Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy
| Autor: | Vsevolod G. Sorokin, Andrei D. Polyanin |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
self-similar solutions
Differential equation General Mathematics Type (model theory) exact solutions additive and multiplicative separable solutions 01 natural sciences Separable space pantograph-type differential equations Computer Science (miscellaneous) Applied mathematics nonlinear reaction–diffusion equations generalized separable solutions 0101 mathematics Engineering (miscellaneous) Scaling Mathematics lcsh:Mathematics 010102 general mathematics Multiplicative function Ode PDEs with proportional delay lcsh:QA1-939 partial functional-differential equations 010101 applied mathematics Nonlinear system functional separable solutions PDEs with varying delay Constant (mathematics) |
| Zdroj: | Mathematics, Vol 9, Iss 511, p 511 (2021) Mathematics Volume 9 Issue 5 |
| ISSN: | 2227-7390 |
| Popis: | We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments of the form w=u(px,t), w=u(x,qt), and w=u(px,qt), where p and q are the free scaling parameters (for equations with proportional delay we have 0< p< 1, 0< q< 1). A brief review of publications on pantograph-type ODEs and PDEs and their applications is given. Exact solutions of various types of such nonlinear partial functional differential equations are described for the first time. We present examples of nonlinear pantograph-type PDEs with proportional delay, which admit traveling-wave and self-similar solutions (note that PDEs with constant delay do not have self-similar solutions). Additive, multiplicative and functional separable solutions, as well as some other exact solutions are also obtained. Special attention is paid to nonlinear pantograph-type PDEs of a rather general form, which contain one or two arbitrary functions. In total, more than forty nonlinear pantograph-type reaction–diffusion PDEs with dilated or contracted arguments, admitting exact solutions, have been considered. Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of nonlinear pantograph-type PDEs. A number of exact solutions of more complex nonlinear functional differential equations with varying delay, which arbitrarily depends on time or spatial coordinate, are also described. The presented equations and their exact solutions can be used to formulate test problems designed to evaluate the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for PDEs with varying delay. The principle of analogy allows finding solutions to other nonlinear pantograph-type PDEs (including nonlinear wave-type PDEs and higher-order equations). |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|