Address: Markusa st. 22, Vladikavkaz,
362027, RNO-A, Russia
Phone: (8672)50-18-06
E-mail: rio@smath.ru
DOI: 10.23671/VNC.2016.4.5992
On the Solutions of Multi-Dimensional Arbitrary order Differential Equation with Mixed Senior Partial Derivative and Power-Law Non-Linearities
Rakhmelevich I. V.
Vladikavkaz Mathematical Journal 2016. Vol. 18. Issue 4.
Abstract: We study the solutions of a multi-dimensional differential equation of arbitrary order containing mixed senior partial derivative and power-law non-linearities on unknown function and its first derivatives. The method of functional separation of variables is applied for examining of this equation. The particular solutions of the equation under consideration are obtained. Some theorems which permit to decrease the order of this equation are proved.
Keywords: partial differential equation, functional separation of variables, power-law non-linearity.
1. Bondarenko B.A. Bazisnye sistemy polinomialnykh i
kvazipolinomialnykh resheniy uravneniy v chastnykh proizvodnykh
[Basic Systems of Polynomial and Quasy-polynomial Solutions of
Partial Differential Equations]. Tashkent, FAN, 1987. 146 p.
2. Zhegalov V.I., Mironov A.N. Differentsialnye uravneniya so
starshimi chastnymi proizvodnymi [Differential Equations with
Highest Partial Derivatives]. Kazan, Kazan Mathematical Society,
2001. 226 p.
3. Utkina E.A. On a Differential Equation with a Higher-Order
Partial Derivative in Three-Dimensional Space. Differential
Equations, 2005, vol. 41, no. 5, pp. 733-738.
4. Mironov A.N. The Riemann Method for Equations with Leading
Partial Derivative in \(R^n\). Siberian Mathematical Journal, 2006, vol. 47,
no. 3, pp. 481-490.
5. Zhegalov V.I., Tikhonova O.A. Factorization of Equations with
Dominating Higher Partial Derivative. Differential Equations, 2005,
vol. 50, no. 1, pp. 66-72.
6. Polyanin A.D., Zaitsev V.F. Handbook of Nonlinear Partial Differential Equations. London, Boca Raton, 2004. 834 pp.
7. Polyanin A.D., Zhurov A.I. The Generalized and Functional
Separation of Variables in Mathematical Physics and Mechanics.
Doklady Mathematics, 2002, vol. 65, no. 1, pp. 129-134.
8. Rakhmelevich I.V. On Application of Variables Separation Method
to Equations of Mathematical Physics Containing Homogeneous
Functions of Derivatives. Vestnik Tomskogo gosudarstvennogo
universiteta. Matematika i mekhanika [Tomsk State University Journal
of Mathematics and Mechanics], 2013, no. 3, pp. 37-44 [in Russian].
9. Rakhmelevich I.V. On Equations of Mathematical Physics Containing
Multi-Homogeneous Functions of Derivatives. Vestnik Tomskogo
gosudarstvennogo universiteta. Matematika i mekhanika [Tomsk State
University Journal of Mathematics and Mechanics]. 2014, no. 1, pp.
42-50 [in Russian].
10. Miller J. (Jr.), Rubel L.A. Functional separation of variables
for Laplace equations in two dimensions . Journal of Physics A,
1993, vol. 26, pp. 1901-1913.
11. Rakhmelevich I.V. On Two-Dimensional Hyperbolic Equations with
Power-Law Non-Linearity on the Derivatives. Vestnik Tomskogo
gosudarstvennogo universiteta. Matematika i mekhanika [Tomsk State
University Journal of Mathematics and Mechanics], 2015, no. 1, pp.
12-19 [in Russian].
