517.9 Methods of functional separation of variables and their application in mathematical physics

Polyanin A. D. (Bauman Moscow State Technical University/Ishlinsky Institute for Problems in Mechanics/MEPhI)


doi: 10.18698/2309-3684-2019-1-6597

A brief review of existing modifications of the method of functional separation of varia-bles is given. A new more general approach is proposed for construction of ex-act solu-tions of nonlinear equations of mathematical physics and mechanics, which is based on implicit transformations of integral type in combination using the split-ting principle. The effectiveness of this approach is illustrated on nonlinear diffusion equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on two or three arbitrary functions (such nonlinear equations are the most difficult to analyze). Many new exact solutions with functional separation of variables and generalized traveling wave type solutions are described. The obtained solutions can be used to test various numerical and approximate analytical methods of mathematical physics

[1] Grundland A.M., Infeld E. A family of non-linear Klein–Gordon equations and their solutions. J. Math. Phys., 1992, vol. 33, pp. 2498–2503.
[2] Miller W. (Jr.), Rubel L.A. Functional separation of variables for Laplace equa-tions in two dimensions. J. Phys. A, 1993, vol. 26, pp. 1901–1913.
[3] Zhdanov R.Z. Separation of variables in the non-linear wave equation. J. Phys. A, 1994, vol. 27, pp. L291–L297.
[4] Galaktionov V.A., Posashkov S.A., Svirshchevskii S.R. Differencial'nye Uravneniya — Differential equation, 1995, vol. 31, no. 2, pp. 253–261.
[5] Galaktionov V.A. Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities. Proc. Roy. Soc. Edinburgh, Sect. A, 1995, vol. 125, no. 2, pp. 225–246.
[6] Andreev V.K., Kaptsov O.V., Pukhnachov V.V., Rodionov A.A. Applications of Group-Theoretical Methods in Hydrodynamics, Dordrecht, Kluwer, 1998, 396 p.
[7] Doyle Ph.W., Vassiliou P.J. Separation of variables for the 1-dimensional non-linear diffusion equation. Int. J. Non-Linear Mech., 1998, vol. 33, no. 2, pp. 315–326.
[8] Pucci E., Saccomandi G. Evolution equations, invariant surface conditions and functional separation of variables. Physica D, 2000, vol. 139, pp. 28–47.
[9] Polyanin A.D. Exact solutions to the Navier–Stokes equations with generalized separation of variables. Doklady Physics, 2001, vol. 46, no. 10, pp. 726–731.
[10] Estevez P.G., Qu C., Zhang S. Separation of variables of a generalized porous medium equation with nonlinear source. J. Math. Anal. Appl., 2002, vol. 275, pp. 44–59.
[11] Estevez P.G., Qu C.Z. Separation of variables in nonlinear wave equations with variable wave speed. Theor. Math. Phys., 2002, vol. 133, № 2, pp. 1490–1497.
[12] Polyanin A.D, Zaitsev V.F, Zhurov A.I. Metody resheniya nelineynykh uravneniy matematicheskoy fiziki i mekhaniki [Solution methods for nonlinear equations of mathematical physics and mechanics]. Moscow, Fizmatlit, 2005, 256 p.
[13] Galaktionov V.A., Svirshchevskii S.R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics. Boca Raton, Chapman & Hall/CRC Press, 2007, 498 p.
[14] Hu J., Qu C. Functionally separable solutions to nonlinear wave equations by group foliation method. J. Math. Anal. Appl., 2007, vol. 330, pp. 298–311.
[15] Jia H., Zhao W.X.X., Li Z. Separation of variables and exact solutions to non-linear diffusion equations with dependent convection and absorption. J. Math. Anal. Appl., 2008, vol. 339, pp. 982–995.
[16] Polyanin A.D., Zaitsev V.F. Handbook of Nonlinear Partial Differential Equa-tions, 2nd Edition. Boca Raton, CRC Press, 2012, 1912 p.
[17] Polyanin A.D., Zhurov A.I. Functional and generalized separable solutions to unsteady Navier–Stokes equations. Int. J. Non-Linear Mech., 2016, vol. 79, pp. 88–98.
[18] 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, vol. 111, pp. 95–105.
[19] Polyanin A.D. Construction of functional separable solutions in implicit formfor non-linear Klein–Gordon type equations with variable coefficients. Int. J. Non-Linear Mech., 2019, vol. 114, pp. 29–40.
[20] Polyanin A.D., Zhurov A.I. Functional separable solutions of two classes of nonlinear mathematical physics equations. Doklady Mathematics, 2019, vol. 99, no. 3, pp. 321–324.
[21] Polyanin A.D. Comparison of the effectiveness of different methods for con-structing exact solutions to nonlinear PDEs. Generalizations and new solutions. Mathematics, 2019, vol. 7, no. 5, 386.
[22] Polyanin A.D. Functional separable solutions of nonlinear reaction-diffusion equations with variable coefficients. Applied Math. Comput., 2019, vol. 347. pp. 282–292.
