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Section
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Mathematics
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Title
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On one approach to solving nonhomogeneous partial differential equations
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Author(-s)
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Rubina L.I.a,
Ul’yanov O.N.ab
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Affiliations
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Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa,
Ural Federal Universityb
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Abstract
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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.
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Keywords
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nonhomogeneous partial differential equations, exact solutions, ODE, systems of ODEs
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UDC
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517.977
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MSC
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35C05, 35C99
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DOI
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10.20537/vm170306
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Received
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3 July 2017
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Language
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Russian
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Citation
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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.
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References
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