Heat Equation

The temperature variation u = u(a:,t) in a rod of length 1r. initially at temperature given by f(x). then
positioned with one end in ice and the other end insulated. can be modeled by the heat equation

c’3;u=6§u, O<x<1r, t>O,
with boundary conditions (BCs)
u(O, t) = O, 6Iu(1r,t) = 0,
and initial condition (IC)
u(a:,0) = f(x).
(3) Show that the general solution satisfying the heat equation and the BCs is given by
cc 2
2 + 1 2 + 1
u(a:,t) = 2A,, s1n 27) exp [- tj
nro
(b) Show that. when fitting the IC, the unknown coefficients An can be determined from f(:r:) via
2 7″ . 2n+ 1
An – 1-r/0 f(a:) s1n dx.
You may use the result
“sin flat sin flat dz: 0’ nsém n,mEN.
0 2 2 1r/2, n = m
[10 marks] – Laplace’s Equation for an annulus
Find a solution 12 = v(r,9) to the following Dirichlet problem for Laplace’s equation in polar coordinates.
1 1
83v+;8rv+r-28§v=0, 1<r<2,-rrgélgrr,
v(1, 9) = sin 46 – cos 9, v(2, 0) = -sin3 0 + 3(cos2 0)(sin 0)
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