Green's Functions with Reflection Conditions

order of the equation

1

2

boundary conditions

Neumann conditions

a

1

′′

u

(t)+u(-t)σ(t),

′

u

(1)

′

u

(-1)0

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]

.