A general representation for the Green's function of second‐order nonlinear differential equations

• Published in: Computational and mathematical methods Vol. 1; no. 4
• Main Authors: Frasca, Marco, Khurshudyan, Asatur Zh
• Format: Journal Article
• Language: English
• Published: Hoboken John Wiley & Sons, Inc 01.07.2019
• Subjects: generalized homogeneity property; Green's functions; Hierarchies; Homogeneity; homogeneous solution; hyperdistributions; Liouville equations; Mathematical analysis; Nonlinear differential equations; Nonlinear equations; nonlinear Green's function; Numerical analysis; Representations; short‐time expansion
• ISSN: 2577-7408; 2577-7408
• DOI: 10.1002/cmm4.1038

In this paper, second order quasi‐linear differential equations are studied, admitting a specific representation for nonlinear Green's function. Specifically, it is shown that, when the nonlinear term possesses the generalized homogeneity property, the corresponding nonlinear Green's function is represented as the product of the Heaviside function and the general solution of the corresponding homogeneous equation subject to nonhomogeneous Cauchy conditions. Typical hierarchies of specific nonlinearities possessing the generalized homogeneity property are derived. The case of Liouville equation, which lacks to possess the generalized homogeneity property, is studied. Numerical analysis of sinh‐Gordon and Liouville equations is carried out for diverse source functions showing the efficiency of the proposed solution. Two open problems leading to a more thorough characterization of nonlinearities possessing the generalized homogeneity property are distinguished.