MATH3271

FW03

Solutions to Assignment 3

1. (a) Solve the two-dimensional Laplace’s equation on the rectangular region 0 < x < a,

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

We need to calculate the coefficients:

(i)   f1(x) = 0, f2(x) = sin(2πx), g1(y) = y(1-y), g2(y) = 0, a = 2, b = 1;

Here An = Dn = 0.

using the results of question 3(a)(i) of Assignment 2.  Hence C2k = 0 while

Thus our solution is

(ii)  f1(x) = f2(x) = 1 - x, g1(y) = cos(2πy), g2(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 C2k = 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 umn(x,y) = sin(mπx/2)sin(nπy/3).  Then umn(0,y) = umn(2,y) = umn(x,0) = umn(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 Emn = 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 ρ2 sinφ cosφ, ρ2 – z2;

In cylindrical polar coordinates

Thus

(ii)  the divergence of zuρ, cosφuz,

In cylindrical polar coordinates

Thus

(iii) the curl of ρ, ρuφ

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, r2 cos2θ;

In spherical polar coordinates

Thus

(ii)  the divergence of r, uφ,

In spherical polar coordinates

Thus

(iii) the curl of cosφur, r sinθuθ

In spherical polar coordinates

Thus

(iv) the Laplacian of 1/r, r2 sin2θ sinφ, rn

In spherical polar coordinates

Thus

4.(a)  In MAPLE the Bessel function Jn(x) is denoted by BesselJ(n,x).  Plot J0(x) and J1(x) on the same graph over the range from 0 to 20.  What can you say about the location of the zeros of J1 relative to those of J0?

See MAPLE output.

(b)  The function BesselJZeros(n,m) calculates the mth zero of Jn(x).  Find the first 10 zeros of J0(x) and store them in an array.  Then evaluate J1 at the 10 roots of J0 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 Am and Bm 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.