On Multiplicative Multi-Dimensional Partial Differential Equations

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.46698/u4295-5273-0507-x On Multiplicative Multi-Dimensional Partial Differential Equations Rakhmelevich, I. V. Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 1. Abstract: The class of multiplicative partial differential equations is considered. The left side of the equation is represented in the form of the product of linear differential expressions of arbitrary order. The right side of the equation is a function of independent variables and unknown function. The solutions with additive, multiplicative and combined separation of variables are received for the equation with one-dimensional linear operators and factorized right side. The initial equation is reduced either to ordinary differential equation or to partial differential equation of lower dimension. It is shown that if the operators in the left side are homogeneous then the equation has the solutions in the form of generalized polynomials. Also the self-similar solutions are founded and the sufficient conditions of their existence are formulated for the equation with homogeneous operators. The traveling wave type solutions are received for the case of operators with constant coefficients and the right side depending on the linear combination of independent variables. It is shown that if the right side of equation depends on several linear combinations on subsets of independent variables, then there are multi-dimensional traveling wave type solutions depending on these linear combinations. A solution is obtained in the form of an expansion with respect to eigenfunctions of the kernels of linear operators in the left side of equation. Also some equations with multi-dimensional linear operators are investigated. In particular, the solutions with separation of variables for the case of factorized multi-dimensional operators, the of multi-dimensional traveling wave type solutions, and the solutions in the form of functions on arguments of more complicated structure are obtained. The cases when the right side of equation contains power and exponential nonlinearities on unknown function are considered. Keywords: multiplicative differential equation, linear differential operator, separation of variables, solution of traveling wave type, self-similar solution Language: Russian Download the full text For citation: Rakhmelevich, I. V. On Multiplicative Multi-Dimensional Partial Differential Equations, Vladikavkaz Math. J., 2021, vol. 23, no. 1, pp.43-59 (in Russian). DOI 10.46698/u4295-5273-0507-x + References 1. Polyanin, A. D. and Zaytsev, V. F. Handbook of Nonlinear Partial Differential Equations, 2nd ed., Boca Raton-London, Chapman and Hall-CRC Press, 2012. 2. Polyanin, A. D. and Zhurov, A. I. Metody razdeleniya peremennykh i tochnye resheniya nelineynykh uravneniy matematicheskoy fiziki [Variables Separation Methods and Exact Solutions of Nonlinear Equations of Mathematical Physics], Moscow, IPMech Publ., 2020 (in Russian). 3. Polyanin, A. D., Zaitsev, V. F., and Moussiaux, A. Handbook of First Order Partial Differential Equations, London, Taylor Francis, 2002. 4. Rakhmelevich, I. V. On the Solutions of Multi-Dimensional Arbitrary Order Differential Equation with Mixed Senior Partial Derivative and Power-Law Non-Linearities, Vladikavkaz Math. J., 2016, vol. 18, no. 4, pp. 41-49 (in Russian). DOI: 10.23671/VNC.2016.4.5992. 5. Rakhmelevich, I. V. On Two-Dimensional Hyperbolic Equations with Power-Law Non-Linearity in the Derivatives, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2015, no. 1 (33), pp. 12-19 (in Russian). DOI: 10.17223/19988621/33/2. 6. Rakhmelevich, I. V. On Some New Solutions of Multi-Dimensional Partial Differential Equation of the First Order with Power-Law Non-Linearities, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2015, no. 3 (35), pp. 18-25 (in Russian). DOI: 10.17223/19988621/35/3. 7. Rakhmelevich, I. V. On the Pseudo-Polynomial Solutions of Two-Dimensional Equation Containing the Production of Partial Derivatives, Belgorod State University Scientific Bulletin. Applied Mathematics and Physics, 2017, vol. 47, no. 13, pp. 45-50 (in Russian). 8. Rakhmelevich, I. V. On Multi-Dimensional Partial Differential Equations with Power Nonlinearities in First Derivatives, Ufa Mathematical Journal, 2017, vol. 9, no. 1, pp. 98-108. DOI: 10.13108/2017-9-1-98. 9. Rakhmelevich, I. V. A Multidimensional Nonautonomous Equation Containing a Product of Powers of Partial Derivatives, Vestnik St. Petersburg University, Mathematics, 2018, vol. 51, pp. 87-94. DOI: 10.3103/S1063454118010090. 10. Miller, W. and Rubel, L. A. Functional Separation of Variables for Laplace Equations in Two Dimensions, Journal of Physics A: Mathematical and General, 1993, vol. 26, no. 8, pp. 1901-1913. DOI: 10.1088/0305-4470/26/8/017. 11. Zhdanov, R. Z. Separation of Variables in the Nonlinear Wave Equation, Journal of Physics A: Mathematical and General, 1994, vol. 27, no. 9, pp. L291-L297. DOI: 10.1088/0305-4470/27/9/009. 12. Grundland, A. M. and Infeld, E. 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