**MATH3271**

**FW03**

**Assignment
3 Due
date: Monday, Nov. 3**

1. (a) Solve the two-dimensional

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

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

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

You may use your
results from previous assignments where applicable.

(b) Plot the tenth partial sum of the solution to
part (a) (i). Use the option axes=BOXED. Note that you can rotate the plot in 3D by
placing the cursor on the plot, holding down the left mouse button and dragging
the display. This enables you to get a
better idea of the shap of the plot.

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.

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

_{}

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

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

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

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

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

(iii) the curl of **ρ**, ρ**u**_{φ}

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

(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;

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

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

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

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}?

(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.

(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. Notice how the magnitude of these
coefficients varies with m.

(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.

**MORE MAPLE**

Here is some MAPLE code that will
calculate the first 10 zeros of J_{0} and store them in an array a:

>for m from 1
to 10 do a[m]:=evalf(BesselJZeros(0,m)); od;

Note the use of
square brackets to denote an array element.

Here is the code to evaluate J_{1} at the zeros of J_{0}:

>for m from 1
to 10 do b[m]:=evalf(BesselJ(1,a[m])); od;

Note that the
function evalf forces a numerical evaluation of the Bessel function.

To
evaluate

_{}

numerically use:

>f:=whatever_function_of_x;

>evalf(int(f,x=a..b));

Note that if you
just write

>int(f,x=a..b);

MAPLE will
attempt to find an analytic solution to the integral.

To plot a two-dimensional function
in polar coordinates use the following commands (only works in MAPLE 6 or later
versions):

>addcoords(z_cylindrical,[z,rho,phi],[rho_{*}cos(phi),rho_{*}sin(phi),z]);

>plot3d(f,rho=a..b,phi=c..d,coords=z_cylindrical,axes=BOXED);

where f has been defined as a function of
rho and phi, a and b are the limits on rho and c and d are the limits on phi
(normally 0 and 2*Pi). In this
assignment f is a function of rho only.
Note that you must have given t a definite value in the Fourier-Bessel
series before plotting.