L=/o U701 T. jcx3. 1L2,Z fr7. )vo. A (a) [20%) Find the steady-state temperature u(x). M,(Y 7( it) X(o) U ç (ock. Partial Differential Equations 3150

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1 it) O 2h1 i / k ov A u(1,t) L=/o [1. (H3. Heat onduction in a Bar) 1 )vo nstructions: This exam is timed for 12 minutes. You will be given extra time to complete Exam Date: Thursday, May 2, 213 textbook. No answer check is expected. Details count 3/4, answers count 1/4. the exam. No calculators, notes, tables or books. Problems use oniy hapters 1, 2, 3, 4, 7 of the jcx3 X(o) U ç (ock 7 4n(o 1L2,Z fr7 M,(Y 7( (d) [2%] Display an answer check for the solution u(x, t) u1(x) + 712(Z, t). the answer n(x, t) = xi (x) + u2(a t) with the numerical values inserted into the series. A (a) [2%) Find the steadystate temperature u(x). = <x<1, t>, = 5, t>, = 8, t>, onsider the heat conduction problem in a laterally insulated bar of length 1 with one and the other end at 8 elsius The imtial ternpeiature along the bai is 1\ /1 (b) [4%] Solve the bar problem with zero elsius temperatures at both ends, but f(x) replaced by f(x) u1(). all the answeru2(z,t). Ut zz(,t) (c) [2%] ompute the numerical values of the Fourier coefficients of u2(x, t). Then display LA Use this page to start your solution. Attach extra pages as needed, then staple. U71 T T / x(x, ) = f(z), <z<1. Final Exam Partial Differential Equations 315

2 cj\ 1 N \J \V ç r N N \ t t \hi V 2c, ci 7 c \.$:) [ \ fl ii (,)) L,. N X j3 jo cj i _J (J 1:, ) c

3 . }.. \: \J 2 fr end at 5 elsius and the other end at 8 elsius. The initial temperature along the bar is OO 1. (R3. Heat onduction in a Bar) onsider the heat conduction problem in a laterally insulated bar of lenh 1 with one 1 oq the exam. No calculators, notes, tables or books. Problems use only hapters 1, 2, 3, 4, 7 of the textbook. No answer check is expected. Details count 3/4, answers count 1/4. nstructions: This exam is timed for 12 minutes. You will be given extra time to complete / 2c Final Exam 2(x,f). Partial Differential Equations 315 / Use this page to start your solution. Attach extra pages as needed, then staple... (cyo (d) [2%.] Display an answer check for the solution ii(x,t) = u(x)+u the answer u(x, t) = x1 (x) + u2(x, t) with the numerical values inserted into the series. (c) [2%] ompute the numerical values of the Fourier coefficients of i2(x, t). Then display replaced by f(x),a (a) [2%] Find the steadystate temperature u1(x). A (b) [4%1 Solve the bar problem with zero elsius temperatures at both ends, hut f(2) all the answer u(1,t) = 8, t >, x(x,) = f(x), <x < 1. u(,t) = 5, t>,?jt ii, <x < 1, t>, ui(x).,:., \&cu (ilfl ( ) 7_ f(x) =3x. Exam Date: Thursday, May 2, 213 :

4 Name. 315 Final Exam S (Fourier Series) () / (a) [3J Find and display the nonzero terms in the Fourier series expansion of f(x), formed as the even 2irperiodic extension of the function fo (x) sin 2 (x) + 4 cos(2x) on < x < x. ompute the Fourier sine series coefficients b for the function g(x), defined as the period 2 odd extension of the function 9(x) = 1 on x < 1. Draw a representative graph for the partial Fourier sum for five terms of the infinite series. (b) 15 %1 { [2j Define ho(x) and let Ji(x) be the 4x odd periodic = extension of ho(x) to the whole real line. ompute the sum f(5.25x) + f(1.5x). 6 a f() qj L 1T / j 2ti \. x ( os() j 77 t (1 ) (if) rf?c ) (: 2( OL 1T 17 1h.:: 2 rr S c (2)2.? 2rr. 2) cç Li(), ( ( ti (2 + ( /) J( your solution. Attach extra pages as needed, then staple. ri. 1cr di ( L 5 c 9 L,tr f1i.< (5 çn

5 9, z 1 ol \1 9 cy n ( _, U) i 1) n , DE \.. J j ii1 1. \ r H 1 ; ç H D \1 9, U 9, c 9, :s, c.4 11 t n N i. \ i AA A, AA A \ t DD t D ( 9, U] :; G p U] N, ± i c1.,_ N \ \ 1i

