**MATH3271**

**FW03**

**Solutions
to Assignment 3 **

1. (a) Solve the two-dimensional

0 < y < b for each of the following sets of conditions.

We need to calculate the coefficients:

_{}

(i) f_{1}(x) = 0, f_{2}(x) = sin(2πx), g_{1}(y)
= y(1-y), g_{2}(y) = 0, a = 2, b = 1;

Here A_{n}
= D_{n }= 0.

_{}

_{}

using the results of question 3(a)(i) of
Assignment 2. Hence C_{2k} = 0
while

_{}

Thus our solution
is

_{}

(ii) f_{1}(x) = f_{2}(x)
= 1 - x, g_{1}(y) = cos(2πy), g_{2}(y) = sin(πy), a =
b =1.

_{}

using the results of question 4(a) of
Assignment 1.

_{}

using the results from Assignment 2,
question 1(a)(iii). Thus C_{2k}
= 0 while

_{}

_{}

_{}

(b) Plot the tenth partial sum of the solution to
part (a) (i). Use the option axes=BOXED.

See MAPLE output.

2. (a) Find the eigenfunctions and eigenvalues of the
two-dimensional Helmholz equation

_{}

on the rectangular domain 0 < x < 2, 0
< y < 3 where the eigenfunctions are zero on the boundaries.

Let u_{mn}(x,y) = sin(mπx/2)sin(nπy/3). Then u_{mn}(0,y)
= u_{mn}(2,y) = u_{mn}(x,0) = u_{mn}(x,3) =0 and

_{}

i.e. _{}.

(b) Using the
results from part (a), solve the Poisson equation

_{}

on the same rectangular domain and boundary
conditions as in part (a).

Let _{}. Then

_{}

Thus

_{}

If we let x = 2t
in the last integral we get

_{}

as in question 1(a)(i). Thus E_{mn} = 0 if m is even or n
≠ 3 and

_{}

and _{}

(c) Plot the
tenth partial sum of the solution to part (b).

See MAPLE output.

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

(i) the gradient of
ρ^{2 }sinφ cosφ, ρ^{2} – z^{2};

In cylindrical polar coordinates

_{}

Thus

_{}_{}

(ii) the
divergence of z**u**_{ρ}, cosφ**u**_{z},

In cylindrical polar coordinates

_{}

Thus

_{}

(iii) the curl of **ρ**, ρ**u**_{φ}

In cylindrical polar coordinates

_{}

Thus

_{}

(iv) the Laplacian of ln(ρ), 1/ρ, ρz
sinφ

In cylindrical polar coordinates

_{}

Thus

_{}

(b) Evaluate the following expresions using the
vector operators in spherical polar coordinates:

(i) the
gradient of r sinθ cosφ, r^{2} cos^{2}θ;

In spherical polar coordinates

_{}

Thus

_{}

(ii) the
divergence of **r**, **u**_{φ},

In spherical polar coordinates

_{}

Thus

_{}

(iii) the curl of cosφ**u**_{r}, r sinθ**u**_{θ}

In spherical polar coordinates

_{}

Thus

_{}

_{ }

(iv) the Laplacian of 1/r, r^{2} sin^{2}θ
sinφ, r^{n}

In spherical polar coordinates

_{}

Thus

_{}

4.(a) In
MAPLE the Bessel function J_{n}(x) is denoted by BesselJ(n,x). Plot J_{0}(x) and J_{1}(x)
on the same graph over the range from 0 to 20. What can you say about the location of the
zeros of J_{1} relative to those of J_{0}?

See MAPLE output.

(b) The function BesselJZeros(n,m)
calculates the m^{th} zero of J_{n}(x). Find the first 10 zeros of J_{0}(x)
and store them in an array. Then
evaluate J_{1} at the 10 roots of J_{0} and store them. See below for sample MAPLE code.

See MAPLE output.

(c) Find the solution of the two-dimensional wave
equation for a circular membrane of radiius

a = 1 and speed c
= 1 with initial conditions f(ρ) = sin(2πρ)
and g(ρ) = cos(3πρ/2).
Use MAPLE to carry out the required integrations and use the results
from parts (a) and (b) to evaluate the Bessel-Fourier coefficients A_{m}
and B_{m} for m from 1 to 10. Note how the magnitude of
these coefficients vary with m.

This is the symmetric case (no
dependence on φ) so the solution is

_{}

where

_{}

See MAPLE output for calculations of
the coefficients.

(d) Plot the tenth partial sum of the Fourier-Bessel series from part
(c) for several different values of the time in the interval from 0 to 1/2.

See MAPLE output.