**MATH3271**

**FW03**

**Solution
to term test 2 **

The**
**total number of marks on this paper is 70.

1. Orthogonal curvilinear coordinates (20 marks)

(a) Evaluate the following expressions using the vector operators in cylindrical polar coordinates.

(i) **Ñ(**r
cos(2j))
=

(ii) **Ñ
**´ (z **u**_{j})
=

(b) Evaluate the following expressions using the vector operators in spherical polar coordinates.

(i) Ñ×
r^{2} sin(2q)
cos(j) **u**_{q}
=

or equivalent.

(ii) Ñ
^{2}cos(2q) =

or equivalent.

2. Method of Frobenius (9 marks)

For each of the following differential equations, determine if x = 0 is an ordinary point, a regular singular point or neither. If it is a regular singular point, solve the indicial equation and state which case of the Frobenius method applies.

(i) x^{2} y¢¢
+ 2xy¢ + (x^{2} - 12)y
= 0

p(x) = 2x/x^{2},
q(x) = (x^{2} - 12)/x^{2} which are singular at x =
0.

xp(x) = 2, x^{2}q(x)
= x^{2} - 12 which have power series expansions about x = 0.
Thus x = 0 is a regular singular point.

The indicial equation is s(s
- 1) + 2s - 12 = 0 with roots s_{1} = 3 and s_{2} =
-4.

Since s_{1} - s_{2}
= 7 this is Frobenius case III.

(ii) y¢¢
+ 2xy¢ + (x^{2} - 12)y
= 0

p(x) = 2x, q(x) = x^{2}
- 12 which have power series expansions about x = 0. Thus x = 0 is
an ordinary point.

3. Poisson's equation in rectangular coordinates (10 marks)

(a) Show that the functions

f_{mn}(x,
y) = sin(mpx/a)sin(npy/b)

are solutions of the Helmholz equation

with boundary conditions f(0, y) = f(a, y) = f(x, 0) = f(x, b) = 0.

Give an explicit expression for the eigenvalues l.

Since sine is zero when its
argument is zero or a multiple of p the
function f_{mn} satisfies the
given boundary conditions. Since

f_{mn}
satisfies Helmholz's equation with eignevalue

(b) Using the results from part (a) derive the solution to Poisson's equation

with boundary conditions u(0, y) = u(a, y) = u(x, 0) = u(x, b) = 0.

Make sure you give expressions for any coefficients in the solution.

Let Then

This is just a double half-range Fourier expansion for f(x, y) so that the coefficients are

(c) __Describe__ how you
would find a solution to Poisson's equation given in part (b) if the
boundary conditions were given by u(0, y) = g(y), u(a, y) = u(x, 0) =
u(x, b) = 0. __You are not required to find an explicit solution
this equation.__

Let u = u_{1} + u_{2}
where u_{1} is a solution of the Poisson equation given in
part (b) with zero boundary conditions while u_{2} is a
solution of Laplace's equation with boundary conditions

u(0, y) = g(y), u(a, y) = u(x, 0) = u(x, b) = 0.

4. Vibration of a circular membrane - general case (20 marks)

Use separation of variables to find a solution to the two-dimensional wave equation

subject to the boundary conditions that u is finite and continuous with the edge of the membrane held fixed.

The initial conditions are

See section 4.3 of the text. Note you were expected to justify dropping the second solution of Bessel's equation and derive explicit solutions for the coefficients. Since g(r,j) = 0 the coefficients of the sin(clt) terms were zero.

5. Steady-state temperature in a cylinder (5 marks)

The solution of Laplace's equation for a cylinder when there is no f dependence

subject to the boundary conditions u(r, 0) = u(r, h) = 0, u(a, z) = f(z) is given by

where I_{0} is the
modified Bessel function.

(a) Write down the expression
for the coefficients B_{n}.

This is just a half-range Fourier series for f(z) so the coefficients are

(b) Find the explicit solution to Laplace's equation when f(z) = sin(4pz/h).

Because of the orthogonality
of the sine function B_{n} = 0 unless n = 4 in which case

6. Short answer questions (6 marks)

(a) Bessel functions of order zero satisfy the orthogonality integral

(i) What is the weight function w(x)? w(x) = x

(ii) Give two different expressions for the parameter l which makes this orthogonality integral true.

la
= a_{n }a zero of J_{0}(x)
or la = b_{n
}a zero of J_{0}¢(x)

(b) The solution to Laplace's equation for the interior of a disk

is given by

What would be the solution exterior to the disk, i.e. for a < r < ¥ ?

Since the solution of Euler's
equation is Ar^{n} + Br^{-n}
in the exterior region we must choose A = 0 to keep the solution
finite at infinity. Thus the solution is

(c) Write down the boundary
conditions for the temperature distribution in a cylinder of radius a
and height h when the top and side is kept at zero degrees and the
bottom is __insulated__.

u(r, j, h) = 0; u(a, j, h) = 0; N.B. j could be omitted.