6 cj Th 4 \j1 c 1) L a ci S N S. c) cj U L J 7Th c3 Th N (/N H

7 Name. 315 Final Exam S213 A 3. (H3. Finite String: Fourier Series Solution) (a) [5%] Display the series formula, complete with derivation details, for the solution / u(x, t) of the finite string problem < x < 2, t >, u(,t) =, z(2,t), t>, u(x,) = f(x), <x<2, ut(x,) = g(x), <x<2. Symbols f and p should not appear explicitly in the series for u(x,t). formula for u(x, t) are product solutions times constants. Expected in the (b) [25%] Display explicit formulas for the Fourier coefficients which contains the symbols f(x), g(x). (c) [25%] Evaluate the Fourier coefficients when f(x) = 1 and g(x). i. i,i X(xYTM i yz 4 xr x (tin 4ccoS(6x)? JjD cd 21 çr,s / Use this page to start your solution. Attach extra pages as needed, then staple. T 4 d. (/ ) t cc(/q) () ) ) x S(/)oc(/q s;(r/

8 i Th J x _ A, 2 r (

9 2 5 D / 3 2r7< F c7 2 H ij tm Th 4:; c 5 5 ç (N, q ç ro 9 1 ( Y 1? (N 1 r. G i R5 o / i. _ )2!! A 9Z U A A A H H H. 3 AA A d > r (N vvv {. ) t > ( ç.5 > (. 1 (.. l) r i.. F r > T!i

10 F r1 a (1 > f / N 5 2f

11 _. 4sin(3xy), u ui Name. 315 Final Exam S (H3. Poisson Problem) Solve for u(x, y) in the Poisson problem 7 2( ) u(o,y) = u + u sin(irx) sin(2iry), <r < 1, < y < 1, u(x, y) = <y<l, on the other 3 boundary edges. Product solution derivations are not expected. Expected solution details: (1) Decomposition into two problems, it = + zl2. Draw figures. ci (2) Eigenfunctions mn and eigenvalues A, of the Helmholtz equation1(z, y)+yy(x, Y) = )(x, y) with zero boundary conditions on the rectangle. (3) Poisson problem defined for x1 (x, y). Solution formula for u1 (a, y). Fourier coefficient formula. (4) Dirichlet problem defined for u2(, y). Product solutions. Solution formula for 212(1, y). Fourier coefficient formula. (5) Display the answer u = + it 2. ) 2 J4 &t(o 1). c) s(ntx) S)h(j /Nfr (1( ç45 ( c1)i) ) ( ) 5 (i(mj) c 5,vL)5/ ( j Sh t Use this page to start your solution. Attach extra pages as needed, then staple.

12 1 S 4 _ L r s H t LA A 3 5 R Sr

13 , +. cz Di ad A AA A D A a ) DO a a a c 1 hal r:i DD az \\ 1i ;:a N a i 4) ) 4>, _c %a U ()V, & F r. N r\r c rr7 ), (1).., 4 L 7 a 4 aç k a L a2 Q D Q j?. HJ. Q aa ) OD 5 a. + V) fr i. 44. ) L $ s S, 41 4 aa a, /a r. c 5 ) R D va aoqn a S NJ aa ad 3 NJ\ N 5

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15 Name. 315 Final Exam S213 1 \ 5. (H7. Heat Equation and Gauss Heat Kernel) Solve the insulated rod heat conduction problem Ut(,t) = u1(x,t), oo<x<oo, t>o, u(x,) = f(x), oo <.x <, 15 O<<1, f(x) = 1 1<r< otherwise & / Hint; Use the heat kernel g = the error function erf() f e_z c 2 dz, and Fourier transform theory definitions to solve the problem. The answer is expressed in terms of the error function. (/) 11 ($)xs) cs (x s) S z s \S 2 L1 r so 1,j iw $( ( ) e ( ) Use this page to start your solution. Attach extra pages as needed, then staple..naersc.r,cn (/) 5 er, cer() e c 2e*3) 2er (7) 2cer().x L3 LA) t(e1?é.

16 Name. 315 Final Exam S (H7. Heat Eqmiation and Gauss Heat Kernel) Solve the insulated rod heat conduction problem cc <x < cc, t >, u(x,) = f(x), cc <x < cxc, 15 <r<1, f@) 1 1 <x < otherwise Hint: Use the heat kernel g the error function erf(x) = e_z 2 dz, and Fourier transform theory definitions to solve the problem. The answer is expressed in terms of the error function. ooc cl 1 & c :: c / zc) Qccç) ç D c :N) Use this page to start your solution. Attach extra pages as needed, then staple.

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