ma3102soln3-11

ma3102soln3-11 - MATH3102 — Assignment 3, 2011 The...

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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) = rota-1&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 l-dimensional 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.) 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ma3102soln3-11 - MATH3102 — Assignment 3, 2011 The...

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