Green's Function

order

1

2

3

4

sign

positive

negative

m

0.1

boundary conditions

periodic

initial

final

This Demonstration shows the graphic of the Green's function

G(t,s)

related to the linear differential equation of order

n=1

,

n=2

,

n=3

, or

n=4

,

(n)

u

(s)+a

n

m

u(s)=σ(s)

,

where

a=±1

and with the following linear boundary conditions:

n=1

: periodic:

u(0)=u(1)

, initial:

u(0)=0

, or final:

u(1)=0

;

n=2

: periodic:

u(0)=u(1),u'(0)=u'(1)

, Dirichlet:

u(0)=u(1)=0

or Neumann:

u'(0)=u'(1)=0

;

n=3

: periodic:

u(0)=u(1),u'(0)=u'(1),u''(0)=u''(1)

;

n=4

: periodic:

u(0)=u(1),u'(0)=u'(1),u''(0)=u''(1),u'''(0)=u'''(1)

, simply supported beam:

u(0)=u(1)=u''(0)=u''(1)=0

, or clamped beam

u(0)=u(1)=u'(0)=u'(1)=0

.

The solution of the boundary value problem (differential equation and boundary conditions) is given by

u(t)=

1

∫

0

G(t,s)σ(s)ds

.

In the code, the expression for the corresponding Green's function is given for an arbitrary interval

[a,b]

.