advertisement

/<E Partial Differential Equations 3150 y Midterm Exam 2 Exam Date: Monday, 22 April 2013 I Instructions: This exam is timed for 50 minutes. You will be given extra time to complete the exam. No calculators, notes, tables or books. Problems use only chapters 1 to 4 of the textbook. No answer check is expected. Details count 3/4, answers count 1/4. 1. (CH3. Finite String: Fourier Series Solution) (a) [75%] Display the series formula without derivation details for the finite string problem Utt(X,t) u(0,t) u(L,t) u(x,0) u(x,0) = = = = (x,t), c u 2 0, 0, f(x), g(x), 0<x<L, t>0, t>0, t>0. 0<x<L, 0<x<L. Symbols f arid g should not appear explicitly in the series for u(x, t). Expected in the formula for u(x, t) are product solutions times constants. (b) [25%] Display an explicit formula for the Fourier coefficients which contains the symbols L, f(x), g(x). Mryk ijj? 4 (r.)(/ CflTTCt/L yrrx/L) - 11 Vc. pfor 4 Y)oyD4 Le, (ri1rx/L- 4t (‘WrTCt/L_- (flY)) Z -t ?rc±-JL-) ‘TiTC r r.Lrr1L)3 Cfl1Y1L’j ALLc - Use this page to start your solution. Attach extra pages as needed, then staple. tE / Name. 3150 Exam 2 S2013 2. (CH3. Heat Conduction in a Bar) Consider the heat conduction problem in a laterally insulated bar of length 2 with one end at 50 Celsius and the other end at zero Celsius. The initial temperature along the bar is given by function f(z), which is a symbol used throughout the problem, devoid of a specific formula. zz 0<x<2, t>0, 2 c , 1 = t >0, u(0,t) = 50, 0, t>0, n(2,t) u(x,0) = f(x), 0<x<2. (x). 1 u c (a) [25%] Find the steady-state temperature A (b) [50%] Solve the bar problem with zero Celsius temperatures at both ends, but f(x) replaced by f(x) ui (x). Call the answer u2(x, t). The answer has Fourier coefficients in integral form, unevaluated to save time. — (c) [25%] Display an answer check for the solution u(x, t) = ui(x) + u2(X, t). 2( - ‘1) -, I ( () —ç Oi -2\ : ,., ) Use this page to start your solution. Attach extra pages as needed, then staple. pi1JIV?A ? 5 3 r o •= 1 ç q” o -’ + (O’fl = f 1 ( o l 0-40= 4 OS 4S- +0 z + (t)’n (o)fl ad zflx€ tfl,xo,+Ilx =‘L’ 5 Jo !rl)o ck\, (rexe , k 0 1L — ng 2 I ( ; fø’YP U1o ii (X)5( :(OMY) (ij C ‘. .V)3 5 ç:. 7’ ky’O) (<-z) (cY%)) o ( o (o)n o(’LTh/ 05 (:‘ 0) i’ 4 ( 9-i) 0 4- flwf.U1Afll _ iktOA Name. I<y 3150 Exam 2 S2013 3. (CH4. Rectangular Membrane) Consider the general membrane problem utt(x, y, t) u(x, y, t) u(x,y,0) u(x,y,0) = = = = c (n(x, y. t) + u(r, y, t)), 2 0 f(x,y), g(x.y), 0 < x <a, 0 <p < b, t on the boundary, 0<x<a, 0<y<b, 0< x <a, 0< p < b. > f o c.) 0, Solve the problem for a = b = c = 1, f(x,y) = 1, g(x,y) = 0. Expected are displays for the normal mqcles, a superposition formula for n(x, y, t), and explicit numerical values EEe generalized rF.cients. \S Y\oôJL Q €d 6L o cL: •(ow\ sç:p&iiOk1 ô( rn ) VG.vo? co )4 \O’V\ *JU \fl 3 E (VVA ()S 0 nD1 CJ).ç. ‘—I So L OY St i C / S ( Y))) (ti Use this page to start your solution. Attach extra pages as needed, then staple. ( 1 iiwc{ ff C I, :1) J D 4- ifj — I \) — — (. ‘V \ :1 cj _ > CD -J ci— g c_i a _i — yE) 0 L-’ 1 ‘ — — 1.— _1 c__i) 0 I I __ r —c3 :: r1 1 —‘ — it J- ._—‘ “__— 0 C j’ 7_< tf) ‘1 D -_- i 0 c__-, L— ,_—_ I G _i) .__—_/ I It 1:1’ 0 I — (______, V) 0 a + P 0 F: 0 1’I 3150 Exam 2 S2013 Name. 4. (CH4. Steady-State Heat Conduction on a Disk) Consider the heat conduction problem on a disk urr(r,0) + ur(r,6) + uü(r,6) z(a,8) =f(O), ?‘ 0 0, 0< r <a, pulse(6,0,7r/2), that is, f(8) ‘ = 1 on 0 < 0 < v/2, OikR3n vdi &j ) iW,c [)1 l”- 0<0 <2, 0<8 <2. Solve for u(r,9) when a 1 and f(0) f(9) 0 on ir/2 < 8 < 2ir. —icW€ = &tc1 roo 9 [VkiI fl v \tAC Jç - 2f k * L - —p 1’ AC Y / ( I o ) - ?Lr) co (ii G) Use this page to start your solution. Attach extra pages as needed, then staple. Ci n i o 7 I ‘S Cr) — -c / - 1/ S 0 —H Cr) ft 0 ., 0 5, —, 0 / ‘I 1 \_— - V N - — ‘‘ 0 5 — 5 C Cr) ) -