Green's functions, linear second‐order differential equations, and one‐dimensional diffusion advection models

2021 ◽  
Author(s):  
Xiao Yu ◽  
Kunquan Lan ◽  
Jianhong Wu
Keyword(s):  
Differential Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Second Order ◽  
One Dimensional ◽  
Second Order Differential Equations ◽  
Dimensional Diffusion

ON A CERTAIN NONLOCAL POTENTIAL

1966 ◽  
Vol 44 (3) ◽  
pp. 629-636 ◽  
Author(s):  
V. de la Cruz ◽  
B. A. Orman ◽  
M. Razavy
Keyword(s):  
Differential Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Second Order ◽  
Numerical Results ◽  
Nonlocal Potential ◽  
Second Order Differential Equations ◽  
Nonlocal Potentials

A solvable example of a class of nonlocal potentials, whose kernels are related to Green's functions of second-order differential equations, is examined. This solvable example is applied to a few standard problems and, in particular, acceptable numerical results are obtained for p–p scattering in the 1S state.

Geometric characterization of separable second-order differential equations

1993 ◽  
Vol 113 (1) ◽  
pp. 205-224 ◽  
Author(s):  
Eduardo Martínez ◽  
José F. Cariñena ◽  
Willy Sarlet
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Geometric Characterization ◽  
One Dimensional ◽  
Practical Procedure ◽  
Second Order Differential Equations ◽  
Necessary And Sufficient ◽  
Second Order Equations

AbstractWe establish necessary and sufficient conditions for the separability of a system of second-order differential equations into independent one-dimensional second-order equations. The characterization of this property is given in terms of geometrical objects which are directly related to the system and relatively easy to compute. The proof of the main theorem is constructive and thus yields a practical procedure for constructing coordinates in which the system decouples.

scholarly journals Nonlinear Green’s Functions for Wave Equation with Quadratic and Hyperbolic Potentials

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Asatur Zh. Khurshudyan
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Wave Equations ◽  
Separation Of Variables ◽  
Linear Equations ◽  
Second Order ◽  
Nonlinear Wave Equations ◽  
One Dimensional ◽  
Exponential Nonlinearity ◽  
Second Order Differential Equations

The advantageous Green’s function method that originally has been developed for nonhomogeneous linear equations has been recently extended to nonlinear equations by Frasca. This article is devoted to rigorous and numerical analysis of some second-order differential equations new nonlinearities by means of Frasca’s method. More specifically, we consider one-dimensional wave equation with quadratic and hyperbolic nonlinearities. The case of exponential nonlinearity has been reported earlier. Using the method of generalized separation of variables, it is shown that a hierarchy of nonlinear wave equations can be reduced to second-order nonlinear ordinary differential equations, to which Frasca’s method is applicable. Numerical error analysis in both cases of nonlinearity is carried out for various source functions supporting the advantage of the method.

scholarly journals Multiple Bounded Positive Solutions to Integral Type BVPs for Singular Second Order ODEs on the Whole Line

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Integral Type ◽  
One Dimensional ◽  
Second Order Differential Equations ◽  
Type Boundary ◽  
The One

This paper is concerned with the integral type boundary value problems of the second order differential equations with one-dimensionalp-Laplacian on the whole line. By constructing a suitable Banach space and a operator equation, sufficient conditions to guarantee the existence of at least three positive solutions of the BVPs are established. An example is presented to illustrate the main results. The emphasis is put on the one-dimensionalp-Laplacian term[ρ(t)Φ(x’(t))]’involved with the functionρ, which makes the solutions un-concave.

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

Fundamental solutions and analysis of an interfacial crack in a one-dimensional hexagonal quasicrystal bi-material

2020 ◽  
Vol 25 (5) ◽  
pp. 1124-1139
Author(s):  
CuiYing Fan ◽  
Shuai Chen ◽  
QiaoYun Zhang ◽  
Ming Hao Zhao ◽  
Bing Bing Wang
Keyword(s):  
Differential Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Crack Opening ◽  
Interfacial Crack ◽  
Crack Tip ◽  
Delta Function ◽  
Numerical Approach ◽  
Fundamental Solutions ◽  
One Dimensional

Using the Stroh formalism, Green’s functions are obtained for phonon and phason dislocations and opening displacements on the interface of a one-dimensional hexagonal quasicrystal bi-material. The integro-differential equations governing the interfacial crack are then established, and the singularities of the phonon and phason displacements at the crack tip on the interface are analyzed. To eliminate the oscillating singularities, we represent the delta function in terms of the Gaussian distribution function in the Green’s functions and the integro-differential equations, which helps reduce these equations to the standard integral equations. Finally, a boundary element numerical approach is also proposed to solve the integral equation for the crack opening displacements, the asymptotic expressions of the extended intensity factors, and the energy release rate in terms of the crack opening displacements near the crack tip. In numerical examples, the effect of the Gaussian parameter on the numerical results is discussed, COMSOL software is used to validate the analytical solution, and the influence of the different phonon and phason loadings on the interfacial crack behaviors is further investigated.

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.

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