[23] Polyanin A.D. Functional separable solutions of nonlinear convection-diffusion equations with variable coefficients. Commun. Nonlinear Sci. Numer. Simulat., 2019, vol. 73, pp. 379–390.
[24] Bluman G.W., Cole J.D. The general similarity solution of the heat equation. J. Math. Mech., 1969, vol. 18, pp. 1025–1042.
[25] Levi D., Winternitz P. Nonclassical symmetry reduction: Example of the Bous-sinesq equation. J. Phys. A, 1989, vol. 22, pp. 2915–2924.
[26] Nucci M.C., Clarkson P.A. The nonclassical method is more general than the di-rect method for symmetry reductions. An example of the Fitzhugh–Nagumo equation. Phys. Lett. A, 1992, vol. 164, pp. 49–56.
[27] Clarkson P.A. Nonclassical symmetry reductions for the Boussinesq equation. Chaos, Solitons & Fractals, 1995, vol. 5, pp. 2261–2301.
[28] Olver P.J., Vorob'ev E.M. Nonclassical and conditional symmetries. In: CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3 (ed. N.H. Ibragimov), Boca Raton, CRC Press, 1996, pp. 291–328.
[29] Clarkson P.A., Ludlow D.K., Priestley T.J. The classical, direct and nonclassical methods for symmetry reductions of nonlinear partial differential equations. Methods Appl. Anal., 1997, vol. 4, no. 2, pp. 173–195.
[30] Olver P.J. Direct reduction and differential constraints. Proc. Roy. Soc. London, Ser. A, 1994, vol. 444, pp. 509–523.
[31] Kaptsov O.V., Verevkin I.V. Differential constraints and exact solutions of non-linear diffusion equations. J. Phys. A: Math. Gen., 2003, vol. 36, pp. 1401–1414.
[32] Dorodnitsyn V.A. On invariant solutions of the equation of non-linear heat con-duction with a source. USSR Comput. Math. & Math. Phys., 1982, vol. 22, no. 6, pp. 115–122.
[33] Kudryashov N.A. On exact solutions of families of Fisher equations. Theor. Math. Phys., 1993, vol. 94, no. 2, pp. 211–218.
[34] Galaktionov V.A. Quasilinear heat equations with first-order sign-invariants and new explicit solutions. Nonlinear Anal. Theor. Meth. Appl., 1994, vol. 23, pp. 1595–621.
[35] Gandarias M.L., Romero J.L., Diaz J.M. Nonclassical symmetry reductions of a porous medium equation with convection. J. Phys. A: Math. Gen., 1999, vol. 32, pp. 1461–1473.
[36] Popovych R.O., Ivanova N.M. New results on group classification of nonlinear diffusion-convection equations. J. Physics A: Math. Gen., 2004, vol. 37, pp. 7547–7565.
[37] Ivanova N.M., Sophocleous C. On the group classification of variable-coefficient nonlinear diffusion-convection equations. J. Comput. Appl. Math., 2006, vol. 197, № 2, pp. 322–344.
[38] Vaneeva O.O., Johnpillai A.G., Popovych R.O., Sophocleous C. Extended group analysis of variable coefficient reaction-diffusion equations with power nonline-arities. J. Math. Anal. Appl., 2007, vol. 330, no. 2, pp. 1363–1386.
[39] Ivanova N.M. Exact solutions of diffusion-convection equations. Dynamics of PDE, 2008, vol. 5, no. 2, pp. 139–171.
[40] Vaneeva O.O., Popovych R.O., Sophocleous C. Enhanced group analysis and exact solutions of variable coefficient semilinear diffusion equations with a power source. Acta Appl. Math., 2009, vol. 106, no. 1, pp. 1–46.
[41] Vaneeva O.O., Popovych R.O., Sophocleous C. Extended group analysis of var-iable coefficient reaction-diffusion equations with exponential nonlinearities. J. Math. Anal. Appl., 2012, vol. 396, pp. 225–242.
[42] Cherniha R.M., Pliukhin O. New conditional symmetries and exact solutions of reaction-diffusion-convection equations with exponential nonlinearities. J. Math. Anal. Appl., 2013, vol. 403, pp. 23–37.
[43] Bradshaw-Hajek B.H., Moitsheki R.J. Symmetry solutions for reaction-diffusion equations with spatially dependent diffusivity. Appl. Math. Comput., 2015, vol. 254, pp. 30–38.
[44] Cherniha R., Serov M., Pliukhin O. Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applica-tions. Boca Raton, Chapman & Hall/CRC Press, 2018, 238 p.
[45] Polyanin A.D., Zaitsev V.F. Handbook of Ordinary Differential Equations: Ex-act Solutions, Methods, and Problems. Boca Raton, CRC Press, 2018, 1496 p.
[46] Ovsiannikov L.V. Group Analysis of Differential Equations. Boston, Academic Press, 1982, 432 p.

Полянин А.Д. Методы функционального разделения переменных и их применение в математической физике. Математическое моделирование и численные методы, 2019, № 1, с. 67–97.

Работа выполнена по теме государственного задания (№ госрегистрации AAAA-A17-117021310385-6) и при частичной финансовой поддержке Российского фонда фундаментальных исследований (проект № 18-29-03228).

Download article

Количество скачиваний: 421