Abstract
The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein–Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schrödinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson–Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.
Highlights
The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint
The direct method for constructing functional separable solutions in implicit form based on Formula (2) is closely related to the method of differential constraints, which is based on the compatibility theory of PDEs [14]
We show below that the direct method of functional separation of variables based on the implicit representation of Solution (2) can, in certain cases, provide more closed-form solutions than the method of differential constraints with the equivalent differential constraint (5)
Summary
Nonlinear PDEs that involve one or more arbitrary functions of the unknown and/or independent variables are clearly the most difficult to analyze and find exact solutions. Such equations have significant generality and are of great practical interest for testing various numerical and approximate analytical methods for solving corresponding initial-boundary value problems. The splitting principle is used for the construction of exact solutions to functional differential equations of Form (3) and the corresponding nonlinear PDEs (1). The efficiency of the described direct method was clearly demonstrated in [10,11], where more than 40 functional separable solutions to nonlinear reaction-diffusion and Klein–Gordon equations with variable coefficients and involving arbitrary functions were obtained.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
