Skip to main content
SpringerLink
Log in

Exact Solutions of the Generalized Richards Equation with Power-Law Nonlinearities

  • PARTIAL DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We consider a generalized Richards equation with power-law nonlinearities modeling filtration in porous media. Conditions are derived under which the problem can be reduced to the linear heat equation or to nonlinear equations with known solutions. The families of explicit exact solutions that can be expressed via elementary functions or Lambert’s \(W \)-function are found. Some examples illustrating the results are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Barry, D.A., Parlange, J.Y., Sander, G.C., and Sivaplan, M., A class of exact solutions for Richard’s equation, J. Hydrol., 1993, vol. 142, pp. 29–46.

    Article  Google Scholar 

  2. Corless, R.M., Gonnet, G.H., Hare, D.E., Jeffrey, D.J., and Knuth, D.E., On the Lambert W function, Adv. Comput. Math., 1996, vol. 5, pp. 329–359.

    Article  MathSciNet  Google Scholar 

  3. Zlotnik, V.A., Wang, T., Nieber, J.L., and šimunek, J., Verification of numerical solutions of the Richards equation using a traveling wave solution, Adv. Water Resour., 2007, vol. 30, pp. 1973–1980.

    Article  Google Scholar 

  4. Akhmetzyanov, A.V., Kushner, A.G., and Lychagin, V.V., Attractors in models of porous media flow, Dokl. Math., 2017, vol. 95, pp. 72–75.

    Article  MathSciNet  Google Scholar 

  5. Witelski, T.P., Intermediate asymptotics for Richard’s equation in a finite layer, J. Eng. Math., 2003, vol. 45, pp. 379–399.

    Article  MathSciNet  Google Scholar 

  6. Broadbridge, P., Daly, E., and Goard, J., Exact solutions of the Richards equation with nonlinear plant-root extraction, Water Resour. Res., 2017, vol. 53, pp. 9679–9691.

    Article  Google Scholar 

  7. Xi Chen and Ying, Dai., An approximate analytical solution of Richards equation with finite boundary, Boundary Value Probl., 2017, vol. 2017, no. 1, article ID 167.

    Article  MathSciNet  Google Scholar 

  8. Barari, A., Omidvar, M., Ghotbi, A.R., and Ganji, D.D., Numerical analysis of Richard’s problem for water penetration in unsaturated soils, Hydrol. Earth Syst. Sci., 2009, vol. 6, pp. 6359–6385.

    Article  Google Scholar 

  9. Farthing, M.W. and Ogden, F.L., Numerical solution of Richard’s equation: A review of advances and challenges, Soil Sci. Soc. Am. J., 2017, vol. 81, pp. 1257–1269.

    Article  Google Scholar 

  10. Yuanyuan Zha, Jinzhong Yang, Jicai Zeng, Chak-Hau, M., Wenhi Zeng, and Liangsheng, Shi., Review of numerical solution of Richardson–Richards equation for variably saturated flow in soils, Wiley Interdiscip. Rev.: Water Resour., 2019, vol. 6, no. 5, pp. 1–23.

    Google Scholar 

  11. Yung, C.M., Verburg, K., and Baveye, P., Group classification and symmetry reductions of the non-linear diffusion–convection equation \(u_{t}=(D(u)u_x)_x+K^{\prime }(u)u_x\), Int. J. Non-Linear Mech., 1994, vol. 29, no. 3, pp. 273–278.

    Article  Google Scholar 

  12. Gandarias, M.L., Romero, J.L., and Diaz, J.M., Nonclassical symmetry reductions of a porous medium equation with convection, J. Phys. A: Math. Gen., 1999, vol. 32, pp. 1461–1473.

    Article  MathSciNet  Google Scholar 

  13. Popovych, R.O. and Ivanova, N.M., New results on group classification of nonlinear diffusion–convection equations, J. Phys. A: Math. Gen., 2004, vol. 37, no. 30, pp. 7547–7565.

    Article  MathSciNet  Google Scholar 

  14. Ivanova, N.M., Exact solutions of diffusion–convection equations, Dyn. PDE, 2008, vol. 5, no. 2, pp. 139–171.

    MathSciNet  MATH  Google Scholar 

  15. Ovsyannikov, L.V., Group properties of nonlinear heat equations, Dokl. Akad. Nauk SSSR, 1959, vol. 125, no. 3, pp. 492–495.

    MathSciNet  MATH  Google Scholar 

  16. Polyanin, A.D. and Zaitsev, V.F., Spravochnik po nelineinym uravneniyam matematicheskoi fiziki. Tochnye resheniya (Handbook of Nonlinear Equations of Mathematical Physics. Exact Solutions), Moscow: Fizmatlit, 2002.

