Page 137 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

SOLUTIONS WITH LAPLACE TRANSFORMS 127
Consequently

τ

u(ς, τ) = f (τ = 0)v(ς, τ) + f (τ − τ )v(ς, τ )dτ
τ =0

∞
2 f (0) (−1) n 2 2
= ς f (τ) + e −n π τ sin(nπς)
π n
n=1
τ
∞
2 (−1) n −n π τ
2
2

+ sin(nπς) f (τ − τ )e dτ
π n
n=1
τ =0

This series converges rapidly for large values of τ. However for small values of τ,it
converges slowly. There is another form of solution that converges rapidly for small τ.
The Laplace transform of v(ζ, τ)can be writtenas
√ √ √ √ √
sinh ς s e ς s − e −ς s 1 e ς s − e −ς s
√ = √ √ = √ √
s sinh s s (e ς s − e − s ) se s 1 − e −2 s
1 ς s −ς s −2 s −4 s −6 s
√
√
√
√
√
= √ e − e 1 + e + e + e + ...
se s
1 √ √
∞
= e −(1+2n−ς) s − e −(1+2n+ς) s
s
n=0
√
=k s
The inverse Laplace transform of e is the complimentary error function, defined by
s
√
k/2 τ
√ 2 2
erfc(k/2 τ) = 1 − √ e −x dx
π
x=0
Thus we have
∞
1 + 2n − ς 1 + 2n + ς
v(ς, τ) = erfc √ − erfc √
2 τ 2 τ
n=0
and this series converges rapidly for small values of τ.
Example 8.4. Next we consider a conduction problem with a convective boundary condition:
u τ = u ςς
u(τ, 0) = 0
u ς (τ, 1) + Hu(τ, 1) = 0
u(0,ς) = ς