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## Archive of Issues

Russia Yekaterinburg
Year
2017
Volume
27
Issue
3
Pages
355-364
 Section Mathematics Title On one approach to solving nonhomogeneous partial differential equations Author(-s) Rubina L.I.a, Ul’yanov O.N.ab Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Ural Federal Universityb Abstract An approach to obtaining exact solutions for nonhomogeneous partial differential equations (PDEs) is suggested. It is shown that if the right-hand side of the equation specifies the level surface of a solution of the equation, then, in this approach, the search of solutions of considered nonhomogeneous differential equations is reduced to solving ordinary differential equation (ODE). Otherwise, searching for solutions of the equation leads to solving the system of ODEs. Obtaining a system of ODEs relies on the presence of the first derivatives of the sought function in the equation under consideration. For PDEs, which do not explicitly contain first derivatives of the sought function, substitution providing such terms in the equation is proposed. In order to reduce the original equation containing the first derivative of the sought function to the system of ODEs, the associated system of two PDEs is considered. The first equation of the system contains in the left-hand side only first order partial derivatives, selected from the original equation, and in the right-hand side it contains an arbitrary function, the argument of which is the sought unknown function. The second equation contains terms of the original equation that are not included in the first equation of the system and the right-hand side of the first equation in the system created. Solving the original equation is reduced to finding the solutions of the first equation of the resulting system of equations, which turns the second equation of the system into identity. It has been possible to find such solution using extended system of equations for characteristics of the first equation and the arbitrariness in the choice of function from the right-hand side of the equation. The described approach is applied to obtain some exact solutions of the Poisson equation, Monge-Ampere equation and convection–diffusion equation. Keywords nonhomogeneous partial differential equations, exact solutions, ODE, systems of ODEs UDC 517.977 MSC 35C05, 35C99 DOI 10.20537/vm170306 Received 3 July 2017 Language Russian Citation Rubina L.I., Ul’yanov O.N. On one approach to solving nonhomogeneous partial differential equations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 3, pp. 355-364. References Starovoitova R.P. Funktsii Grina (Green's functions), Мoscow: Mir, 1982, 90 p. Cole J.D. Perturbation methods in applied mathematics, London: Ginn-Blaisdell, 1968, 260 p. Translated under the title Metody vozmushchenii v prikladnoi matematike, Мoscow: Mir, 1972, 276 p. Kudryavtsev A.G., Sapozhnikov O.A. Determination of the exact solutions to the inhomogeneous Burgers equation with the use of the Darboux transformation, Acoustical Physics, 2011, vol. 57, issue 3, pp. 311-319. DOI: 10.1134/S1063771011030080 Polyanin A.D., Zaitsev V.F., Handbook of nonlinear partial differential equations, 2nd edition, Boca Raton-London-New York: Chapman & Hall/CRC Press, 2012, 1912 p. DOI: 10.1201/b11412 Rubina L.I., Ul’yanov O.N. A geometric method for solving nonlinear partial differential equations, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2010, vol. 16, no. 2, pp. 209-225 (in Russian). Rubina L.I., Ul’yanov O.N. Solution of nonlinear partial differential equations by the geometric method, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2012, vol. 18, no. 2, pp. 265-280 (in Russian). Rubina L.I., Ul’yanov O.N. 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