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WOLFRAM|DEMONSTRATIONS PROJECT

Green's Functions with Reflection Conditions

This Demonstration plots the Green’s function
G(t,s)
for the linear differential equation with reflection of order 1,
′
u
(t)+au(-t)+bu(-t)=h(t)
,
and order
2
,
″
u
(t)+au(-t)=h(t)
,
where you can vary the parameters
a
and
b
. These equations are coupled with one of the following linear boundary conditions:
Order 1:
initial condition:
u(-1)=0
,
final condition:
u(1)=0
,
periodic condition:
u(1)=u(-1)
,
antiperiodic conditions:
u(1)+u(-1)=0
.
Order 2:
Dirichlet conditions:
u(-1)=u(1)=0
,
mixed conditions:
u(-1)=u′(1)=0
,
Neumann conditions:
′
u
(-1)=
′
u
(1)=0
,
periodic conditions:
′
u
(-1)-
′
u
(1)=u(-1)-u(1)=0
,
antiperiodic conditions:
′
u
(-1)+
′
u
(1)=u(-1)+u(1)=0
,
The solution of the boundary value problem (differential equation and boundary conditions) is given by
u(t)=
1
∫
-1
G(t,s)h(s)ds
.
In the code, the expression for the corresponding Green’s function is given for the arbitrary interval
[-1,1]
.
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