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 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) 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 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 ρ² sinφ cosφ, ρ² – z²;

In cylindrical polar coordinates

Thus

(ii) the divergence of zu_ρ, cosφu_z,

In cylindrical polar coordinates

Thus

(iii) the curl of ρ, ρu_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, r² cos²θ;

In spherical polar coordinates

Thus

(ii) the divergence of r, u_φ,

In spherical polar coordinates

Thus

(iii) the curl of cosφu_r, r sinθu_θ

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 J_n(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 m^th zero of J_n(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 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.