Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays
The study gives a brief overview of publications on exact solutions for functional PDEs with delays of various types and on methods for constructing such solutions. For the first time, second-order wave-type PDEs with a nonlinear source term containing the unknown function with proportional time del...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/3/516 |
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| author | Andrei D. Polyanin Vsevolod G. Sorokin |
| author_facet | Andrei D. Polyanin Vsevolod G. Sorokin |
| author_sort | Andrei D. Polyanin |
| collection | DOAJ |
| description | The study gives a brief overview of publications on exact solutions for functional PDEs with delays of various types and on methods for constructing such solutions. For the first time, second-order wave-type PDEs with a nonlinear source term containing the unknown function with proportional time delay, proportional space delay, or both time and space delays are considered. In addition to nonlinear wave-type PDEs with constant speed, equations with variable speed are also studied. New one-dimensional reductions and exact solutions of such PDEs with proportional delay are obtained using solutions of simpler PDEs without delay and methods of separation of variables for nonlinear PDEs. Self-similar solutions, additive and multiplicative separable solutions, generalized separable solutions, and some other solutions are presented. More complex nonlinear functional PDEs with a variable time or space delay of general form are also investigated. Overall, more than thirty wave-type equations with delays that admit exact solutions are described. The study results can be used to test numerical methods and investigate the properties of the considered and related PDEs with proportional or more complex variable delays. |
| first_indexed | 2024-03-11T09:35:42Z |
| format | Article |
| id | doaj.art-8ed9c5354a414524aa1a0eb18bc72038 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T09:35:42Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-8ed9c5354a414524aa1a0eb18bc720382023-11-16T17:20:43ZengMDPI AGMathematics2227-73902023-01-0111351610.3390/math11030516Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex DelaysAndrei D. Polyanin0Vsevolod G. Sorokin1Ishlinsky Institute for Problems in Mechanics RAS, 101 Vernadsky Avenue, bldg 1, 119526 Moscow, RussiaIshlinsky Institute for Problems in Mechanics RAS, 101 Vernadsky Avenue, bldg 1, 119526 Moscow, RussiaThe study gives a brief overview of publications on exact solutions for functional PDEs with delays of various types and on methods for constructing such solutions. For the first time, second-order wave-type PDEs with a nonlinear source term containing the unknown function with proportional time delay, proportional space delay, or both time and space delays are considered. In addition to nonlinear wave-type PDEs with constant speed, equations with variable speed are also studied. New one-dimensional reductions and exact solutions of such PDEs with proportional delay are obtained using solutions of simpler PDEs without delay and methods of separation of variables for nonlinear PDEs. Self-similar solutions, additive and multiplicative separable solutions, generalized separable solutions, and some other solutions are presented. More complex nonlinear functional PDEs with a variable time or space delay of general form are also investigated. Overall, more than thirty wave-type equations with delays that admit exact solutions are described. The study results can be used to test numerical methods and investigate the properties of the considered and related PDEs with proportional or more complex variable delays.https://www.mdpi.com/2227-7390/11/3/516nonlinear wave-type equationsPDEs with proportional delaydelay Klein–Gordon equationsPDEs with variable delaypartial functional-differential equationsreductions and exact solutions |
| spellingShingle | Andrei D. Polyanin Vsevolod G. Sorokin Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays Mathematics nonlinear wave-type equations PDEs with proportional delay delay Klein–Gordon equations PDEs with variable delay partial functional-differential equations reductions and exact solutions |
| title | Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays |
| title_full | Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays |
| title_fullStr | Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays |
| title_full_unstemmed | Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays |
| title_short | Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays |
| title_sort | reductions and exact solutions of nonlinear wave type pdes with proportional and more complex delays |
| topic | nonlinear wave-type equations PDEs with proportional delay delay Klein–Gordon equations PDEs with variable delay partial functional-differential equations reductions and exact solutions |
| url | https://www.mdpi.com/2227-7390/11/3/516 |
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