Solving PDE’s by Eigenfunction Expansion
Some of these problems are difficult and you should ask questions (either after class or in
my office) to help you get started and after starting, to make sure you are proceeding
correctly.
1. Consider the initial value problem for the heat equation
0 < x < 1, t > 0,
/ t uÝx, tÞ = D / xx uÝx, tÞ,
uÝx, 0Þ = fÝxÞ 5 L 2 Ý0, 1Þ
with BC’s
uÝ0, tÞ = uÝ1, tÞ = 0
ÝiÞ
/ x uÝ0, tÞ = uÝ1, tÞ = 0
ÝiiÞ
/ x uÝ0, tÞ = / x uÝ1, tÞ = 0
ÝiiiÞ
a) Find the eigenvalues and eigenfunctions in each of the 3 cases
b) Try to determine which problem has the solution reaching the steady state soonest.
hint: The more negative the eigenvalues, the faster the components of the solution
decay to zero. So, look at the smallest eigenvalue in each case and note that this term
dictates the decay to zero.
Are the steady states the same in all 3 cases?
2. Consider the initial value problem for the heat equation with convection,
0 < x < 1, t > 0,
/ t uÝx, tÞ = D / xx uÝx, tÞ + V / x uÝx, tÞ,
uÝx, 0Þ = fÝxÞ 5 L 2 Ý0, 1Þ
uÝ0, tÞ = uÝ1, tÞ = 0.
a) Let
uÝx, tÞ = e ax+bt wÝx, tÞ
and show
/ t u = e ax+bt ß/ t wÝx, tÞ + b wà
/ x u = e ax+bt ß/ x wÝx, tÞ + a wà
/ xx u = e ax+bt ß/ xx wÝx, tÞ + 2a / x w + a 2 wà
b) show
/ t wÝx, tÞ + b w = Dß/ xx wÝx, tÞ + 2a / x w + a 2 wà
+Vß/ x wÝx, tÞ + a wà
c) choose the parameters a and b such that wÝx, tÞ solves
/ t wÝx, tÞ = D / xx wÝx, tÞ
wÝx, 0Þ = e ?ax fÝxÞ 5 L 2 Ý0, 1Þ
wÝ0, tÞ = wÝ1, tÞ = 0.
d) Solve for w and then find uÝx, tÞ. How does the solution when V ® 0 differ from the
solution when V = 0?
3. Consider the wave equation with dispersion,
0 < x < 1, t > 0,
/ tt uÝx, tÞ = a 2 / xx uÝx, tÞ ? d 2 uÝx, tÞ,
uÝx, 0Þ = fÝxÞ 5 L 2 Ý0, 1Þ / t uÝx, 0Þ = 0,
uÝ0, tÞ = uÝ1, tÞ = 0.
a) expand the solution in series of the form
K
uÝx, tÞ = > u n ÝtÞd n ÝxÞ
n=1
1
and find the ODE problem solved by u n ÝtÞ.
b) Find u n ÝtÞ and uÝx, tÞ.
c) Compare uÝx, tÞ to the solution obtained in the case d = 0, which is
K
uÝx, tÞ = > f n cosÝn^atÞ sinÝn^xÞ
n=1
=
1
2
=
1
2
K
> f n ßsinÝn^Ýx + atÞ + sinÝn^Ýx ? atÞà
n=1
*fÝx + atÞ + *fÝx ? atÞ
where *f denotes a periodic extension of the function f, defined on ß0, 1à.
4. Suppose u(x,t) solves:
/ t uÝx, tÞ = / xx uÝx, tÞ,
uÝx, 0Þ = 0,
uÝ0, tÞ = T 0 ,
0 < x < 1,
t>0
0 < x < 1,
uÝ1, tÞ = T 1 , t > 0.
(a) Use the change of dependent variable
vÝx, tÞ = uÝx, tÞ ? T 0 Ý1 ? xÞ ? T 1 x
to convert this to a problem in vÝx, tÞ with homogeneous boundary conditions
(b) Solve the problem for vÝx, tÞ and use the solution to give an expression for u(x,t).
(c) Explain the statement that vÝx, tÞ represents the transient part of the response, uÝx, tÞ,
and identify the steady state part of uÝx, tÞ.
5. Consider the Sturm-Liouville problem:
? X”ÝxÞ = VXÝxÞ, 0 < x < 1,
X v Ý0Þ + aXÝ0Þ = 0
X v Ý1Þ + bXÝ1Þ = 0
In each of the following cases, determine all the eigenvalues and eigenfunctions:
ÝaÞ a = 0, b = 1
ÝbÞ a = 6, b = 3
Hint: ÝiÞ if V = ?W 2 < 0 then XÝxÞ = A cosh Wx + B sinh Wx, X v ÝxÞ = WA sinh Wx + WB cosh Wx,
and the boundary conditions imply
WÝa?bÞ
A = ?ÝW/aÞB,
Tanh W = W 2 ?ab
With a = 6, b = 3, plot Tanh W, and
WÝa?bÞ
W 2 ?ab
vs W on the same axes:
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One nonzero root near W = 6 provides a negative e-value of V ?1 u ?36
ÝiiÞ if V = 0 then XÝxÞ = Ax + B and the boundary conditions imply A = B = 0 provided
a ? b + ab ® 0
ÝiiiÞ if V = W 2 > 0 then XÝxÞ = A cos Wx + B sin Wx, X v ÝxÞ = ?WA sin Wx + WB cos Wx,
and the boundary conditions imply
WÝa?bÞ
A = ?ÝW/aÞB,
Tan W = W 2 +ab
Plot Tan W, and
WÝa?bÞ
W 2 +ab
vs W on the same axes:
infinitely many nonzero roots W n
There are an infinite number of nonzero roots W n and as n increases, W n approaches n^.
Use the same approach for the cases:
ÝcÞ a = 3, b = 6
ÝdÞ a = 4, b = ?16
6. A Radon source is placed at one end of a thin, air-filled tube of length L. The opposite
end of the tube is sealed against the passage of Radon and a Radon detector is placed
at the sealed end. Initially the tube is free of Radon, and as Radon diffuses down the tube,
some of the Radon escapes through the sides of the tube due to leakage. This situation
can be modelled by
/ t CÝx, tÞ = D / xx CÝx, tÞ ? K CÝx, tÞ,
CÝx, 0Þ = 0,
0 < x < L, t > 0,
0 < x < L,
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CÝ0, tÞ = C 0 > 0,
/ x CÝL, tÞ = 0,
Here
and
t > 0,
t > 0.
CÝx, tÞ denotes the concentration of Radon at (x,t),
D
denotes the diffusivity of Radon in air,
K
denotes a positive constant related to the Radon leakage.
(a) Find CÝL, tÞ for t > 0.
(b) How does the limit of CÝL, tÞ as t tends to infinity in the case K > 0
differ from the limit in the case K = 0?
7. Consider the problem:
/ xx uÝx, yÞ + / yy uÝx, yÞ = 0,
uÝ0, yÞ = fÝyÞ,
0 < x, y < 1
0<y<1
/ x uÝ1, yÞ = 0,
0<y<1
uÝx, 0Þ = 0,
0<x<1
uÝx, 1Þ = gÝxÞ,
0<x<1
(a) Split this into two subproblems and identify the eigenfunctions for each problem. Why is
it necessary to split the problem in two parts?
(b) Solve each of the subproblems and explain why uÝx, yÞ equals the sum of the two
subproblem solutions.
(c) If fÝ1Þ ® gÝ0Þ then uÝ0, 1Þ must assume two different values at the same point. Does the
solution uÝx, yÞ really do this, and if so, how?
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