Exact Solutions in the Implicit Form of Nonlinear Mass and Heat Transfer Equations with Variable Coefficients
https://doi.org/10.1134/S2304487X19040084
Abstract
Various classes of nonlinear mass and heat transfer equations with variable coefficients, c(x)ut = ⌠a(x)f(u)uxIx = b(x)g(u)ux, which admit exact solutions, are considered. The main attention is focused on nonlinear equations of a sufficiently general form, which contain several arbitrary functions that depend on the unknown function u and the spatial variable x. It is important to note that the exact solutions of nonlinear partial differential equations that contain arbitrary functions and are, therefore, sufficiently general, are of the greatest practical interest for testing various numerical and approximate analytical methods to solve corresponding initial-boundary value problems. The method used to find exact solutions is based on the representation of the solution in the implicit form⌠ h(u)du = ξ(t) + η(x), where the functions h(u), ξ(t), η(x) are determined further by analyzing resulting functional-differential equations. Examples of specif-ic reaction–diffusion type equations and their exact solutions are given. Many new generalized traveling wave solutions and functional separable solutions are described.
Keywords
mass and heat transfer equations,
nonlinear convection–diffusion,
equations,
PDEs with variable coefficients,
exact solutions in implicit form,
functional separable solutions,
generalized traveling-wave solutions
Supporting agencies: The work was carried out on the subject of the state assignment (state registration No. AAAA-A17- 117021310385-6) and with partial financial support from the Russian Foundation for Basic Research (project no. 18-29-10025)
About the Author
A. D. Polyanin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences; National Research Nuclear University MEPhI (Moscow Engineering Physics Institute); Bauman Moscow State Technical University
Russian Federation
Russian Federation
119526
115409
105005
Moscow
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Review
For citations:
Polyanin A.D. Exact Solutions in the Implicit Form of Nonlinear Mass and Heat Transfer Equations with Variable Coefficients. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2019;8(5):415-427. (In Russ.) https://doi.org/10.1134/S2304487X19040084
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