ON MULTIDIMENSIONAL NONLINEAR EQUATIONS ASSOCIATED WITH LAPLACE AND HEAT CONDUCTION EQUATIONS BY MEANS OF FUNCTIONAL SUBSTITUTIONS 

Zhuravlev Viktor Mikhaylovich, Doctor of physical and mathematical sciences, sub-department of theoretical physics, Ulyanovsk State University (42 Lva Tolstogo street, Ulyanovsk, Russia), zhvictorm@gmail.ru

51-71, 532.51, 538.93

10.21685/2072-3040-2016-4-8

Background. The article considers the application of the method of functional substitutions of Cole-Hopf type to multidimensional problems of wave dynamics and hydromechanics. Using additional transformations, solutions to the Liouville equation are developed in a two-dimensional coordinate space while analyzing the selfmapping of the solution space and the mapping of the Laplace equation.
Materials and methods. A method of researching the equations under consideration is the method of functional substitutions of Cole-Hopf type. In terms of the aggregate of simple basic differential correlations of one auxiliary function and an additional equation for it, the given method enables to solve new nonlinear equations associated with the said equation, the solutions of which are developed in the form of differential substitutions. An elementary example of the given approach is a ColeHopf substitution for the Burgers equation. The given approach appears to be efficient for a series of problems. The present study uses multidimensional extension of the method of functional substitutions allowing to obtain a series of useful results concerning dynamics of liquid in 3D space.
Results. It is shown that in 2D space there exist infinite recurrent chains of selftransformations of Laplace and Liouville equations associated with an ascending order of derivatives of one initial function being a solution of the Laplace equation. The form of nonlinear equations associated as well with the Helmholtz equation in a 2D coordinate space is calculated using substitutions and mappings. Then, the approach under consideration is applied to multidimensional equations of heat conduction, and there is established a connection of solutions to the said equations with solutions of multidimensional equations for viscous compressible and incompressible fluids in the potential flow class. The above-said allows to specify a method for calculation of precise solutions of the Navier-Stokes equations on the basis of heat conduction equations’ solutions. In conclusion, the article considers the problem of calculation using substitutions of multidimensional equations associated with Laplace and heat conduction equations. The work shows that such equations may be reduced to heterogeneous equations of Liouville type, the heterogeneity of which is associated with properties of unit vector fields in a corresponding coordinate space.
Conclusions. The multidimensional variant of the method of functional substitutions, developed in the work, makes it possible to obtain correlations useful for ap-plied problems. Such correlations link solutions of sinple euqations of Laplace, heat conduction and Helmholtz types with nonlinear equations related to wave dynamics and hydromechanics. The results obtained expand the field of application of the method of functional substitutions for problems of the oretical and mathematical physics.

exactly integrable nonlinear equations, generalized Hopf-Cole substitution, multi-dimensional Laplace equation, Navier-Stokes equations, Liouville equation

1. Zhuravlev V. M., Nikitin A. V. Nelineynyy mir [The nonlinear world]. 2007, vol. 5, no. 9, pp. 603–611.
2. Zhuravlev V. M., Zinov'ev D. A. Pis'ma v Zhurnal eksperimental'noy i teoreticheskoy fiziki [Letters to the Journal of experimental and theoretical physics]. 2008, vol. 87, no. 5, pp. 314–318.
3. Zhuravlev V. M.,. Zinov'ev D. A Pis'ma v Zhurnal eksperimental'noy i teore-ticheskoy fiziki [Letters to the Journal of experimental and theoretical physics]. 2008, vol. 88, no. 3, pp. 194–197.
4. Zhuravlev V. M. Teoreticheskaya i matematicheskaya fizika [Theoretical and mathematical physics]. 2009, vol. 159, no. 1, pp. 58–71.
5. Zhuravlev V. M., Zinov’ev D. A. Physics of Wave Phenomena. 2010, vol. 18, no. 4, pp. 245–250.
6. Zhuravlev V. M., Zinov’ev D. A. Physics of Wave Phenomena. 2011, vol. 19, no. 4, pp. 313–317.
7. Semenov E. I. Sibirskiy matematicheskiy zhurnal [Siberian mathematical journal]. 2008, vol. 48, no. 1, pp. 207–217.
8. Semenov E. I. Sibirskiy matematicheskiy zhurnal [Siberian mathematical journal]. 2003, vol. 44, no. 4, pp. 863–869.
9. Dodd R., Eylbek Dzh., Gibbon Dzh., Morrio Kh. Solitony i nelineynye volnovye uravneniya [Solitons and nonlinear wave equations]. Moscow: Mir, 1988, p. 694.
10. Zhuravlev V. M. Teoreticheskaya i matematicheskaya fizika [Theoretical and mathematical physics]. 1999, vol. 120, no. 1, pp. 3–19.
11. Zhuravlev V. M. Izvestiya vuzov. Ser.: Prikladnaya nelineynaya dinamika [University proceedings. Series: Applied nonlinear dynamics]. 2001, vol. 9, no. 6, pp. 115–128.
12. Zhuravlev V. M. Nelineynye volny v mnogokomponentnykh sistemakh s dis-persiey i diffuziey [Nonlinear waves in multidimensional systems with dispersion and diffusion]. Ulyanovsk: Izd-vo UlGU, 2001, 252 p.