EXACT SOLUTIONS OF NONLINEAR TRANSPORT EQUATIONS WITH PROPORTIONAL DELAY
https://doi.org/10.26583/vestnik.2024.351
EDN: OHSXWJ
Abstract
Nonlinear transport equations with proportional delay, allowing exact solutions, are considered. More than thirty equations with proportional delay and a constant transfer coefficient, or with a transfer coefficient of power-law, exponential or logarithmic form depending on the desired function, have been described. The kinetic functions of all equations under consideration contain free parameters and in most cases also contain arbitrary functions. Exact additive, multiplicative, generalized and functional separable solutions, as well as traveling-wave and self-similar solutions are obtained. Most exact solutions contain free parameters. Over twenty more complex nonlinear transport equations with arbitrary arguments, allowing exact solutions, are also presented. The equations considered and their exact solutions can be used in the formulation of test problems to assess the accuracy of numerical methods
References
1. Zaidi A.A., Van Brunt B., Wake G.C. Solutions to an advanced functional partial differential equation of the pantograph type. Proceedings of the Royal Society A, 2015. Vol. 471. 20140947. DOI: 10.1098/rspa. 20140947.
2. Rey A.D., Mackey M.C. Bifurcations and traveling waves in a delayed partial differential equation. Chaos: An Interdisciplinary Journal of Nonlinear Science, 1992. Vol. 2. Pp. 231–244.
3. Mackey M.C., Rudnicki R. Global stability in a delayed partial differential equation describing cellular replication. Journal of Mathematical Biology, 1994. Vol. 33. Pp. 89–109.
4. Dyson J., Villella-Bressan R., Webb G.F. A semilinear transport equation with delays. International Journal of Mathematics and Mathematical Sciences, 2003. Vol. 32. Pp. 2011–2026.
5. Solodushkin S.I., Yumanova I.F., De Staelen R.H. First order partial differential equations with time delay and retardation of a state variable. Journal of Computational and Applied Mathematics, 2015. Vol. 289. Pp. 322–330.
6. Mackey M.C., Rudnicki R. A new criterion for the global stability of simultaneous cell replication and maturation processes. Journal of Mathematical Biology, 1999. Vol. 38. Pp. 195–219.
7. Meleshko S.V., Moyo S. On the complete group classification of the reaction–diffusion equation with a delay. Journal of mathematical analysis and applications, 2008. Vol. 338. Pp. 448–466.
8. Long F.-S., Meleshko S.V. On the complete group classification of the one-dimensional nonlinear Klein- Gordon equation with a delay. Mathematical Methods in the Applied Sciences, 2016. Vol. 39. No. 12. Pp. 3255–3270.
9. Lobo J.Z., Valaulikar Y.S. Group analysis of the one dimensional wave equation with delay. Applied Mathematics and Computation, 2020. Vol. 378, iss. C, 125193. DOI: 10.1016/j.amc.2020.125193.
10. Polyanin A.D., Zhurov A.I. Functional con-straints method for constructing exact solutions to delay reaction-diffusion equations and more complex nonlinear equations. Communications in Nonlinear Science and Numerical Simulation, 2014. Vol. 19. Pp. 417–430.
11. Polyanin A.D., Zhurov A.I. The functional constraints method: Application to non-linear delay reaction-diffusion equations with varying transfer coefficients. International Journal of Non-linear Mechanics, 2014. Vol. 6. Pp. 267–277.
12. Polyanin A.D., Sorokin V.G. Tochnye reshenija nelinejnykh reakcionno-diffuzionnykh uravnenij giperbolicheskogo tipa s zapazdyvaniem [Exact solutions of nonlinear delay reaction-diffusion equations of hyperbolic type]. Vestnik NIYaU «MIFI», 2014. Vol. 3. No. 2. Pp. 141–148 (in Russian).
13. Polyanin A.D., Sorokin V.G. Tochnye reshenija nelinejnykh uravnenij v chastnykh proizvodnykh s peremennym zapazdyvaniem tipa pantografa [Exact Solutions of Nonlinear Partial Differential Equations with Pantograph Type Variable Delay]. Vestnik NIYaU MIFI, 2020. Vol. 9. No. 4. Pp. 315–328 (in Russian).
