work is the method of non-linear functional substitutions, which is a development of the method of functional substitutions, which was previously used to construct solutions to Burgers-type equations. The method of non-linear functional substitutions is applicable to a wider range of problems, including non-linear wave equations and non-linear equations of parabolic type. Results. The study develops the general scheme of the method and gives specific examples of its application to the calculation of the Bäcklund transformations, as well as the construction of exact solutions for a wide range of nonlinear diffusion equations. New exact solutions of equations of the diffusion type are found and the methodology for applying the method in practice is indicated. Conclusions. The developed approach demonstrates its versatility and efficiency for solving and analyzing nonlinear problems in wave dynamics and various diffusion processes.

Keywords

• functional substitution method
• exact solutions of nonlinear wave and diffusion equations
• nonlinear diffusion processes and models