Test 3 - Test 3, 427K, 12 /03/09 PRINTED NAME: Every...

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Unformatted text preview: Test 3, 427K, 12 /03/09 PRINTED NAME: Every prootem is worth on squat number of points for a total of 100 points. You must show your work; answers without substantiation do not count. Answers must appear in the boat provided! Some Laplace Chansforms x t to. -- ‘7 Ila) Suppose y’ = f (t, y) is to be solved numerically on the interval [3, 8] with initial condition y(3) m 1 and with step size h. ' _' ‘ State how the local and global errors depend on it and state the TOTAL number of evaluations needed, for the Euler, Improved Euler, and Runge—Kutta methods, respectively ‘[~ stands for “is proportional to”]. __ p _ . i e~ :' e~ '3“ " 1110) In problem Ila) assume now specifically that f (13,31) m t, + y and h = 0.2-[end again that to = 3 and ye :2 l]. Approximating the solution using the Henri method, also called the improved Euler method, with this step size h m 0.2, compute the point (t1, yl) you arrive at after the first .[AEI step. . “W ‘ | '1 v ‘ “is,” r in r. a. j e {Mfg—m 2 ‘ ) ‘ I f z‘ a ’ r r' ' "a _fl[§r_iv;\i:l ” g: \' {'5 \I I) is 314,9, :2» . J‘s! " \ ‘ {yth l 5/ r l r \. « ‘3? i - w {MM 31:12 M; W ’u " P X} f m. .4».- f a” 9 J; a”? w 2v was» I 3,5.»4’9‘ rm “gang mg“ r“ £2 y {is 6“ W M ‘J 3:» a g.” 3 w ’ ‘ ‘6: M; if; } "3‘1 Mid a W as i g " A? * SUM” “i” «in ... L t? Answer: t; m f}? '91 Z 6;? p» a «as; r 5‘ MW III) (a) State the Fourier convergence theorem, including formulas for the coefficients, and (1)) describe the Gibbs phenomenon. Answer: the Fourier series of a piecewise differentiable function f : [-L, L] —-> R is defined as - :‘ufi a , “3“: figs {5! {7:1541 x ‘3 "- “7 {‘- N ,2 u .y *- utr. 9,] {ax ¢ '9 .z . {V (53, 3" w i w ‘ ~ ’ m- ' W ’= ' "q‘ ' ’ “ i a? " ;= i 391"»; I, C ‘5' ("H 5“ f: 4 4’- £5 may??? . A L» » yr. ‘ > 5’ ’c‘ 59”?" ' _. I, _:‘ é w fir *" “Ti-511m sir H erfifl’AyM’ Vet‘s“ \ {1; § I 6 - eta-{‘2 if} . 3 fly” ;y$‘h’n(’ (v 4‘“ , / a A" r . - fix or '5 (g: H . , (I A. 64 9 1 fl 5’ (b) The Gibbs phenomenon concerns . . . and says that [make a. drawing with comments if words fail you] égfiéd‘zmi +A5 ggflffié‘ayg ‘5 Vie-2'6!" ‘; - ‘3 , .. {A awi‘ (€11 e ewe graffiti? 9 define 1%: 6M x» m Fem-9M W51? 7'71 55" M Wan—immmwwwmvmmwna. IV) (3.) Compute the Fourier series flm) of the function f defined by [ for —7r 3 a: S —7T/2 7r/2 for mfl/Zfimgfl ' f0r0<x37r/2 for arr/2 3 a: < 7r. drawing it might hem] m .12“ @577 V) Solve the heat conduction problem at m 2am for a. bar of iength 7r that is kept at temperature ~71" / 200 at its left end and at +7r/200 at its right end at ali times, and that has initial temperature u(05m):{ x forOSxSrr/Z 7r/2 for 7r/2 g a: g 7r. / g It, I“ N f3. 4‘ f x MW, ‘ X __ g1}! f? - ‘ I if in » i ‘ ~9fl§e 0 “r em fl‘é w. 2 (*X “t 73‘) g? 4/ e w a ‘ ’r‘\ ‘ MW swam may“) h A?“ " n we é . W ‘ . + 5;“ u :7 2&6 72"} _ ML. ””""§?XE?“”" A .n, m, “if”. _ I 4” Q fig as: Answer: u(:13,t) g: X! m. MEN >m[2€m(3§ér\ § 1 B 6 t Test 3, 427K W, 12/83/69 Every problem is worth an equai mum You must Show your work; answers without substantiation do not count. Answers must appear in the bow provided! Some Laplace Transforms 7(3) :: 973:“ + €322th “’54 , , “33 . " g x g a J 5:“ ESE“? j i524) PRINTED NAME: M Rm “£53 of oints for a totai of 100 points. Answer: y :2: E“ w Ila) Suppose y’ 2 f (t, y) is to be solved numerically on the interval [2, 6] with initial eorrdition y(2) 2 1 and with step size b. State how the local and global errors depend on h and state the TOTAL number of evaluations needed, for the Euler, Improved Euler, and RurigemKutta methods, respectively [N stands for “is preportional to”). 1 * Ilb) In problem Ila) assume now specifically that f (23,11) m the y and h = 0.2 [and again that ii to r: 2 and ye = l]. Approximating the solution using the u method, also called the improved Euler method, with this step size h m 0.2, compute the poi Oh, ya} you arrive at after the first step. Mm) tint} L“ {p— to + eaa ‘ Answer: t1 :2 l 2_ yl 2 E 1H) State the Fourier convergence theorem, including formuias for the coefficien‘ss, and (‘0) describe the Gibbs phenomenon. Answer: the Fourier series of a piecewise difierentiable function f : [WL, L} ——> R is defined as Where :3 H C) .‘3 and on I K/\ F. :3; “é. %' 74 3 ii F‘fi-s. \3 “‘2ka WW Ae- WM“‘§WQB@\SJ CM“ “43" i - r -‘ w -i a I} r» r ’ 'w “Q'W'Q no..-” inww‘wmm imJ \rflw‘ifi { w‘ éim'fifi' W”) mafia" /i/ Ph’h’)‘; W firm/M M fife {mime {9 gm” WW“, (‘0) The Gibbs phenomenon concerns . . . . and says that [make a drawing with comments if words fail you} E i . :33} - . 43L; MP"? E/iigmor. ‘-;; Want» r--ma.w«% W a ‘ mfimwfiw Commit?“ cJ-iikiieximcflx ( fl ,9 Moe/Nani», cilia-mi} IV) (3.) Compute the Fourier series fix) of the function f defined by {drawing it might heip] ms)? 1.5;“ :3 for ww/Zgzcgw/Q §?7’§"‘"W‘fEE?7§“§&“§E¥r~-\ \k‘h‘g M {—77/2 for «war 5 a: g —7r/2 N (b) Determine f(7r) (giving reasons“) V) Solve the heat conduction problem at : Qum for a bar of length 7r that is kept at temperature +7r/200 at its left end and at ~7r/2OC at its right end at .31} times, and that has initial temperatu; _ 7r/2 f0r0<$§7r/2 u(0’m)~{7T——$ forw/2S3:<7r. \ '2“ J” K O{ ~ TV 7;” v ’3: w _ fr»— h a“ x 3 . H‘H‘” awn-gr.“de " - ’ u. «h 246' I ._._ U I ‘ i m! Answer: “(33:73) 2(. )_,X +. Z (hm c QM . W(WK,) Jr 2 gm: Jr n a a r/szfiK Ngz’fi f e} EWK > ...
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Test 3 - Test 3, 427K, 12 /03/09 PRINTED NAME: Every...

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