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