Partial Differential Equations with MATLAB

Nonhomogeneous Problems and Green’s Functions 471

So we have

G(x;ξ)=

⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩

−

y 1 (ξ)y 2 (x)

r(ξ)W(ξ)

, ifa≤ξ≤x(orξ≤x≤b),

−

y 1 (x)y 2 (ξ)

r(ξ)W(ξ)

, ifx≤ξ≤b(ora≤x≤ξ).

It’s easy to see thatGsatisfies Properties (1), (2), (4) and (5), above. What
about the jump in the derivative? We have

Gx(ξ+,ξ)−Gx(ξ−,ξ)=

y 1 ′(ξ)y 2 (ξ)−y 1 (ξ)y′ 2 (ξ)

r(ξ)W(ξ)

=−

1

r(ξ)

.

We put everything together in a theorem.

Theorem 10.1Given the regular 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,

suppose thatλis not an eigenvalue and suppose thaty 1 andy 2 are solutions
of the associated homogeneous equation, satisfying

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

Then the solution of the problem is given by

y=

∫b

a

G(x, ξ)f(ξ)dξ,

whereGreen’s functionGis given by

G(x;ξ)=

⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩

−

y 1 (ξ)y 2 (x)

r(ξ)W(ξ)

, ifa≤ξ≤x(orξ≤x≤b),

−

y 1 (x)y 2 (ξ)

r(ξ)W(ξ)

, ifx≤ξ≤b(ora≤x≤ξ).

Further,Gsatisfies the following properties:

1. G(x;ξ)=G(ξ;x)for allx, ξin[a, b]. (It turns out that this symmetry is
   a result of the problem’s being self-adjoint. It is often called the property
   ofreciprocityorMaxwell’s reciprocity.∗∗)

∗∗After James Clark Maxwell.