Green’s functions for higher order nonlinear equations

Marco Frasca () and Asatur Zh. Khurshudyan
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Marco Frasca: Via Erasmo Gattamelata 3, 00176 Rome, Italy
Asatur Zh. Khurshudyan: Department on Dynamics of Deformable Systems and Coupled Fields, Institute of Mechanics, National Academy of Sciences of Armenia, Yerevan, Armenia3Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, P. R. China

International Journal of Modern Physics C (IJMPC), 2018, vol. 29, issue 10, 1-13

Abstract: The well-known Green’s function method has been recently generalized to nonlinear second-order differential equations. In this paper, we study possibilities of exact Green’s function solutions of nonlinear differential equations of higher order. We show that, if the nonlinear term satisfies a generalized homogeneity property, then the nonlinear Green’s function can be represented in terms of the homogeneous solution. Specific examples and a numerical analysis support the advantage of the method. We show how, for the Bousinesq and Kortweg–de Vries equations, we are forced to introduce higher order Green’s functions to obtain the solution to the inhomogeneous equation. The method proves to work also in this case supporting our generalization that yields a closed form solution to a large class of nonlinear differential equations, providing also a formula easily amenable to numerical evaluation.

Keywords: Higher order nonlinear equations; nonlinear Green’s function; Korteweg–de Vries equation; Boussinesq equation; method of lines (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1142/S0129183118501048

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