Using Simple Solutions of Nonlinear Equations of Mathematical Physics to Construct More Complex Solutions
https://doi.org/10.1134/S2304487X20050119
Abstract
A number of simple, but quite efficient, methods for constructing exact solutions of nonlinear partial differential equations that do not require special training and require a small amount of intermediate calculations are described. These methods are based on the following two main ideas: (i) simple exact solutions can serve as the basis for constructing more complex solutions of the equations under consideration; (ii) exact solutions of some equations can serve as the basis for constructing solutions of more complex equations. In particular, a method for constructing complex solutions based on simple solutions using translation and scaling transformations is proposed. It is shown that quite complex solutions can be obtain in some cases by adding terms to simpler solutions. Situations where a more complex composite solution can be constructed using similar simple solutions (nonlinear superposition of solutions) are considered. A method for constructing complex exact solutions of linear equations by introducing a complex parameter into more simple solutions is described. The efficiency of the proposed methods is illustrated by a large number of particular examples. Nonlinear heat conduction equations, reaction–diffusion equations, nonlinear wave equations, equations of motion in porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, gas dynamics equations, Navier–Stokes equations, etc. are considered. In addition to exact solutions of ordinary partial differential equations, some exact solutions of nonlinear functional-differential equations of the pantograph type with partial derivatives that, in addition to the required function, also contain functions with stretching or shrinking independent variables are described. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of such functional-differential equations.
Keywords
exact solutions,
nonlinear partial differential equations,
reaction-diffusion equations,
nonlinear wave equations,
pantograph type functional-differential equations,
solutions with generalized separation of variables
Supporting agencies: The work was carried out with the support of the Ministry of Science and Higher Education of the Russian Federation (draft state assignment no. AAAA-A20- 120011690135-5 and no. 0723-2020-0036) and grants RFBR No. 18-29-10025 and No. 18-01-00890
About the Authors
A. D. Polyanin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences
Russian Federation
Russian Federation
119526
Moscow
A. V. Aksenov
Moscow State University; National Research Nuclear University MEPhI (Moscow Engineering Physics Institute);Keldysh Institute of Applied Mathematics, Russian Academy of Sciences
Russian Federation
Russian Federation
119992
115409
125047
Moscow
References
1. Ovsiannikov L. V., Group Analysis of Differential Equations, New York: Academic Press, 1982.
2. Bluman G. W., Cole J. D. Similarity Methods for Differential Equations. New York: Springer, 1974.
3. Olver P. J., Applications of Lie Groups to Differential Equations, 2nd ed., Springer, 1993.
4. Ibragimov N. H. (Ed.) CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, Symmetries, Exact Solutions and Conservation Laws. Boca Raton: CRC Press, 1994.
5. Andreev V. K., Kaptsov O. V., Pukhnachov V. V., Rodionov A. A., Applications of Group-Theoretical Methods in Hydrodynamics, Dordrecht: Kluwer, 1998.
6. Bluman G. W., Cole J. D. The general similarity solution of the heat equation // Journal of Mathematics and Mechanics. 1969. V. 18. № 11. P. 1025–1042.
7. Levi D., Winternitz P. Nonclassical symmetry reduction: Example of the Boussinesq equation // J. Phys. A. 1989. V. 22. P. 2915–2924.
8. Nucci M. C., Clarkson P. A. The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh–Nagumo equation // Phys. Lett. A. 1992. V. 164. P. 49–56.
9. Cherniha R., Serov M., Pliukhin O. Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applications. Boca Raton: Chapman & Hall/CRC Press, 2018.
10. Clarkson P. A., Kruskal M. D. New similarity reductions of the Boussinesq equation // Journal of Mathematical Physics. 1989. V. 30. № 10. P. 2201–2213.
11. Polyanin A. D. Comparison of the effectiveness of different methods for constructing exact solutions to non-linear PDEs. Generalizations and new solutions // Mathematics. 2019. V. 7. № 5. P. 386.
