mid2sol

# mid2sol - Math 257/316 Midterm 2 Section 101 7 November...

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Unformatted text preview: Math 257/316, Midterm 2, Section 101 7 November 2011 Instructions. The exam lasts 50 minutes. Calculators are not allow-3d. A formula, sheet is attached. @% hawk—511. Consider the following problem for the heat equation with a source term, and non—homogeneous Neu- mann boundary conditions: U¢=um+l 0<w<1, t>0 ‘ ant-(0‘15) = 0, 1t;,,(1,t) = 2 Mn, 0) 2 3:2 + 1 —— 2cos{7r:r;). ﬂ (3.) Find a particular solution of the PDE and boundary conditions of the form v(a:, t) = (1.352 +brr+ct [2 (b) Find the solution u(m,t) of the full problem. a (c) Write a ﬁnite difference approximation to the PDE (Lo. u; = um. + 1)1 and aiso write finite difference approximations for the boundary conditions. In case you need it, the Taylor expansion formula is f(:1:+ Am) 2 f(.1:) + f’(:1:)/_\m + %f”(m)(A2:)2 + O((A:L‘)3). (a) V: Qx’lt—EX-ﬁ- of?) “a Vic:v)‘~3b+\ 7') C- : 20“” \jk:?_0\3(. tic I0 0: Vxﬂo)t\= b o 01: VKCMF 20095 =’LO\ =3 oc—i =) Q23 So ‘VCJL)L‘\:. xii—3%! (is) («uni-g. obi,“ :: VLocllclJr wLaLJJA '50 Wt: mm; 00L“: £30 3 DD Jen/1e WX(O)P\= 0: whom] 2: so wtxjm -,: 92;: ;. QACDS (“WAX Q wbcﬂh : 91(me -VL)LJOB “bl = x}? i ~Zc<=s LENA ‘0‘} WA Ic- relﬁﬂg T— i" UosUTDQ 0;; l—- Emma 2' when: 95’ 4» i on‘t’shmd , , . 7 7 ' {a} FAWWMMM “I 3,:qu x Oct) 1: +25%} *QLxJ m M4 in) 1: uwa‘mZ Q+ubr4\$au P\'Zubklt\ A t a at} LAAXL m .50 6.. ¥ :1.» “(’17 U)! of PO L; LS QLM’ E+Ak)—Ubbi’s _ obey—Ax) J.Uuaﬁx)£]—deﬁa\+'l [:4- JM. 6C5 .‘ Ab RAJLSL o - 3::(Oi‘k\ g {AIDL 1.") 1.: 2.25103“ *5 oar/.5393.- ULI’Ain’\ 1:) mt, E 2. Use the method of separation of variables to solve the following initial boundary value [)I‘OlJlL‘IEl for the M 1 wave equation with 20m Ncumann BCSZ uu=um 0<ac<rr, t>0 um(0,t) = 0 = um(7r, t) u(m,0) = O, ut(a:,0) ={ 5 ,m f; 50):) _ U i; a 0(a)); = )(LqT Uh :9 j; ; >5. : ﬁx 9% ngLm: KY)“: .«xXCxE 3 :=> k :- xh= hi) hi 0mg.” XII-0‘ : - X: K6”): COSLnJLB _..JI . t _ A ' T prawn“: T’Lﬂ: .anu,\ 2» ‘nzo :3 \ :0 ,2) [C+\:_’%:i. a “M D T”: M111) Tm: maﬁa l' \gr‘S\\’\Cﬁ 3' '3/‘\W\ Old'lN‘M g . _._. g ' a) g 0 (xi—‘4 _ ‘l' ‘l‘ Ulnar-,an + gquL‘ Cosinx\ " in\\'\"\\ COAJ\'\\M,“;: ‘2’ H:\ L (:38;— +_ g hlrsh GOLD-a) Inst f :1) 80 = Symon - [42“: W 1. (Al g4 ﬁll) “an: 2817 (SK/L Cubhwﬁoln -:: éslAUf/LB :\ ...
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• Fall '08
• PEIRCE
• Boundary value problem, Partial differential equation, finite difference approximation, Boundary conditions, mann boundary conditions, DD Jen/1e WX

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mid2sol - Math 257/316 Midterm 2 Section 101 7 November...

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