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# 105936 - KWS(6 Echlﬁqeﬂy “ﬁg f Third Test 427Ky(OD...

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Unformatted text preview: KWS (6 Echlﬁqeﬂy “ﬁg f Third Test, 427Ky, 05/06/2010 (OD PRINTED NAM CNL ; Every problem is worth an equal number of points for a total of 10 You must show your work; answers without substantiation do not count. Answers must appear in the box provided! No or the wrong answer in the answer born results in no credzt! L/S2 >753 v 8: 4% +LISl/[Q 4% Jr W5) = 6'“; [JUHZPE L15 2% tzs+'>*5‘*” “W9 + as m) + m) ’ 5’s — m if- (QHUHS—rh \/[£> [£1,52qu JH> ’85 vi; ; C [L[\$1+Lls+l> _: e-L\S+8‘S Wei a” 18\$ +22 o J. em I: +(ﬂC—i ‘75:;Ii) El CS+V1 )(‘>f\/1>i KT: \ 45 K3 -\ z , n1 ,’*!B*C'/>‘1C>> a: Meta/Q 40mm? /W ,_. . I (Ha) Suppose y’ = f (t,y) is to be solved numerically on the interval [3,8] with initial condition y(3) = 2 and with various step sizes h. State how the local and global errors depend on h and state the TOTAL number of eval- uations needed, for the Euler, Improved Euler, and Runge—Kutta methods, respectively [N stands for “is bounded by a multiple of”]. m — I 21-! a ll (IIb) In problem (Ha) assume now speciﬁcally that I f (t, y) = tﬂ—I: and that the step size is ﬁxed at h = 0.2 [and again that to = 3 and yo Apprbximating the solution using the Heun method, also called the improved Euler method, with this step size h = 0.2, compute the point (t1,y1) you arrive at after the ﬁrst step. __ 1% £(n+h3=a+.z (354(9) ‘3" wlfhm lean) -; .7. +é.3L{ 3 1+ (433 ‘1 * (5+0 Jr {5 we 9» “ cw ’ amt (01+0(,577>/ = 2+,9 afﬁf‘é/ it”. ( / :2 {-13% / /. —; . CHW,’ 5 P/ fw’ 3.2 Answer: t1 = (Ilia) How are the Fourier coefﬁcients of a function f : [~L, L] —> R deﬁned? {as A»? if StQLQ COS (n:K> ’9 S1» Coca i ff 0 (3‘5 x i g“ 5%} g Hm» .s) rowan gm o I, (IIIb) How is the Fourier series of such f deﬁned? (IIIC) State Fourier’s Theorem a 6k ‘ ‘ "I‘ 11L)- Four??? gen}, ) wkt’C/Lw “(PM WM“st PlﬂCLWIS-A Cm W QWV‘CVW‘ 9% 964 61,9 cob Egrck'); CW3”; *0 VAL 0kng "36 M W rcﬁhif’ hngﬂ (W343 dvli' o» lmealA‘fg‘m—iymié: (IIId) Descrihe Gibbs’ phenomenon r , m- QM 4W1)“ J5“ ° m Mum ; 1;va WXMMMﬂCmM'ﬂGa‘FFké‘4 "' '* Gibbs! {WWW 9“? 514V)" 5‘3 will now» J jW‘W’ ZOV (IV) Let fIv(33) ={ (a) Draw the function f. (0) Compute the Fourier series » A ’7 A (:05 g; 2 Bnéfnszm £WOO eff/.4— ?“ h 5 Jr “1‘ A 3 L’M‘vﬁw" s \ :z gin m>31 - wwilw 3%ng a ‘* 3 c {49:} 5 L “#1”. 0% £65,135er gmma‘buvb '5th a , 5% hp I“? (\z 00 i o ; 5—? (4 \v‘ MI nTT 3 w 7‘ Answer 21—? Z, (J 5" (nth-Iv > " M h 3 2113/3 +1 for ~3Sx§ —3/2 0 for —3/2<\$<3/2 2m/3—1 for3/2Smg3 ﬂy of the function fIV. . (VjLet :1: forOSmS3/2 3—30 for3/2Sa:<3 Find the solution of the heat conduction equation WW '7- M = 5’ that satisﬁes u‘(t,0) = 0 \7’t > 0,, u(t, 3) = 3 Vt > 0, and u(0, :17); fV(x) Vm E 09 r w MUTIX7: §Ch . grn(ngx> 9/ 3/1, 2 r 33 X 5‘“ '3 (A?! OQX +5- (E’K>5(ﬂm/ﬁ o a 3 ut(t, :12) = 5umx(t, x) Ch: 3’/z '3 2" ' 8 X 605 £171. m “1 sin m’g‘ y 3/1. :L. 3 MT 6 a + ﬁn, *‘é‘ \D + "1 ﬂ. -9 ? S/Qh n»ﬁL 3— 3% + #7?” ‘b E. 3 : 3"” I'M," > Mfr" 2/ [0, 3). “353/ C05 M'- ' v : rm 3 m [A7 X r q Aht’ ﬂVI (A: nnx ,\ /« OQI 2 ,3X (03 157V. 4. "Q/ (93" w: ﬂag“ (R u 3) r 3 r x 3 L on ~ ch (3 0 a (205 (m\ ALA T)" 3 5 Ja‘ ’ ‘ _. X “X Slh ﬂ 6 co£(%)}+ (3 > ‘ hu / D ’0 on” , /60\$ "(l-32> S h?) _’ 7 3' ' 'y‘ M-U\L l Ftﬁt 2.40; r/gX-> : 5—K : hf!» yup/[ﬁx 0M: aha/11L ix 3 3 '3 _B<5’Y>C0\$[ﬂ>} ,échOS h“ 3 s/wni' 3/ 3 _ ,3 \ 1’ " D iiﬁlcosg‘p) J :n/L‘L’i ) hq 2. huff-t a 3/ Z O ’ O , O , j” 91‘“ rx‘Tl'x 9\ ﬂ 3% ...
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