Page 98 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

88 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
and the orthogonality of sin(nθ)and J n (λ mn η)

2π 2π 1
1
2
2
f (η, θ)ηJ n (λ mn η)sin(nθ)dθdη = K nm sin (nθ)dθ ηJ (λ mn η)dη
n
η=0
θ=0 θ=0 r=0 (5.86)
K nm 2
= J n+1 (λ mn )
4

2π
4 1
K nm = f (η, θ)ηJ n (λ nm η)sin(nθ)dθdη (5.87)
2
J n+1 (λ nm ) η=0
θ=0
Problems
1. The conduction equation in one dimension is to be solved subject to an insulated surface
at x = 0 and a convective boundary condition at x = L. Initially the temperature is
u(0, x) = f (x), a function of position. Thus

u t = αu xx
u x (t, 0) = 0
ku x (t, L) =−h[u(t, L) − u 1 ]

u(0, x) = f (x)

First nondimensionalize and normalize the equations. Then solve by separation of
2
variables. Find a specific solution when f (x) = 1 − x .
2. Consider the diffusion problem

u t = αu xx + q(x)
u x (t, 0) = 0

u x (t, L) =−h[u(t, L) − u 1 ]

u(0, x) = u 1

Define time and length scales and define a u scale such that the initial value of the
dependent variable is zero. Solve by separation of variables and find a specific solution
for q(x) = Q, a constant. Refer to Problem 2.1 in Chapter 2.