Green's function for second order differential equations with piecewise constant arguments

2006 ◽  
Vol 64 (8) ◽  
pp. 1812-1830 ◽  
Author(s):  
Pinghua Yang ◽  
Yuji Liu ◽  
Weigao Ge
Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1953
Author(s):  
Sebastián Buedo-Fernández ◽  
Daniel Cao Labora ◽  
Rosana Rodríguez-López ◽  
Stepan A. Tersian

We find sufficient conditions for the unique solution of certain second-order boundary value problems to have a constant sign. To this purpose, we use the expression in terms of a Green’s function of the unique solution for impulsive linear periodic boundary value problems associated with second-order differential equations with a functional dependence, which is a piecewise constant function. Our analysis lies in the study of the sign of the Green’s function.


2018 ◽  
Vol 29 (07) ◽  
pp. 1850054 ◽  
Author(s):  
Asatur Zh. Khurshudyan

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.


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