12. Polyanin A. D., Zaitsev V. F., Zhurov A. I., Metody reshenija nelinejnyh uravnenij matematicheskoj fiziki i mehaniki (Solution Methods for Nonlinear Equations of Mathematical Physics and Mechanics), Moscow, Fizmatlit, 2005 (in Russian).
13. Polyanin A. D., Zaitsev V. F. Handbook of Nonlinear Partial Differential Equations, 2nd ed. Boca Raton: CRC Press, 2012.
14. Polyanin A. D., Zhurov A. I., Metody razdelenija peremennykh i tochnye reshenija nelinejnykh uravnenij matematicheskoj fiziki (Methods of separating variables and exact solutions for nonlinear mathematical physics equations), M.: IPMech RAS, 2020 (in Russian).
15. Partial Differential Equations in Mechanics and Physics. Boca Raton: Chapman & Hall/CRC, 2007.
16. Polyanin A. D. Construction of exact solutions in implicit form for PDEs: New functional separable solutions of non-linear reaction-diffusion equations with variable coefficients // Int. J. Non-Linear Mech. 2019. V. 111. P. 95–105.
17. Polyanin A. D. Functional separation of variables in nonlinear PDEs: General approach, new solutions of diffusion-type equations // Mathematics. 2020. V. 8. № 1. P. 90.
18. Polyanin A. D., Zhurov A. I. Separation of variables in PDEs using nonlinear transformations: Applications to reaction-diffusion type equations // Applied Math. Letters. 2020. V. 100. 106055.
19. Sidorov A. F., Shapeev V. P., Yanenko N. N., Method of Differential Constraints and Its Applications in Gas Dynamics, Novosibirsk: Nauka, 1984 (in Russian).
20. Kudryashov N. A., Analytical Theory of Nonlinear Differential Equations, Moscow–Izhevsk: Institut kompjuternyh issledovanii, 2004 (in Russian).
21. Kudryashov N. A., Methods of Nonlinear Mathematical Physics, Dolgoprudnyi: Izd. Dom Intellekt, 2010 (in Russian).
22. Conte R., Musette M., The Painlevé Handbook, Springer, 2008.
23. Schlichting H., Gersten K. l., Boundary-Layer Theory, Ninth Ed., Springer-Verlag, 2017.
24. Boussinesq J. Recherches théorique sur l’écoulement des nappes d’eau infiltrées dans le sol et sur le débit des sources // J. Math. Pures Appl. 1904. V. 10. № 1. P. 5–78.
25. Guderley K. G., The Theory of Transonic Flow, Oxford: Pergamon, 1962.
26. Aksenov A. V., Sudarikova A. D., Chicherin I. S. The surface tension effect on viscous liquid spreading along a superhydrophobic surface // Journal of Physics: Conference Series. 2017. V. 788. 01200.
27. Aksenov A. V., Sudarikova A. D., Chicherin I. S., Effect of the surface tension on the spreading of a viscous Liquid along a superhydrophobic surface. I. Plane-parallel motion, Vestnik Natsional’nogo issledovatel’skogo yadernogo universiteta ”MIFI”, 2016, vol. 5, no. 6, pp. 489–496 (in Russian).
28. Barenblatt G. I., Zel’dovich Ya. B., On dipole-type solutions in problems of nonstationary filtration of gas under polytropic regime, Prikl. Math. & Mech., 1957, vol. 21, pp. 718–720 (in Russian).
29. Titov S. S., A method of finite-dimensional rings for solving nonlinear equations of mathematical physics, In: Aerodynamics, Ed. T. P. Ivanova, Saratov: Saratov Univ., 1988, pp. 104–110 (in Russian).
30. Polyanin A. D., Zaitsev V. F. Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems. Boca Raton–London: CRC Press, 2018.
31. Pavlovskii Yu. N., Investigation of some invariant solutions to the boundary layer equations, Zhurn. Vychisl. Mat. i Mat. Fiziki, 1961, vol. 1, no. 2, pp. 280–294 (in Russian).
