This preview shows pages 1–16. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 *" “Ti511m 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 Vie2'6!" ‘;  ‘3 , .. {A awi‘ (€11 e ewe grafﬁti? 9 define 1%: 6M x» m Fem9M 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‘ﬁs.
\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» rma.w«% W a ‘ mﬁmwﬁw Commit?“ cJiikiieximcﬂx ( ﬂ ,9
Moe/Nani», ciliami} 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‘”
awngr.“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 > ...
View Full
Document
 Spring '09
 Bischler

Click to edit the document details