Page 23 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

THE FOURIER METHOD: SEPARATION OF VARIABLES 13
We first note that the problem has a fundamental length scale, so that if we define another
space variable ξ = x/L, the partial differential equation can be written as

−2
ρcu t = L ku ξξ 0 <ξ < 1 t < 0 (2.3)
2
Next we note that if we define a dimensionless time-like variable as τ = αt/L , where α = k/ρc
is called the thermal diffusivity,wefind

u τ = u ξξ (2.4)
We now proceed to nondimensionalize and normalize the dependent variable and the boundary
conditions. We define a new variable

U = (u − u 1 )/(u 0 − u 1 ) (2.5)

Note that this variable is always between 0 and 1 and is dimensionless. Our boundary value
problem is now devoid of constants.

U τ = U ξξ (2.6)
U(τ, 0) = 0
U ξ (τ, 1) = 0 (2.7)
U(0,ξ) = 1

All but one of the boundary conditions are homogeneous. This will prove necessary in our analysis.

2.1.2 Separation of Variables
Begin by assuming U = (τ) (ξ). Insert this into the differential equation and obtain

(ξ) τ (τ) = (τ) ξξ (ξ). (2.8)

Next divide both sides by U = ,

τ ξξ 2
= =±λ (2.9)

The left-hand side of the above equation is a function of τ only while the right-hand side is a
function only of ξ. This can only be true if both are constants since they are equal to each other.
2
λ is always positive, but we must decide whether to use the plus sign or the minus sign. We
have two ordinary differential equations instead of one partial differential equation. Solution
2
2
for gives a constant times either exp(−λ τ)orexp(+λ τ). Sinceweknowthat U is always
between 0 and 1, we see immediately that we must choose the minus sign. The second ordinary