Solutions to term test 1
The total number of marks on this paper is 60 but grades will be calculated out of 50.
Note the following properties of periodic functions.
If f(x) is T-periodic then it is nT-periodic where n is a positive integer.
If f(x) and g(x) are both T-periodic then so is their sum, difference, product and quotient.
is independent of a.
1. (6 marks)
(a) What is the period of the function cos(x+p) + cos(3x)?
cos(x+p) = -cos(x) is 2p -periodic
cos(3x) is 2p/3 -periodic and hence 2p -periodic.
Therefore cos(x+p) + cos(3x) is 2p -periodic.
This function is even since cosine is an even function.
(b) What is the period of the function cos(2px)sin(5px)?
cos(2px) has period 1and hence is 2-periodic
sin(5px) has period 2/5 and hence is 2-periodic
Therefore cos(2px)sin(5px) is 2-periodic.
This function is odd since cosine is even and sine id odd.
2. (4 marks)
(a) Write down the appropriate boundary conditions for the heat equation for a bar of length L when one end is kept at 0 degrees while the other end is allowed to radiate.
Boundary condition for the end kept at zero degrees is u(x,0) = 0
Boundary condition for the radiating end is
(b) Explain briefly what is meant by the Gibbs phenomenon with regard to Fourier series.
A finite Fourier series overshoots a function near a singularity.
3. (9 marks) Suppose that f(x) is a 2p-periodic function with Fourier coefficients a0, an and bn, n = 1, ¥. If the function F(x) = f(x+p) has Fourier coefficients A0, An and Bn n = 1, ¥, show that A0 = a0, An = (-1)nan.
From the formula sheet
where we have made the substitution s = x+p and used the property of the integral of a periodic function stated above to justify the change of limits. Similarly
using the same substitution as above and the fact that
cos(np(s-p)/p) = cos( nps/p -np) = (-1)n cos( nps/p)
4. (8 marks) Find the half-range sine expansion for cos(2px) for 0 < x < 1.
The half-range sine expansion is given by
Thus b2k = 0 while
Note that when n = 2 the second term in the integral above is zero but this gives the same result.
5. (7 marks) Define completely the function f*(x) and G(x) in d'Alembert's solution of the one-dimensional wave equation for 0 < x < 1 with initial conditions
u(x, 0) = 2x + x2
ut(x, 0) = cos(3px).
f* is the odd extension of 2x + x2 and must be periodic of period 2. Thus
f*(x) = 2x + x2 if 0 < x <1; 2x - x2 if -1 < x < 0 and f*(x+2) otherwise.
Similarly g*(x) = cos(3px) if 0 < x <1; cos(3px) if -1 < x < 0 and g*(x+2) otherwise.
Now taking a = 1 in the definition of G we have
and G(x) = G(x+2) otherwise.
6. (13 marks) Use separation of variables to derive the solution of the one-dimensional wave equation
subject to the boundary conditions u(0, t) = 0, u(p, t) = 0
and the initial conditions u(x, 0) = 3 sin(2x), ut(x, 0) = 5 sin(3x).
Give explicit expressions for the Fourier coefficients in the solution.
See the text for the derivation of the separated solution. Now for L = p
so that applying the initial conditions we have
Thus an = 0 if n ¹ 2 while a2 = 3. Also
Thus bn = 0 if n¹ 3 while b3 = 5/3c giving u(x,t) = 3 cos(2ct) sin(2x) +(5/3c) sin(3ct) sin(3x).
7. (13 marks) Use separation of variables to derive the solution of the two-dimensional heat equation
for a rectangular plate if the edges are kept at zero degrees and the initial temperature distribution is given by f(x, y) = 12 sin(2px/a)sin(4py/b), 0 < x < a, 0 < y < b.
See the text for the derivation of the separated solution
The initial condition gives
Thus Amn = 0 unless m = 2 and n = 4 when A24 = 12 giving
u(x, y, t) = 12 sin(2p x/a) sin(4p y/b) exp(-l242 t)