Partial Differential Equation Problem

Partial Differential Equation Problem
1. The steady concentration of a reactant ?(r, ?) outside a disk of radius 1
(r > 1) obeys the following equation:
0 = ?2? =
1
r
?
?r (r
??
?r ) + 1
r
2
?
2?
??2
The following boundary conditions apply:
ˆr · ??(1, ?) = f(?)(1 + a?(1, ?))
where f(?) is an even function whose Fourier expansion is
ˆf(?) = 1
2
F0 +
X8
i=1
Fn cos(n?)
(a) Write down an expression for the general solution (before applying boundary
conditions) for a steady solution as
?(r, ?) = X8
i=-8
gn(r)(An cos(n?) + Bn sin(n?))
Then apply the boundary conditions and deduce which boundary conditions
would vanish.
(b) For a = 0, show that a steady solution ?(r, ?) exists only if F0 = 0 and
find the steady solution in this case.
(c) For a 6= 0, show that the Fourier coefficients An for a steady solution
?(r, ?) satisfy the equation
X8
n=0
MmnAn = -Fm
where Mmn is the m × n terms of a m × n matrix that depends on the
coefficients Fm’s.
Given the solution, write down the nonlinear criterion that F0 must satisfy
if a steady solution is to exist. (The criterion requires a matrix inversion, which
you cannot do explicitly since the matrix is infinite; just write it symbolically
as a matrix inverse)
(d) For a = 1 and Fn = dn,1, numerically approximate F0 to five digits.
dn,1 = 1 for n = 1 and 0 otherwise.
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