Nonlinear implicit Green’s functions for numerical approximation of partial differential equations: Generalized Burgers’ equation and nonlinear wave equation with damping

Asatur Zh. Khurshudyan
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Asatur Zh. Khurshudyan: Department on Dynamics of Deformable Systems and Coupled Fields Institute of Mechanics, National Academy of Sciences of Armenia, Armenia2Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, P. R. China

International Journal of Modern Physics C (IJMPC), 2018, vol. 29, issue 07, 1-14

Abstract: A representation formula for second-order nonhomogeneous nonlinear ordinary differential equations (ODEs) has been recently constructed by M. Frasca using its Green’s function, i.e. the solution of the corresponding nonlinear differential equation with a Dirac delta function instead of its nonhomogeneity. It has been shown that the first-order term–the convolution of the nonlinear Green’s function and the right-hand side, analogous to the Green’s representation formula for linear equations — provides a numerically efficient solution of the original equation, while the higher order terms add complementary corrections. The cases of square and sine nonlinearities have been studied by Frasca. Some new cases of explicit determination of nonlinear Green’s function have been studied previously by us. Here, we gather nonlinear equations and their explicitly determined Green’s functions from existing references, as well as investigate new nonlinearities leading to implicit determination of nonlinear Green’s function. Some transformations allowing to reduce second-order nonlinear partial differential equations (PDEs) to nonlinear ODEs are considered, meaning that Frasca’s method can be applied to second-order PDEs as well. We perform a numerical error analysis for a generalized Burgers’ equation and a nonlinear wave equation with a damping term in comparison with the method of lines.

Keywords: Frasca’s method; short tie expansion; nonlinear Green’s function; generalized separation of variables; traveling wave; nonlinear wave; generalized Burgers’ equation; nonlinear wave equation with damping; method of lines (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1142/S0129183118500547

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