14. Polyanin A.D., Zhurov A.I. Exact separable solutions of delay reaction–diffusion equations and other nonlinear partial functional-differential equations. Communications in Nonlinear Science and Numerical Simulation, 2014. Vol. 19. No. 3. Pp. 409–416.
15. Polyanin A.D., Zhurov A.I. Generalized and functional separable solutions to nonlinear delay Klein–Gordon equations. Communications in Nonlinear Science and Numerical Simulation, 2014. Vol. 19. No. 8. Pp. 2676–2689.
16. Polyanin A.D., Zhurov A.I. Nonlinear delay reaction-diffusion equations with varying transfer coefficients: Exact methods and new solutions. Applied Mathematics Letters, 2014. Vol. 37. Pp. 43–48.
17. Polyanin A.D., Zhurov A.I. New generalized and functional separable solutions to nonlinear delay reaction–diffusion equations. International Journal of Non-linear Mechanics, 2014. Vol. 59. Pp. 16–22.
18. Polyanin A.D., Sorokin V.G. Postroenie tochnykh reshenij nelinejnykh uravnenij matemati-cheskoj fiziki s zapazdyvaniem s pomoshh'ju reshenij bolee prostykh uravnenij bez zapazdyvanija [Construction of exact solutions for nonlinear equations of mathematical physics with delay using solutions of simpler equations without delay]. Vestnik NIYaU MIFI, 2020. Vol. 9. No. 2. Pp. 115–128 (in Russian).
19. Polyanin A.D., Sorokin V.G. Nonlinear pantograph-type diffusion PDEs: Exact solutions and the principle of analogy. Mathematics, 2021. V. 9 (5). 511. DOI: 10.3390/math9050511.
20. Polyanin A.D., Sorokin V.G. A method for constructing exact solutions of nonlinear delay PDEs. Journal of Mathematical Analysis and Applications, 2021. V. 494 (2). 124619. DOI: 10.1016/jmaa.2020. 124619.
21. Polyanin A.D., Sorokin V.G. Construction of exact solutions to nonlinear PDEs with delay using solutions of simpler PDEs without delay. Communications in Nonlinear Science and Numerical Simulation, 2021. V. 95(1). 105634. DOI: 10.1016/ j.cnsns.2020.105634.
22. Polyanin A.D., Sorokin V.G. Reductions and exact solutions of nonlinear wave-type PDEs with proportional and more complex delays. Mathematics, 2023. Vol. 11. 516. DOI: 10.3390/math11030516.
23. Polyanin A.D., Sorokin V.G. Exact solutions of reaction-diffusion PDEs with anisotropic time delay. Mathematics, 2023. Vol. 11. 3111. DOI: 10.3390/ math11143111.
24. Polyanin A.D., Sorokin V.G., Zhurov A.I. Differencial'nye uravnenija s zapazdyvaniem: Svojstva, metody, reshenija i modeli [Differential equations with delay: Properties, methods, solutions and models]. Moscow, IPMech RAS Publ., 2022. 464 р. (in Russian).
25. Tanthanuch J. Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay. Communications in Nonlinear Science and Numerical Simulation, 2012. Vol. 17. Pp. 4978–4987.
26. Polyanin A.D. Postroenie reshenij nelinejnykh uravnenij matematicheskoj fiziki s pomoshh'ju tochnykh reshenij bolee prostykh uravnenij [Construting solutions to nonlinear equations of mathematiсal physiсs using exaсt solutions to simpler equations]. Vestnik NIYaU MIFI, 2024. Vol. 13. No. 2. Pp. 66–75 (in Russian).
27. Polyanin A.D., Zaitsev V.F. Handbook of Nonlinear Partial Differential Equations, 2nd ed. Boca Raton, CRC Press, 2012.
Review
For citations:
Sorokin V.G. EXACT SOLUTIONS OF NONLINEAR TRANSPORT EQUATIONS WITH PROPORTIONAL DELAY. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2024;13(4):251-262. (In Russ.) https://doi.org/10.26583/vestnik.2024.351. EDN: OHSXWJ
Views:
98
Email this article 