Exact Solutions of Nonlinear Telegraph Equations with Variable Coefficients
https://doi.org/10.56304/S2304487X1902010X
Abstract
Various classes of nonlinear telegraph equations with variable coefficients c(x)un + d(x)uτ = [a(x)ux]x + b(x)ux + p(x)ux + p(x)f(u), which allow exact solutions with a functional separation of variables of the form u = U(z), z = ϕ , (x, t ). , have been described. It has been shown that the source function f(u) and any four of the five coefficients a(x), b(x), c(x), d(x), p(x) of these equations can be chosen arbitrarily, and the remaining coefficient is expressed in terms of them. The properties have been studied of the overdetermined system of differential equations for the function ф(x, t) and some its solutions have been obtained. Examples of particular equations and their exact solutions are given. Some exact generalized traveling-wave solutions of more complex nonlinear telegraph equations with delay of the form c(x)un + d(x)uτ = [a(x)ux]x + b(x)ux + p(x)ux + p(x)f(u, w), w = u(x,t - τ), where τ > 0 is the delay time and f(u, w) is an arbitrary function of two arguments, are also obtained.
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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For citations:
Polyanin A.D. Exact Solutions of Nonlinear Telegraph Equations with Variable Coefficients. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2019;8(3):213-225. (In Russ.) https://doi.org/10.56304/S2304487X1902010X
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