Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays
Abstract
:Contents
- 1. Introduction
- 1.1. Delay ODEs
- 1.2. Delay First-Order PDEs
- 1.3. Delay Reaction-Diffusion PDEs
- 1.4.Delay Wave-Type PDEs
- 2. Exact Solutions: Definition and Construction Methods
- 2.1. Reductions. Term ‘Exact Solution’ for Nonlinear PDEs with Proportional Delay
- 2.2. Construction of Exact Solutions of Nonlinear PDEs with Proportional Delay
- 3. Exact Solutions of Nonlinear Wave-Type PDEs with Proportional Arguments
- 3.1. Equations with Constant Speed
- 3.2. Equations with Variable Speed
- 4. Exact Solutions to Nonlinear Wave-Type PDEs with Variable Delays of General Form
- 4.1. Equations with Constant Speed
- 4.2. Equations with Variable Speed
- 5. Brief Conclusions
- References
1. Introduction
1.1. Delay ODEs
1.2. Delay First-Order PDEs
1.3. Delay Reaction–Diffusion PDEs
1.4. Delay Wave-Type PDEs
2. Exact Solutions: Definition and Construction Methods
2.1. Reductions. Term ‘Exact Solution’ for Nonlinear PDEs with Proportional Delay
- (i)
- Elementary functions, functions appearing in the PDE (this is necessary when the PDE contains arbitrary functions), and indefinite or/and definite integrals;
- (ii)
- Solutions of usual non-delay ODEs or systems of non-delay ODEs;
- (iii)
- Solutions of ODEs with constant or variable delay (including proportional delay), or systems of such equations.
2.2. Construction of Exact Solutions to Nonlinear PDEs with Proportional Delay
3. Exact Solutions to Nonlinear Wave-Type PDEs with Proportional Arguments
3.1. Equations with Constant Speed
3.2. Equations with Variable Speed
4. Exact Solutions to Nonlinear Wave-Type PDEs with Variable Delays of General Form
4.1. Equations with Constant Speed
4.2. Equations with Variable Speed
5. Brief Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Polyanin, A.D.; Sorokin, V.G. Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays. Mathematics 2023, 11, 516. https://doi.org/10.3390/math11030516
Polyanin AD, Sorokin VG. Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays. Mathematics. 2023; 11(3):516. https://doi.org/10.3390/math11030516
Chicago/Turabian StylePolyanin, Andrei D., and Vsevolod G. Sorokin. 2023. "Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays" Mathematics 11, no. 3: 516. https://doi.org/10.3390/math11030516
APA StylePolyanin, A. D., & Sorokin, V. G. (2023). Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays. Mathematics, 11(3), 516. https://doi.org/10.3390/math11030516