Partial Differential Equations with MATLAB

468 An Introduction to Partial Differential Equations with MATLAB©R

Finally, we write

yp=−

sinkx

k

[

coskL

sinkL

∫L

0

f(ξ)sinkξ dξ−

∫L

x

f(ξ)coskξ dξ

]

+

coskx

k

∫x

0

f(ξ)sinkξ dξ.

Since this representation is different forξ<xandξ>x,webreakupthe
first integral atxand rewrite

yp=

∫x

0

f(x)

[

coskxsinkξ

k

−

sinkxcoskLsinkξ

ksinkL

]

dξ

+

∫L

x

f(ξ)

[

sinkxcoskξ

k

−

sinkxcoskLsinkξ

ksinkL

]

dξ

=

∫L

0

f(ξ)G(x;ξ)dξ,

where

G(x;ξ)=

⎧

⎪⎪

⎨

⎪⎪

⎩

coskxsinkξ

k

−

sinkxcoskLsinkξ

ksinkL

, if 0≤ξ≤x

sinkxcoskξ

k

−

sinkxcoskLsinkξ

ksinkL

, ifx≤ξ≤L.‡

Notice thatGsatisfies the boundary conditions; however, this is obvious if we
rewriteGas

G(x;ξ)=

⎧

⎪⎪

⎨

⎪⎪

⎩

1

ksinkL

sinkξsink(L−x), if 0≤ξ≤x

1

ksinkL

sinkxsink(L−ξ), ifx≤ξ≤L§

(see Exercise 8b).
So let’s list the important properties of this particular Green’s function
(which, as it turns out, all Green’s functions will possess, although the dis-
continuity in the derivative may behave differently).

1.Symmetry: G(x;ξ)=G(ξ;x) for allx, ξin [0,L].

2.Continuity: Gis continuous on [0,L] and, specifically, at the point

ξ=x.

3.Derivative discontinuous: The derivative

Gx(x;ξ)=

⎧

⎪⎪

⎨

⎪⎪

⎩

−

1

sinkL

sinxξcosk(L−x), if 0<ξ<x,

1

sinkL

coskxsink(L−ξ), ifx<ξ<L,