MATH0056
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MATH0056
1. (1) Consider the diffusion equation
ut = kuxx , u(t = 0) = 1 + sin(πx/l), u(x = 0) = u(x = l) = 1,
on 0 < x < l and t > 0, with k > 0 a constant. Show that the Laplace transform u(s, x) satisfies
kuxx - su = -1 - sin(πx/l),
and write down relevant boundary conditions for u.
(2) Determine u, and hence show that
u(x, t) = 1 + f (t) sin(πx/l),
for some function f (t) that you should determine.
(3) Consider instead the wave equation
utt = k2 uxx , u(t = 0) = 1+sin(πx/l), u2 (t = 0) = 0; u(x = 0) = u(x = l) = 1, on 0 < x < l and t > 0, with k > 0 a constant. Find u and show that
u(x, t) = 1 + g(t) sin(πx/l),
for some function g(t) that you should determine. Over what time period is the initial condition recovered (such that u(x, t) = u(x, 0))?
(20 marks)
2. Consider Chebyshev’s equation
(1 - x2 )T22 (x) - xT2 (x) + n2 T (x) = 0,
where x e R and n > 0 is an integer.
(1) Classify the points x = 0, x = -1 and x = 1 as either ordinary points or regular singular points of Chebyshev’s equation.
(2) By seeking a solution of the form Tn (x) = ak xk , show that the coef- ficients ak satisfy the recurrence relationship
ak+2 = Λ(k, n)ak ,
for k > 0, for some Λ(k, n) that you should determine.
(3) For a given n, we now make the choice that a0 = 1 and a1 = 0 if n is even, otherwise we choose a0 = 0 and a1 = 1 if n is odd. Using the recurrence relation, show that Tn (x) is then a polynomial in x, consisting of only even powers of x if n is even, and only of odd powers of x if n is odd.
(4) Find the standard Sturm–Liouville form of Chebyshev’s equation. Deduce
that
Wn (x) =!1 - x2
is a second particular solution of Chebyshev’s equation. Show that Wn (x) and Tn (x) are linearly independent.
(20 marks)
3. The Bessel function Jn (x) of order n, where n is an integer, is a solution of Bessel’s equation
x J2n(22)(x) + xJn(2)(x) +╱x2 - n2、Jn (x) = 0 V x > 0.
The generating function formula is
G(x, t) = exp ╱ ╱t - 、、 = ntn Jn (x).
(1) Show that
x
and that
Jn(2)(x) = (Jn / 1 (x) - Jn+1(x)) .
Let Fn (x) = Jn+1(x)/Jn / 1 (x). Further let the sequence {jnk } denote the real positive zeroes of the Bessel function Jn (x). Show that Fn (jnk ) is a constant that you should determine.
(2) Use these identities to find an expression for Jn / 1 (x)Jn+1(x) in terms of Jn and Jn(2). Hence deduce that
, ╱ Jn (x)2 - Jn / 1 (x)Jn+1(x)、、= xJn (x)2 .
(3) Use the above results to show that
←0 1 rJn (jnk r) dr2 = .
(20 marks)
4. The displacement of the membrane on a circular drum is modelled by the wave equation
utt = ∆u
in the disc of radius R > 0 defined by ,(x, y) : x2 + y2 < R、. The solution is assumed to be axisymmetric, so that u = u(r, t) in polar coordinates. The boundary condition u(R, t) = 0 is prescribed on the boundary of the disc for all t > 0, and the initial displacement u(r, 0) = f (r) and initial velocity ut (r, 0) = g(r) are specified at the initial time t = 0. Furthermore, the solution is assumed to be oscillatory with respect to time.
(1) Using the information stated at the bottom of the problem when appropri- ate, show that the solution is of the form
o
u(r, t) = J0 (λk r) [αk cos(λk t) + βk sin(λk t)] ,
k=1
where J0 (x) is the Bessel function of index zero, and where the αk , βk and λk are real constants for all k > 1. You should determine the values of the λk as part of your solution.
(2) Using the information stated at the bottom of the problem when appropri- ate, and by carefully explaining any orthogonality results used, express the coefficients αk and βk in terms of integrals involving the functions f(r) and g(r).
(3) The sounds that the drum produces are related to the values of the λk , with higher values of the λk corresponding to higher-pitched sounds. Based on your answer in (a), explain why larger drums produce lower-pitched sounds than smaller drums.
You may state without proof that J0 (x) is the only solution (up to a multiplica- tive constant) of Bessel’s equation of index zero
xw22 + w2 + xw = 0,
that is nonsingular at x = 0. You may also use without proof that J0 (x) has infinitely many positive real zeros {j0k} , as well as the identity
←0 1 xJ0 (j0kx)J0 (j0jx)dx = J1 (j0k)2 V k, j e 宏>0 ,
where δkj is the Kronecker delta.
(20 marks)
2022-04-13