# 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]