Page 90 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

80 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
The problem can be broken down into five problems. u = u 1 + u 2 + u 3 + u 4 + u 5 .

u 1t = u 1xx + u 1yy
u 1 (0, x, y) = g(x, y) (5.29)
u 1 = 0, all boundaries

u 2xx + u 2yy = 0
u 2 (0, y) = f 1 (y) (5.30)
u 2 = 0 on all other boundaries

u 3xx + u 3yy = 0
u 3 (a, y) = f 2 (y) (5.31)
u 3 = 0 on all other boundaries

u 4xx + u 4yy = 0
u 4 (x, 0) = f 3 (x) (5.32)
u 4 = 0 on all other boundaries

u 5xx + u 5yy = 0
u 5 (x, b) = f 4 (x) (5.33)
u 5 = 0 on all other boundaries

5.1.1 Time-Dependent Boundary Conditions
We will explore this topic when we discuss Laplace transforms.

Example 5.4 (A finite cylinder). Next we consider a cylinder of finite length 2L and radius
r 1 . As in the first problem in this chapter, there are two possible length scales and we choose
r 1 . The cylinder has temperature u 0 initially. The ends at L =±L are suddenly insulated while
the sides are exposed to a fluid at temperature u 1 . The differential equation with no variation
in the θ direction and the boundary conditions are

α
u t = (ru r ) r + u zz
r
u z (t,r, −L) = u z (t,r, +L) = 0
ku r (r 1 ) + h[u(r 1 ) − u 1 (r 1 )] = 0 (5.34)

u(0,r, z) = u 0
u is bounded