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Green’s function solution of nonlinear wave equation depending on the absolute value of the unknown function

Department on Dynamics of Deformable Systems and Coupled Fields, Institute of Mechanics, National Academy of Sciences of Armenia, 24B Baghramyan Ave., 0019 Yerevan, Armenia

Institute of Natural Sciences, Shanghai Jiao Tong University, 800 Dong Chuan Road, 200240 Shanghai, P. R. China

E-mail Address: khurshudyan@mechins.sci.am

E-mail Address: khurshudyan@sjtu.edu.cn

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and

Martiros Khurhsudyan

International Laboratory for Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics, (TUSUR) Tomsk, Russia

Research Division, Tomsk State Pedagogical University, 634061 Tomsk, Russia

CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei, P. R. China

School of Astronomy and Space Science, University of Science and Technology of China, Hefei, P. R. China

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https://doi.org/10.1142/S0219887818501797Cited by:4 (Source: Crossref)

Abstract

This paper is devoted to possibilities of a semi-analytical approximation of nonlinear wave equations with nonlinearities depending on the absolute value of the unknown function, which arise in different areas of physics and mechanics. The main difficulty of analysis of such equations is that the derivation of their rigorous solution is highly sophisticated, while their numerical solution requires burdensome computational costs. Using the traveling wave ansatz, we first reduce the wave equation to a nonlinear ordinary differential equation. Then, applying Frasca’s method, we construct its general solution in terms of the nonlinear Green’s function. For particular nonlinearities, it is shown that the first-order approximation of the Green’s function solution is numerically comparable with the solution obtained by the well-known numerical method of lines. The contribution of the higher order terms is studied for a particular nonlinearity.

Keywords:

AMSC: 34B27, 35D35, 35C05, 35C07, 35C15, 35G20, 35G25, 35Q75

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