Page 151 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

141

CHAPTER 9

Sturm–Liouville Transforms

Sturm–Liouville transforms include a variety of examples of choices of the kernel function
K(s, t) that was presented in the general transform equation at the beginning of Chapter 6. We
first illustrate the idea with a simple example of the Fourier sine transform, which is a special
case of a Sturm–Liouville transform. We then move on to the general case and work out some
examples.

9.1 A PRELIMINARY EXAMPLE: FOURIER SINE TRANSFORM
Example 9.1. Consider the boundary value problem

u t = u xx x ≤ 0 ≤ 1

with boundary conditions

u(0, t) = 0

u x (1, t) + Hu(1, t) = 0

and initial condition

u(x, 0) = 1

Multiply both sides of the differential equation by sin(λx)dx and integrate over the
interval x ≤ 0 ≤ 1.

1 1
d u d
2
sin(λx) dx = u(x, t)sin(λx)dx
dx 2 dt
x=0 x=0
Integration of the left hand side by parts yields

1
2 1
d du d
[sin(λx)]u(x, t)dx + sin(λx) − u [sin(λx)]
dx 2 dx dx 0
x=0