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Green's Functions with Reflection Conditions

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This Demonstration plots the Green’s function for the linear differential equation with reflection of order 1,

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,

and order ,

,

where you can vary the parameters and . These equations are coupled with one of the following linear boundary conditions:

Order 1:

initial condition: ,

final condition: ,

periodic condition: ,

antiperiodic conditions: .

Order 2:

Dirichlet conditions: ,

mixed conditions: ,

Neumann conditions: ,

periodic conditions: ,

antiperiodic conditions: ,

The solution of the boundary value problem (differential equation and boundary conditions) is given by .

In the code, the expression for the corresponding Green’s function is given for the arbitrary interval .

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Contributed by: Alberto Cabada, José Ángel Cid, F. Adrián F. Tojo, and Beatriz Máquez-Villamarín (December 2014)
Open content licensed under CC BY-NC-SA

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Details

The way to compute a Green's function for a problem with reflection is described in [1].

You can download a notebook for computing other Green's functions with reflection from [2].

References

[1] A. Cabada and F. Adrián F. Tojo, "An Algebraic Method of Obtaining the Green's Function for Some Reducible Functional Differential Equations." arxiv.org/abs/1411.5507.

[2] F. Adrián F. Tojo, A. Cabada, J. A. Cid, and B. Máquez-Villamarín, "Green's Functions with Reflection" from Wolfram Library Archive—A Wolfram Web Resource. library.wolfram.com/infocenter/MathSource/9087.

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