Page 100 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

90 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
or, with

c

g c (n α, x) = f (ς)cos[n α(ς − x)]dς (5.91)
ς=0
we have
∞

f (x) = g c (n α, x) α (5.92)
n=1
As c approaches infinity we can imagine that α approaches dα and n α approaches α,
whereupon the equation for f (x) becomes an integral expression

∞ ∞
2
f (x) = f (ς)cos[α(ς − x)]dς dα (5.93)
π
ς=0 α=0
which can alternatively be written as

∞

f (x) = [A(α)cos α x + B(α)sin α x]dα (5.94)
α=0
where
∞
2
A(α) = f (ς)cos ας dς (5.95)
π
ς=0
and

∞
2

B(α) = f (ς)sin ας dς (5.96)
π
ς=0
Example 5.8 (Transient conduction in a semi-infinite region). Consider the boundary value
problem

u t = u xx (x ≥ 0, t ≥ 0)
u(0, t) = 0 (5.97)

u(x, 0) = f (x)

This represents transient heat conduction with an initial temperature f (x) and the boundary
at x = 0 suddenly reduced to zero. Separation of variables as T(t)X(x) would normally yield a