Page 153 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

STURM–LIOUVILLE TRANSFORMS 143
and this is simply

∞
sin(λ n x)

u(x, t) = # U(λ n , t)
# 2
sin(λ n ) #
n=1
#
with λ n given by the transcendental equation above. The final solution is therefore
∞
2(1 − cos λ n )
2
−λ t
u(x, t) = 1 sin(λ n x)e n
λ n − sin(2λ n )
n=1 2
9.2 GENERALIZATION: THE STURM–LIOUVILLE TRANSFORM:
THEORY
Consider the differential operator D

D[ f (x)] = A(x) f + B(x) f + C(x) f a ≤ x ≤ b (9.1)
with boundary conditions of the form

N α [ f (x)] x=a = f (a)cos α + f (a)sin α
(9.2)

N β [ f (x)] x=b = f (b)cos β + f (b)sin β
where the symbols N α and N β are differential operators that define the boundary conditions.
For example the differential operator might be

D[ f (x)] = f xx

and the boundary conditions might be defined by the operators

N α [ f (x)] x=a = f (a) = 0

and

N β [ f (x)] x=b = f (b) + Hf (b) = 0

We define an integral transformation

b

T[ f (x)] = f (x)K(x,λ)dx = F(λ) (9.3)
a