Partial Differential Equations with MATLAB

472 An Introduction to Partial Differential Equations with MATLAB©R

2. G(x;ξ)is continuous ona≤x≤b.


3. Gx(x;ξ)has a jump discontinuity atx=ξgiven by

Gx(ξ+;ξ)−Gx(ξ−;ξ)=−

1

r(ξ)

.††

4. GxandGxxare continuous ona<x<ξandξ<x<band satisfy the
   associated homogeneous ODE on these intervals.


5. Gsatisfies the boundary conditions.
   The above construction guarantees the existence of Green’s function, while
   uniqueness is left to Exercise 10.

Theorem 10.2For the Sturm–Liouville problem of Theorem 10.1, Green’s
function exists and is unique.

So we may compute Green’s functions either directly, using variation of
parameters, or by constructing them via the properties given in Theorem
10.1. Let’s look at an example.

Example 2Find Green’s function for the BVP

y′′=−f,

y(0) =y(L)=0

in three different ways.

First, sinceλ= 0 is not an eigenvalue of the problem

y′′+λy=0, 0 <x<L,

y(0) =y(L)=0,

we are guaranteed thatGexists. We begin by finding the solutionsz 1 =1and
z 2 =xof the associated homogeneous equation. Now,y 1 =z 2 =xsatisfies
the boundary condition atx= 0, whiley 2 =z 2 −Lz 1 =x−Lsatisfies the
other; further,y 1 andy 2 are linearly independent. So Green’s function will
be

G(x;ξ)=

⎧

⎪⎪

⎨

⎪⎪

⎩

−

y 1 (ξ)y 2 (x)

rW

, if 0≤ξ≤x,

−

y 1 (x)y 2 (ξ)

rW

, ifx≤ξ≤L,

=

⎧

⎪⎪

⎨

⎪⎪

⎩

−

ξ(x−L)

L

, if 0≤ξ≤x,

−

x(ξ−L)

L

, ifx≤ξ≤L,

††Again,seeExercise6.