Construction of Exact Solutions of Nonlinear Differential Equations by the Splitting Method
https://doi.org/10.1134/S2304487X20010071
Abstract
Various classes of nonlinear ordinary differential equations are considered. To construct exact solutions in an implicit form, the splitting method based on the generalized separation of variables is used. The main attention is paid to nonlinear equations of a sufficiently general form that contain one or more arbitrary functions. It is important to note that the exact solutions of nonlinear differential equations that depend on arbitrary functions and, therefore, are sufficiently general are of the greatest practical interest for testing numerical and approximate methods for solving various problems. Examples of particular nonlinear equations and their exact solutions are given. In some cases, it is possible to find general solutions of the equations or lower their order. The approach used can be generalized to nonlinear partial differential equations. New exact solutions with functional separation of variables are obtained for reaction-diffusion type equations.
Keywords
nonlinear ordinary differential equations,
reaction-diffusion equations,
exact solutions in an implicit form,
generalized separation of variables,
functional separation of variables,
splitting method
Supporting agencies: The work was carried out on the topic of the state task (state registration number AAAA-AA20-120011690135-5) and with partial financial support from the Russian Foundation for Basic Research (project No. 18-29-10025)
About the Authors
A. D. Polyanin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences
Russian Federation
Russian Federation
119526
Moscow
L. V. Linchuk
Peter the Great St. Petersburg Polytechnic University; Herzen State Pedagogical University of Russia
Russian Federation
Russian Federation
195251
191186
St. Petersburg
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Review
For citations:
Polyanin A.D., Linchuk L.V. Construction of Exact Solutions of Nonlinear Differential Equations by the Splitting Method. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(1):32-44. (In Russ.) https://doi.org/10.1134/S2304487X20010071
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