Page 34 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

24 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
the time derivative term. We now have

U τ = U ξξ + 1 (2.78)
U(0,ξ) = 0
U(τ, 0) = 0 (2.79)

U(τ, 1) = 0

2.1.20 Relocating the Nonhomogeneity
We have only one nonhomogeneous condition, but it’s in the wrong place. The differential
equation won’t separate. For example if we let U(ξ, τ) = P(ξ)G(τ) and insert this into the
partial differential equation and divide by PG,wefind

G (τ) P (ξ) 1

= + (2.80)
G P PG
The technique to deal with this is to relocate the nonhomogenous condition to the initial
condition. Assume a solution in the form U = W(ξ) + V (τ, ξ). We now have

V τ = V ξξ + W ξξ + 1 (2.81)

If we set W ξξ =−1, the differential equation for V becomes homogeneous. We then set
both W and V equal to zero at ξ = 0and 1and V (0,ξ) =−W(ξ)

W ξξ =−1 (2.82)
W(0) = W(1) = 0 (2.83)

and

V τ = V ξξ (2.84)
V (0,ξ) =−W(ξ)

V (τ, 0) = 0 (2.85)
V (τ, 1) = 0

The solution for W is parabolic

1
W = ξ(1 − ξ) (2.86)
2