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 uj) =

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

(i) Ñ× r2 sin(2q) cos(j) uq =

or equivalent.

(ii) Ñ 2cos(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) x2 y¢¢ + 2xy¢ + (x2 - 12)y = 0

p(x) = 2x/x2, q(x) = (x2 - 12)/x2 which are singular at x = 0.

xp(x) = 2, x2q(x) = x2 - 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 s1 = 3 and s2 = -4.

Since s1 - s2 = 7 this is Frobenius case III.

(ii) y¢¢ + 2xy¢ + (x2 - 12)y = 0

p(x) = 2x, q(x) = x2 - 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

fmn(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 fmn satisfies the given boundary conditions. Since

fmn 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 = u1 + u2 where u1 is a solution of the Poisson equation given in part (b) with zero boundary conditions while u2 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 I0 is the modified Bessel function.

(a) Write down the expression for the coefficients Bn.

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 Bn = 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 = an a zero of J0(x) or la = bn a zero of J0¢(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 Arn + 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.

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