# A multi-dimensional non-autonomous non-linear partial differential equation with senior partial derivative

### Abstract

We study the solutions of a multi-dimensional non-autonomous partial differential equation of arbitrary order which contains the senior partial derivative, an arbitrary nonlinearity with respect to an unknown function, and power nonlinearities with respect to its first derivatives. The separation of variables is applied for the investigation of this equation. We consider the cases when the right-hand side of the equation can be represented as a product of functions depending on some subsets of independent variables and, in particular, of functions of one variable. This equation is reduced either to ordinary differential equations or to partial differential equations of the lower dimension. We obtain particular solutions of the power, exponential, and logarithmic form, as well as a particular solution of the polynomial form, study the dependence of these solutions on the equation parameters, and find their existence conditions. We consider separately the case of a nonlinear non-autonomous equation of Bianci’s type, the equation with only first derivatives with respect to every independent variable, and obtain the exact solutions to it.

### References

[1]

Bondarenko B. A., The Basic Systems of Polynomial and Quasypolynomial Solutions of Partial Differential Equations [in Russian], FAN, Tashkent (1987).

[2]

Zhegalov V. I. and Mironov A. N., Differential Equations with Senior Partial Derivatives [in Russian], Kazan. Mat. Obshch., Kazan (2001).

[3]

Zhegalov V. I. and Tikhonova O. A., “Factorization of equations with dominating higher partial derivative,” Differ. Equ., 50, No. 1, 66–72 (2014).

[4]

Mironov A. N., “The Riemann method for equations with leading partial derivative in $R^n$,” Sib. Math. J., 47, No. 3, 481–490 (2006).

[5]

Utkina E. A., “On a differential equation with a higher-order partial derivative in three-dimensional space,” Differ. Equ., 41, No. 5, 733–738 (2005).

[6]

Rakhmelevich I. V., “On two-dimensional hyperbolic equation with power non-linearity on the derivatives [in Russian],” Vestn. Tomsk. Gos. Univ., Mat., Mekh., No. 1 (33), 12–19 (2015).

[7]

Rakhmelevich I. V., “On the solutions of the arbitrary order differential equation with mixed senior partial derivative and power non-linearities [in Russian],” Vladikavkaz. Mat. Zhurn., 18, No. 4, 41–49 (2016).

[8]

Rakhmelevich I. V., “Two-dimensional non-autonomous hyperbolic equation of the second order with power non-linearities [in Russian],” Vestn. Tomsk. Gos. Univ., Mat., Mekh., No. 49, 52–60 (2017).

[9]

Rakhmelevich I. V., “A multidimensional nonautonomous equation containing a product of powers of partial derivatives,” Vestn. St. Peterburg. Univ., Mat., 51, No. 1, 87–94 (2018).

[10]

Galaktionov V. A., Posashkov S. A., and Svirshchevskii S. R., “Generalized separation of variables for differential equations with polynomial right-hand sides [in Russian],” Differents. Uravn., 31, No. 2, 253–261 (1995).

[11]

Polyanin A. D. and Zaytsev V. F., Handbook of Nonlinear Partial Differential Equations, 2nd ed., CRC Press, Boca Raton; London (2012).

[12]

Polyanin A. D. and Zhurov A. I., “Generalized and functional separation of variables in mathematical physics and mechanics [in Russian],” Dokl. Akad. Nauk, 382, No. 5, 606–611 (2002).

[13]

Polyanin A. D., Zaytsev V. F., and Zhurov A. I., Methods of Solving Nonlinear Equations of Mathematical Physics and Mechanics [in Russian], Fizmatlit, Moscow (2005).

[14]

Rakhmelevich I. V., “On application of variable separation method to mathematical physics equations containing homogeneous functions of derivatives[in Russian],” Vestn. Tomsk. Gos. Univ., Mat., Mekh., No. 3 (23), 37–44 (2013).

[15]

Rakhmelevich I. V., “On equations of mathematical physics containing multi-homogeneous functions of derivatives [in Russian],” Vestn. Tomsk. Gos. Univ., Mat., Mekh., No. 1 (27), 42–50 (2014).

[16]

Grundland A. M. and Infeld E., “A family of non-linear Klein–Gordon equations and their solutions,” J. Math. Phys., 33, No. 7, 2498–2503 (1992).

[17]

Miller J., Jr. and Rubel L. A., “Functional separation of variables for Laplace equations in two dimensions,” J. Phys. A, 26, 1901–1913 (1993).

[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., 111, 95–105 (2019).

[19]

Polyanin A. D., “Comparison of the effectiveness of different methods for constructing exact solutions to nonlinear PDEs. Generalizations and new solutions,” Mathematics, 7, No. 5, 386 (2019).

[20]

Zhdanov R. Z., “Separation of variables in the non-linear wave equation,” J. Phys. A, 27, L291–L297 (1994).

*Mathematical notes of NEFU*, 28(1), pp. 37-50. doi: https://doi.org/10.25587/SVFU.2021.55.86.004.

This work is licensed under a Creative Commons Attribution 4.0 International License.