Green's function solution of second order differential equations

1980 ◽  
Vol 11 (3) ◽  
pp. 447-451
Author(s):  
P. S. Thapliyal
Keyword(s):  
Differential Equations ◽  
Green’S Function ◽  
Green's Function ◽  
Second Order ◽  
Function Solution ◽  
Second Order Differential Equations

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

2018 ◽  
Vol 29 (07) ◽  
pp. 1850054 ◽  
Author(s):  
Asatur Zh. Khurshudyan
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Green’S Function ◽  
Green's Function ◽  
Nonlinear Wave Equation ◽  
Burgers Equation ◽  
Representation Formula ◽  
Second Order ◽  
Generalized Burgers Equation

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.

Note on the Uniqueness of the Green'S Functions Associated With Certain Differential Equations

1950 ◽  
Vol 2 ◽  
pp. 314-325 ◽  
Author(s):  
D. B. Sears
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Order Differential Equation ◽  
Green’S Functions ◽  
Second Order ◽  
Second Order Differential Equation

Conditions to be imposed on q(x) which ensure the uniqueness of the Green's function associated with the linear second-order differential equation

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

scholarly journals New oscillation criteria for second-order neutral differential equations with distributed deviating arguments

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Osama Moaaz ◽  
Choonkil Park ◽  
Elmetwally M. Elabbasy ◽  
Waed Muhsin
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
The Other ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Deviating Arguments ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Continuous Delay ◽  
New Criteria

AbstractIn this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.

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