Skip to main content
SpringerLink
Log in

Reduction Method and New Exact Solutions of the Multidimensional Nonlinear Heat Equation

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

Abstract

A nonlinear multidimensional heat equation with a power-law coefficient is studied. It is proposed to construct its exact solutions by the multidimensional reduction method based on the use of a special ansatz. As a result of the reduction, the problem is reduced to solving systems of matrix-vector algebraic equations that determine the dependence on the spatial variables and integrating ordinary differential equations that determine the dependence on time. For a number of examples with various values of the exponents, explicit expressions are obtained in terms of elementary functions for exact multidimensional solutions, including those anisotropic in the spatial variables. The exact solutions found can be useful when constructing approximate solutions of boundary value problems for the nonlinear heat equation using numerical methods leading to the need to solve high-dimensional systems of equations.

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. King, J.R., Exact multidimensional solutions to some nonlinear diffusion equations, Quart. J. Mech. Appl. Math., 1993, vol. 46, no. 3, pp. 419–436.

    Article  MathSciNet  Google Scholar 

  2. Pukhnachev, V.V., Multidimensional exact solutions of nonlinear diffusion equation, Prikl. Mekh. Tekh. Fiz., 1995, vol. 36, no. 2, pp. 23–31.

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  4. 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 

  5. Polyanin, A.D. and Zaitsev, V.F., Nelineinye uravneniya matematicheskoi fiziki. V 2-kh ch. Ch. 1. (Nonlinear Equations of Mathematical Physics in 2 Parts. Part 1), Moscow: Fizmatlit, 2017.

    Google Scholar 

  6. Polyanin, A.D. and Zhurov, A.I., Metody razdeleniya peremennykh i tochnye resheniya nelineinykh uravnenii matematicheskoi fiziki (Methods of Separation of Variables and Exact Solutions of Nonlinear Equations of Mathematical Physics), Moscow: Izd. IPMekh RAN, 2020.

    Google Scholar 

  7. Polyanin, A.D. and Zhurov, A.I., Separation of Variables and Exact Solutions to Nonlinear PDEs, Boca Raton–London–New York: CRC Press, 2022.

    MATH  Google Scholar 

  8. Kosov, A.A. and Semenov, E.I., Multidimensional exact solutions to the reaction-diffusion system with power-law nonlinear terms, Sib. Math. J., 2017, vol. 58, no. 4, pp. 619–632.

    Article  MathSciNet  Google Scholar 

  9. Kosov, A.A. and Semenov, E.I., On exact multidimensional solutions of a nonlinear system of reaction–diffusion equations, Differ. Equations, 2018, vol. 54, no. 1, pp. 106–120.

    Article  MathSciNet  Google Scholar 

  10. Galaktionov, V.A. and Samarskii, A.A., Methods of constructing approximate self-similar solutions of nonlinear heat equations. I, Sb. Math., 1983, vol. 46, no. 3, pp. 291–321.

    Article  Google Scholar 

  11. Ulrich Olivier Dangui-Mbani, Liancun Zheng, and Xinxin Zhang, On analytical solutions for the nonlinear diffusion equation, Am. J. Eng. Res., 2014, vol. 3, no. 9, pp. 224–232.

    Google Scholar 

Download references

Funding

This work was financially supported in part by the Russian Foundation for Basic Research, project no. 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. Reduction Method and New Exact Solutions of the Multidimensional Nonlinear Heat Equation. Diff Equat 58, 187–194 (2022). https://doi.org/10.1134/S0012266122020057

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266122020057

Search

Navigation