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Green's Functions with Reflection Conditions

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