Generalized separation of variables and exact solutions of nonlinear equations

Barannyk, A. F.; Barannyk, T. A.; Yuryk, I. I.

doi:10.1007/s11253-011-0475-2

Ukr Math J

2011

DOI: 10.1007/s11253-011-0475-2

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Generalized separation of variables and exact solutions of nonlinear equations

A. F. Barannyk¹,

T. A. Barannyk

I. I. Yuryk

Abstract: We consider a generalized procedure of separation of variables in nonlinear hyperbolic-type equations and Korteweg-de-Vries-type equations. We construct a broad class of exact solutions of these equations that cannot be obtained by the Lie method and method of conditional symmetries.

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2018

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Cited by 2 publications

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New Green’s functions for some nonlinear oscillating systems and related PDEs

Khurshudyan

2018

Int. J. Mod. Phys. C

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During the past three decades, the advantageous concept of the Green’s function has been extended from linear systems to nonlinear ones. At that, there exists a rigorous and an approximate extension. The rigorous extension introduces the so-called backward and forward propagators, which play the same role for nonlinear systems as the Green’s function plays for linear systems. The approximate extension involves the Green’s formula for linear systems with a Green’s function satisfying the corresponding nonlinear equation. For the numerical evaluation of nonlinear ordinary differential equations, the second approach seems to be more convenient. In this article we study a hierarchy of nonlinear partial differential equations that can be approximated by the second approach. Green’s functions for particular nonlinearities are derived explicitly. Numerical error analysis in the case of exponential nonlinearity for different source functions supports the advantage of the approach.

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New Green’s functions for some nonlinear oscillating systems and related PDEs

Khurshudyan

2018

Int. J. Mod. Phys. C

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Nonlinear Green’s Functions for Wave Equation with Quadratic and Hyperbolic Potentials

Khurshudyan

2018

Advances in Mathematical Physics

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The advantageous Green's function method that originally has been developed for nonhomogeneous linear equations has been recently extended to nonlinear equations by Frasca. This article is devoted to rigorous and numerical analysis of some secondorder differential equations new nonlinearities by means of Frasca's method. More specifically, we consider one-dimensional wave equation with quadratic and hyperbolic nonlinearities. The case of exponential nonlinearity has been reported earlier. Using the method of generalized separation of variables, it is shown that a hierarchy of nonlinear wave equations can be reduced to secondorder nonlinear ordinary differential equations, to which Frasca's method is applicable. Numerical error analysis in both cases of nonlinearity is carried out for various source functions supporting the advantage of the method.

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Generalized separation of variables and exact solutions of nonlinear equations

Abstract: We consider a generalized procedure of separation of variables in nonlinear hyperbolic-type equations and Korteweg-de-Vries-type equations. We construct a broad class of exact solutions of these equations that cannot be obtained by the Lie method and method of conditional symmetries.

Cited by 2 publications

References 6 publications

New Green’s functions for some nonlinear oscillating systems and related PDEs

New Green’s functions for some nonlinear oscillating systems and related PDEs

Nonlinear Green’s Functions for Wave Equation with Quadratic and Hyperbolic Potentials

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