    Google Scholar 

  17. Polyanin, A.D. and Zaitsev, V.F., Nelineinye uravneniya matematicheskoi fiziki. V 2-x ch. Ch. 1 (Nonlinear Equations of Mathematical Physics. In 2 Parts. Part 1), Moscow: Yurait, 2017.

    Google Scholar 

  18. Polyanin, A.D. and Zaitsev, V.F., Nelineinye uravneniya matematicheskoi fiziki. V 2-x ch. Ch. 2 (Nonlinear Equations of Mathematical Physics. In 2 Parts. Part 2), Moscow: Yurait, 2017.

    Google Scholar 

  19. Cherniha, R., Serov, M., and Pliukhin, O., Nonlinear Reaction–Diffusion–Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applications, Boca Raton: Chapman & Hall/CRC Press, 2018.

    MATH  Google Scholar 

  20. Polyanin, A.D., Comparison of the effectiveness of different methods for constructing exact solutions to nonlinear PDEs. Generalizations and new solutions, Mathematics, 2019, vol. 7, no. 386, pp. 1–19.

    Google Scholar 

  21. Galaktionov, V.A. and Posashkov, S.A., New exact solutions of parabolic equations with quadratic nonlinearities, Zh. Vychisl. Mat. Mat. Fiz., 1989, vol. 29, no. 4, pp. 497–506.

    MathSciNet  MATH  Google Scholar 

  22. Polyanin, A.D., Zaitsev, V.F., and Zhurov, A.I., Metody resheniya nelineinykh uravnenii matematicheskoi fiziki i mekhaniki (Methods for Solving Nonlinear Equations of Mathematical Physics and Mechanics), Moscow: Fizmatlit, 2005.

    Google Scholar 

  23. Galaktionov, V.A and Svirshchevskii, S.R., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Boca Raton–London–New York: Chapman & Hall/CRC Press, 2007.

    MATH  Google Scholar 

  24. Rudykh, G.A. and Semenov, E.I., Construction of exact solutions of the one-dimensional nonlinear diffusion equation by the method of nonlinear invariant subspaces,Izv. Irkutsk. Gos. Univ. Ser. Mat., 2013, vol. 6, no. 4, pp. 69–84.

    MATH  Google Scholar 

  25. Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., and Mikhailov, A.P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii (Blow-Up Regimes in Problems for Quasilinear Parabolic Equations), Moscow: Nauka, 1987.

    Google Scholar 

  26. Dubinov, A.E., Dubinova, I.D., and Saikov, S.K., \(W \)-funktsiya Lamberta i ee primenenie v matematicheskikh zadachakh fiziki (Lambert’s \(W \)-Function and Its Application to Mathematical Problems in Physics), Sarov: FGUP “TFYaTs-VNIIEF”, 2006.

  27. Kosov, A.A. and Semenov, E.I., The Lambert function and exact solutions of nonlinear parabolic equations, Russ. Math., 2019, vol. 63, no. 8, pp. 10–16.

    Article  Google Scholar 

  28. Barenblatt, G.I., On limit self-similar motions in the theory of nonstationary gas filtration in a porous medium and the boundary layer theory, Prikl. Mat. Mekh., 1954, vol. 18, no. 4, pp. 409–414.

    Google Scholar 

  29. Barenblatt, G.I., Podobie, avtomodel’nost’, promezhutochnaya asimptotika (Similarity, Self-Similarity, and Intermediate Asymptotics), Leningrad: Gidrometeoizdat, 1978.

    Google Scholar 

  30. Rudykh, G.A. and Semenov, E.I., New exact solutions to the one-dimensional equation of nonlinear diffusion, Sib. Mat. Zh., 1997, vol. 38, no. 5, pp. 1130–1139.

    Article  Google Scholar 

  31. Rudykh, G.A. and Semenov, E.I., On new exact solutions to the one-dimensional equation of nonlinear diffusion with a source (sink), Zh. Vychisl. Mat. Mat. Fiz., 1998, vol. 38, no. 6, pp. 971–977.

    MathSciNet  MATH  Google Scholar 

Download references

Funding

This work was supported by the Russian Foundation for Basic Research, projects nos.19-08-00746 and 20-07-00397.

Author information

Authors and Affiliations

  1. Matrosov Institute for System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Irkutsk, 664033, Russia

    A. A. Kosov & E. I. Semenov

Authors
  1. A. A. Kosov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. I. Semenov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. A. Kosov or E. I. Semenov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kosov, A.A., Semenov, E.I. Exact Solutions of the Generalized Richards Equation with Power-Law Nonlinearities. Diff Equat 56, 1119–1129 (2020). https://doi.org/10.1134/S0012266120090025

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266120090025

Search

Navigation