32. Loitsyanskiy L. G., Mechanics of Liquids and Gases, New York: Begell House, 1995.
33. Dorodnitsyn V. A., On invariant solutions of the equation of non-linear heat conduction with a source, USSR Comput. Math. & Math. Phys. 1982, vol. 22, no. 6, pp. 115–122.
34. Hall A. J., Wake G. C. A functional differential equation arising in the modelling of cell growth // J. Aust. Math. Soc. Ser. B. 1989. V. 30. P. 424–435.
35. Hall A. J., Wake G. C., Gandar P. W. Steady size distributions for cells in one dimensional plant tissues // J. Math. Biol. 1991. V. 30. P. 101–123.
36. Derfel G., van Brunt B., Wake G. C. A cell growth model revisited // Functional Differential Equations. 2012. V. 19. № 1–2. P. 71–81.
37. Zaidi A. A., Van Brunt B., Wake G. C. Solutions to an advanced functional partial differential equation of the pantograph type // Proc. R. Soc. A. 2015. V. 471. 20140947.
38. Efendiev M., van Brunt B., Wake G. C., Zaidi A. A. A functional partial differential equation arising in a cell growth model with dispersion // Math. Meth. Appl. Sci. 2018. V. 41. № 4. P. 1541–1553.
39. Ambartsumyan V. A. On the fluctuation of the brightness of the Milky Way // Dokl. Akad. Nauk SSSR. 1944. V. 44. P. 223–226.
40. Dehghan M., Shakeri F. The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics // Phys. Scripta. 2008. V. 78. № 6. 065004.
41. Ajello W. G., Freedman H. I., Wu J. A model of stage structured population growth with density depended time delay // SIAM J. Appl. Math. 1992. V. 52. P. 855–869.
42. Mahler K. On a special functional equation // J. London Math. Soc. 1940. V. 1. № 2. P. 115–123.
43. Ferguson T. S. Lose a dollar or double your fortune // In: Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, V. III (eds. L. M. Le Cam et al.). P. 657–666. Berkeley, Univ. California Press, 1972.
44. Robinson R. W. Counting labeled acyclic digraphs // In: New Directions in the Theory of Graphs (ed. F. Harari). P. 239–273. New York: Academic Press, 1973.
45. Gaver D. P. An absorption probablility problem // J. Math. Anal. Appl. 1964. V. 9. P. 384–393.
46. Zhang F., Zhang Y. State estimation of neural networks with both time-varying delays and norm-bounded parameter uncertainties via a delay decomposition approach // Commun. Nonlinear Sci. Numer. Simul. 2013. V. 18. № 12. P. 3517–3529.
47. Ockendon J. R., Tayler A. B. The dynamics of a current collection system for an electric locomotive // Proc. R. Soc. Lond. A. 1971. V. 332. P. 447–468.
48. Polyanin A. D., Sorokin V. G., Exact solutions of non-linear partial differential equations with variable delay of the pantograph type, Vestnik Natsional’nogo issledovatel’skogo yadernogo universiteta ”MIFI”, 2020, vol. 9, no. 4, pp. 315–328 (in Russian).
49. Dobrokhotov S. Yu., Tirozzi B., Localized solutions of one-dimensional non-linear shallow-water equations with velocity, Russian Mathematical Surveys, 2010, vol. 65, no. 1, pp. 177–179.
50. Aksenov A. V., Dobrokhotov S. Yu., Druzhkov K. P., Exact step-like solutions of one-dimensional shallow-water equations over a sloping bottom, Mathematical Notes, 2018, vol. 104, no. 6, pp. 915–921.
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For citations:
Polyanin A.D., Aksenov A.V. Using Simple Solutions of Nonlinear Equations of Mathematical Physics to Construct More Complex Solutions. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(5):420-437. (In Russ.) https://doi.org/10.1134/S2304487X20050119
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