Partial Differential Equations with MATLAB

Nonhomogeneous Problems and Green’s Functions 469

has a jump discontinuity atx=ξ. In particular, for this Green’s func-

tion it is

Gx(ξ+;ξ)−Gx(ξ−,ξ)

=−

1

sinkL

sinkξcosk(L−ξ)−

1

sinkL

coskξsink(L−ξ)

=−1(why?).

4.Satisfies homogeneous equation:Ghas a continuous second deriva-

tive on 0<x<ξandξ<x<Land satisfies the associated homoge-

neous equation there. (But not atx=ξ,ofcourse.)

5.Satisfies boundary conditions: Gsatisfies both boundary condi-

tions, for each value ofξ.

Let’snoteherethatwemayrewriteProperty3as

lim

→ 0

∫ξ+

ξ−

(y′′+k^2 y)dx=− 1

(see Exercise 6) in anticipation of the two- and three-dimensional ver-

sions of this property.

Guided by the above example, let’s construct the Green’s function for the
general nonhomogeneousregular Sturm–Liouville problem

(ry′)′+(q+λw)y=−f(x), a<x<b,

a 1 y(a)+a 2 y′(a)=b 1 y(b)+b 2 y′(b)=0.

Here, as before,w,q,randr′are continuous ona≤x≤bandw(x)>0,
r(x)>0ona≤x≤b.Also,fis continuous ona≤x≤b.(Theresultcan
be extended to many of the important singular problems, as well.)
To begin, and guided by Example 1, we assume thatλis not an eigenvalue
of the associated homogeneous problem, so that the latter has a unique solu-
tion. (We deal withλ= an eigenvalue later.) Further, as we’d like Green’s
function to satisfy the boundary conditions, and we notice from Example 1
that sink(L−x) satisfies the boundary condition atx=L,andsinkxthat
atx= 0, we try to do the same here. So, given any two linearly independent
solutionsz 1 ,z 2 of the homogeneous problem, it turns out that we may always
find constantsc 1 ,c 2 ,c 3 ,c 4 such that

y 1 =c 1 z 1 +c 2 z 2 ,y 2 =c 3 z 1 +c 4 z 2

are linearly independent and satisfy the boundary condition atx=aand
x=b, respectively. (Why? By the way, cany 1 , for example, satisfyboth
boundary conditions? Why or why not?)