412m2_solns - MATH 412: Midterm 2 Thursday, April 18, 2008...

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Unformatted text preview: MATH 412: Midterm 2 Thursday, April 18, 2008 NAME: This exam consists of 3 problems, with points as 1abefled. Please write each answer Clearly and legibly. Don’t forget to show all of your work! fill M fl- (3 1. (32 pts) For the function h(33) = cc + 1 with a: E (—1,1), find the first two non—zero terms in the full Fourier series or 2. (32 pts) Consider the following general statement of Fourier decomposition (the rep— resentation of a function as an infinite sum of orthogonal eigenfunctions X j) fm=2aam (a) Using an inner product (f, g) but Without Specifying its particular form, derive an abstract expression for Ag; in terms of f and the Xj. (b) By taking the inner product of f with itself, derive the relation with its Fourier coefficients known as Parseval’s identity. (c) NOW take the inner product of f with another function 9(a"), and derive another relation involving the Fourier coefficients of both functions. 222% <><,»¢> an //»¢//é 55’. “2 04:7/ fi/ Lyn/8'15 “SQ L95 4€f>:4§4%2%~6> [2M Oweaé/A/V/W/Q 3. (36 pts} Find the spatial eigenfunctions and eigenvalues, and the general series solution for the following PDE (which is like the diffusion equation with an increasing diffusion constant), then answer the questions that follow {at = Kt2uw, a: 6 (0,12), t > 0 Zyfiimt) :0" 4?“ My [Z 7L) :0 What is the largest eigenvalue? The smallest? What is it about the PDE that allows for a series solution? we was-772:3 a) XT’:L¢PX”7’ 31;”: X4 WA F/Tt; ‘ - leRT 7: (>0 LE”; “Tit/th; 1&3 A ) SPQh‘q/ 9B3: {—3 C5 [6:45 7“» :14 A17€AK7L5 Some Useful Formulae and Identities: 1. Green’s identity: For 9 C IR“ fwAudV=—/Vu°del/+/ w(Vu-fi)dA o 52 so Where it is the outward normal to 59. 2. Divergence Theorem: Let 13 be a smooth vector field defined in 9. Then fv-FdV:/13.ndA 9 an where n is the outward normal to 89. 3. Definitions of Fourier Series coefficients: The full Fourier series for the function fir) on the interval (—3, E) is fir) : % + Z An cos(mr;z:/€) + Bn sin(mr:r:/€) 1121 where l 3 A0 = 2 An = élfflw) cosmms/E) dr and 1 e ‘ Bn 2 E Sin(mT:t/€)d$ 4. gonservation Law in IR”: Let p be the density for some conserved quantity M , and F be a vector field representing the flux associated with M , both defined in S2. Then at each point in Q —; ...
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This note was uploaded on 07/23/2008 for the course MATH 412 taught by Professor Belmonte during the Spring '08 term at Pennsylvania State University, University Park.

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412m2_solns - MATH 412: Midterm 2 Thursday, April 18, 2008...

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