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Unformatted text preview: MATH3102 — Assignment 3, 2011 The assignment consists of the starred questions with marks as shown.
Due 3pm Thursday, 6 October.
Hand to the tutor, or place in the assignment box on Level 4 of Bldg 67 before that time. 1._ The steady state temperature distribution T(r, 6) in a circular disc of radius a is
' given by Laplace’s equation V2T= 33< 8T) + r81" TE 1 WT ﬁwﬂ” Here (7", 6) are the usual polar variables in the plane. Use separation of variables to ﬁnd T(7", 6) for an imposed temperature distribution T(a, 6) = 6(27r — 6) around the
edge of the disc. ' 2. Show that the eigenvalues and eigenfunctions of the Sturm~Liouville problem i + Mag/(:5) = 0, y(1) = 0, y(e)'= 0, are given by An = 1 + n27r2, yn(x) = 33* sin[mrln(a:)], n = 1, 2, 3,‘ . . .. Hint: Change the independent variable to u = ln cc. V f sinz dz
22(1 + 2'?)
around a square of side—length 4 centered on the origin, with sides parallel to the
coordinate axes. 3. Evaluate the integral 4. (a) * (3 marks.) Using the Cauchy residue theorem show that for any integer m
ii 2””1 dz = 2m 6mg,
0 Where C’ is a circle of radius unity centered at z = 0. Use this result to deduce
that, if you are given T(z) = 2 z‘”_2 Ln,
then
L = i f zn+1T(z) dz
n 2m 0 7
_ 1 n+2 I
(n+2)Ln _ 27”sz T(z)dz, Where T’ stands for the derivative of T(z) with respect to z. (b) * (4 marks.) You are given 1 dw wm+1 dz zn+1T(z)T(w), Lan — I;an = .
(271—1)2 00 cm 5. 10. 11. where Cw is a circle of radius unity centered at z = w and Co a circle of radius
unity centered at w = O, and 0/2
(2 — NJ)“ 2T('w)
(Z — 7102 TM
2 — w T(z)T(w) = + + with c being a constant and T’ (to) being the derivative of T(w) with respect
to w. Show that 1
—cn<n2 — 1) 6am. Lan — Lan = (n — m)Ln+m + 12 * (6 marks.) Suppose that a: > O, and Im(p) > O. Show that
/.00 eiAm
—00 A — p What are the corresponding results for x < 0, lm(p) > O; for 93 > O, Im(p) < 0 and
for at < 0, Im(p) < 0? dA : 2m 6”? . Hint: Consider the semicircular contours FU, F L of radius R, centre 0, in the upper
and lower halves of the complex A—plane, and the contour F0 along the :c—axis from
a: = —R to a: = R, and use Cauchy’s Theorem. To show carefully in the ﬁrst case that the integral on FU goes to zero as R —> 00, you Will need to set /\ = R e” there
and then note that sin(6) Z 26/7T, for O S 6 g 7r/2. Compute the Fourier transform F = .7: [ f] and inverse transform f‘1[F] when
f(:1:) = (am sin(ba:) Where a > O and b is real. . * (3 marks.) Compute the Fourier transform F = FM and inverse transform .73‘1[F] when f(x) = 6““2 sin(b:z:). . Show that by writing sinac = (6“C — e‘i“) / 2, breaking the integral in two, and indenting the
contour around the pole at the origin. Show that if f0(33) = 6—932/2, f1 = x 642/2, f2(a:) = (2x2 — 1) e‘wz/Z’ and f3(a:) =
(2$3 — 6—362/2, then = f0, = "ifl, —f2 and = ifg. Compute the inverse Fourier transform of sin(ap)
(p2 + 1X]?  325' where or is real. (Write sin(ozp) = (em? — e'mp)/22' and then break the inversion
integral into two parts.) F07) = Use the Fourier analysis presented in class for the temperature of a long rod to ﬁnd
the solution for the initial temperature distribution a T($,0)=m. Note in particular the form of the solution as a —> 0. 12. 13. 14. 15. 16. Use the Fourier analysis presented in class for the temperature of a long rod to ﬁnd
the solution for the initial temperature distribution T 0 eﬂm
(x: ) — m '
Find the Green’s function G(CL‘, y) for the problem
u”(:r) = —f(a:), O < x < 1, u(0) = 0 = u(1), where f is any prescribed function, by looking for a solution in the form = + and ChOOSiIlg A’(93) + DUB/(2:) = 0 along the way. Check that
(X139) = G(y,l‘),
G(y+,y) = (Ky—w),
Gmm(x7y) : _6("r _ y)‘ Then check that the use of the Green’s function gives the correct solution in the
case f = :13, by solving the ODE directly in that case. Consider the PDE G(0,y) = 0 = 00,9),
Gx(y+,y)  Cruz/1y) = 1, amt) _ 1 amt)
at —a 8x , oo<cc<oo, t>0,
with initial condition
6”“ m < 0
wmﬁl‘{0 x>0 Here Or aé 0 is a constant. You may assume ¢(x,t) —> O as —> oo fort > 0.
Let \Il(/\, t) denote the Fourier transform of ¢(m, t). (a) Use the Fourier transform of the PDE to show that
mm) = rota1&2
and evaluate ‘ll()\, 0).
(b) Invert the Fourier transform using suitable contours to ﬁnd ib(x, t).
(c) Use the Fourier transform convolution theorem to obtain 77b($, t). Check that your answer is equal to that obtained in Part * (5 marks.) Find the Fourier transform F = 7" when f(a:) = 6—6434, a > 0.
Work out the inversion and show f(:c) = F‘1[F]. * (4 marks.) Consider the ldimensional wave equation 8211 2 3211, _=c_. , 8162 8952 in —oo < a: < 00 and t > 0, where u(a:, O) = f(:r) and 8146929) = 0. (You may assume that u(x,t) —> 0 and 61g?” —> 0 as —> 00.) Use the Fourier transform to show
that the solution of the initial value problem is given by u(:c,t) = %f(x+ct) +—:f(x—ct). DAATH3\WL SQLQ‘B ‘ixip “1M i . \ a!
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 Three '11
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 Applied Mathematics

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