MATHS 361 Tutorial 2
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DEPARTMENT OF MATHEMATICS
MATHS 361
Tutorial 2
The aim of this tutorial is to practice calculating Fourier series expansions and to do more practice on the method of separation of variables.
1. For each of the following functions determine whether the function is odd, even or neither. For any functions that are neither odd nor even, find fe and fo .
(a) _ -x-
(b) sinh(x) = (ex _ eox )/2
(c) xeox
(d) ln(1 + x2 )
2. Using pen and paper, calculate the Fourier coefficients a0 , an and bn for the function f (x) = 2 _ -x- defined on the interval _1 < x < 1.
3. (Optional)1 Download the matlab notebook fourierseries.mlx from CANVAS and save it somewhere in your home directory. This file uses matlab symbolic calculations to compute the Fourier coefficients for a defined function and plots partial sums of the Fourier series.
By adapting the matlab code in this file, do the following tasks.
(a) Plot the graph of the function f (x) = 2 _ -x- on the interval [_1, 1]. Also
plot the sum of the first 2N +1 terms of the Fourier series for the following choices of N : N = 1, 3, 10, 30. Plot all the graphs on the same figure. Note: Some of the terms may be zero; when you count terms, include the zero terms in your count.
(b) Repeat (a) for the function f (x) = xeox on the interval [_1, 1].
(c) What do you notice about convergence of the two Fourier series? Which Fourier series converges faster? Is this what you expected? Why?
(d) Do you see any evidence for Gibbs’ phenomenon in either example? Ex- plain your answer.
4. Let L be a positive real number. Using paper and pen, calculate
0 L cos │ 
、cos ╱ 
← dx.
5. This problem concerns the heat conduction problem considered in lectures. The temperature u of a rod of length 1 satisfies
ut = u北北 0 < x < 1, t > 0.
The end x = 0 is held at temperature 0 and the end x = 1 is perfectly insulated. Hence the boundary conditions are,
u(0, t) = 0, u北 (1, t) = 0.
(a) Considering the boundary conditions, sketch the steady-state solution.
Then solve the steady state problem to check whether your sketch is cor- rect.
(b) Using the method of separation of variables, show that the following func-
tion is a solution to the PDE plus boundary conditions for all n = 0, 1, 2, 3, . . . .
un (x, t) = sin ││n + 
、πx、eo((n+
)π)2 t .
(c) Show that the sine functions satisfy the following orthogonality property: 0 1 sin ││n + 
、πx、sin ││m + 
、πx、dx = ,
(d) Given that the initial condition is u(x, 0) = f (x), show that
u(x, t) = n
cnun (x, t) = n
cn sin ││n + 
、πx、eo((n+
)π)2 t
where
cn = 2 0 1 f (x) sin ││n + 
、πx、dx
You can use the orthogonality property that you just proved, and you can also assume that f is sufficiently smooth that you can interchange the order of summation and integration when using a series representation of f .
6. Challenge question: If the Fourier coefficients of a periodic function f (x) are an (n = 0, 1, 2, . . . ) and bn (n = 1, 2, 3, . . . ), what are the Fourier coefficients of the shifted periodic function f (x _ c), where c is some positive constant?
2023-03-24