Exact Solutions of Nonlinear Partial Differential Equations with Pantograph Type Variable Delay
https://doi.org/10.1134/S2304487X20040069
Abstract
Nonlinear partial differential equations with variable delay of pantograph type are studied. These equations, in addition to the unknown function u = u(x, t), also contain functions with stretching of one or several independent variables of the form u(px, t), u(x, qt), or u(px, qt), where p and q are the scaling parameters (0 < p < 1, 0 < q < 1). Exact solutions of various classes of such equations are described for the first time. Examples of nonlinear partial differential equations with variable delay of pantograph type that allow self-similar solutions are given (note that partial differential equations with constant delay do not have self-similar solutions). Additive, multiplicative, and generalized separable solutions, as well as solutions of a more complex form, are obtained. Special attention is paid to nonlinear partial differential equations of pantograph type of a fairly general form that contain arbitrary functions. In total, more than 40 nonlinear equations with variable delay of pantograph type, admitting exact solutions, are considered. It is shown that some equations can be generalized to the case of delay, which arbitrarily depends on time. The described equations and their exact solutions can be used to formulate test problems designed to check the adequacy and assess the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for partial differential equations with variable delay of pantograph type.
Keywords
nonlinear pantograph-type differential equations,
partial differential equations with variable delay,
reaction-diffusion equations,
wave equations,
exact solutions,
self-similar solutions
Supporting agencies: The work was carried out on the topic of state Denmark (State Registration No. AAAA-A20-120011690135-5) 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
V. G. Sorokin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences
Russian Federation
Russian Federation
119526
Moscow
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For citations:
Polyanin A.D., Sorokin V.G. Exact Solutions of Nonlinear Partial Differential Equations with Pantograph Type Variable Delay. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(4):315-328. (In Russ.) https://doi.org/10.1134/S2304487X20040069
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