Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

1.

a) Solve the following problem:

PDE u_{t} = 4 u_{xx}, for 0 < t, 0 < x < 1

BC u_{x}(t,0) = 0, u_{x}(t,1) = 0, for 0 < t

IC u(0,x) = 2 x - x^{2}, for 0 < x < 1

ANSWER:

u(t,x) = 2/3 - 4 Sum exp(-4 n^{2}^{2} t) cos(n
x)/(n^{2} ^{2}).

The way to get this solution is to recall that the separated solutions with these BC are either constants or of the form constant exp(-4 n^{2} ^{2} t) cos(n x)/(n^{2}. The general solution is a sum of these solutions, with coefficients we could call a_{n}. When we put t = 0, we get a Fourier cosine series, and the coefficients are the Fourier cosine coefficients for 2 x - x^{2}. Then use the standard formula for these coefficients.

b) Find the maximum value of u(t,x) for 0 <= t <= 1, 0 <= x <= 1:

We know by the maximum principle that the maximum value occurs either at t=0, t = 1, x = 0, or x = 1, so only these values need to be considered. Physical reasoning can be used, if you wish, to eliminate all possibilities for t > 0 (the rod is insulated and has no sources, so heat cannot appear at the ends while the temperature relaxes to equilibrium). Considering t=0, we find:

ANSWER:

The maximum temperature occurs at x = 1, t = 0

The maximum temperature is u_{max} = 1

2. Some background information:

There is a complete, orthonormal set of functions denoted
_{n}(x), for
-infinity < x < infinity, which are eigenfunctions for the
ordinary differential equation

- _{n}'' + x^{2}
_{n} = (2n+1) _{n}, n = **0**, 1, 2,
....

You may use the notation _{n} in the answer to this problem.

Consider the following PDE:

PDE u_{tt} = grad^{2} u - x^{2} u, for 0 < t, 0 < x <
1, 0 < y <

BC u(t,x,0) = u(t,x, ) = 0, for 0 < t

Find the normal mode with the lowest frequency of vibration (include the time dependence):

ANSWER:

This is a straightforward separation of variables, albeit with a new function. The function Y satisfies the usual eigenvalue equation with 0 Dirichlet BC, so the spatial part of the separated solutions are of the form
_{n} sin(m y), n = 0, 1, ...; m = 1, 2, .... If the full solution is of the form T_{nm}(t)
_{n}(x) sin(m y), then substituting into the PDE shows that

T_{nm}''(t) = _{nm} T_{nm}(t),

where _{nm} = (2 n + 1) + m^{2}. The solutions are sines and cosines of _{nm}^{1/2}. The lowest frequency occurs when m = 0, n = 1:

u(t,x,y) = (A_{01} cos(2^{1/2} t) + B_{01} sin(2^{1/2} t) ) _{0}(x) sin(y),

or, if you prefer,

u(t,x,y) = (A_{01} cos(2^{1/2} t) + B_{01} sin(2^{1/2} t) ) exp(-x^{2}/2) sin(y).

Find the general solution:

ANSWER:

u(t,x,y) = Sum of (A_{nm} cos((2n+1+m^{2})^{1/2} t) + B_{nm} sin((2n+1+m^{2})^{1/2} t) )
_{n}(x) sin(m y), for n = 0, 1, ...; m = 1, 2, ....

3. A slice of pizza is shaped like a sector in cylindrical coordinates,

0 < r < 20 cm,

0 < < /3 radians

0 < z < 1 cm.

It has come to thermal equilibrium while sitting on a student's computer monitor, so that the temperature on its surface is

u(r, , 0) = 30

u(r, , 1) = 20

u(r,0,z) = u(r, /3,z) = 30 - 10 z

u(20, ,z) = 30 - 10 z - 5

This is a low quality pizza consisting of a homogeneous material (independent of position)

a) The partial differential equation for a homogeneous material at thermal equilibrium is Laplace's equation,

grad^{2} u = 0

b) Answer the following questions.

Are there useful simplifications involving the boundary conditions? If so, what are they?

Be specific and put the answer here: Most of the boundary conditions can be made *homogeneous* by redefining the unknown as

v(t,**x**) = u - (30 - 10 z).

The only BC which remains nonhomogeneous is the one at r = 20.

Are there useful separated solutions? If so, write the specific ordinary differential equations that the separated solutions satisfy below. Include boundary conditions.

write v(r,, z) = R(r) Q() Z(z). Then:

- Q'' = Q,

(0) = (/3) = 0

- Z'' = Z,

Z(0) = Z(1) = 0

R'' + (1/r) R' = (/r^{2} + ) R,

R(0) bounded

(We don't do anything about the nonhomogeneous boundary before putting the separated solutions together.)

c) Solve the differential equations with the given boundary conditions.

The equation for R is our old friend Bessel's equation, but the eigenvalues
are now determined by the conditions on Q, which lead to eigenfunctions of the
form sin(3 m ) and
eigenvalues _{m} = 9 m^{2}.

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