{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Times" 0 14 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2 " -1 257 1 {CSTYLE "" -1 -1 "Courier" 0 11 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Unit 2: Infinite Series" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Chapter \+ 10: Fourier Series" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Section 10.5: periodically driven damped oscillator" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "C opyright" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Copyright * 2001 by Addison Wesley Longman, Inc." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All rights reserved. \+ No part of this publication may be reproduced, stored in a retrieval s ystem, or transmitted, in any form or by any means, electronic, mechan ical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(plots):\nwith(inttrans):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "interface(showassumed=0);\nassume(n,integer); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "alias(Y=laplace(y(t),t, s)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 499 "PX := proc(h, g)\n local f, var, d;\n if type(h, procedure) then RETURN(subs(\n \+ \{'D' = rhs(g) - lhs(g), 'F' = h, 'L' = lhs(g)\}, proc(\n x::a lgebraic)\n local y;\n y := floor((x - L)/D); F(x - \+ y*D)\n end))\n fi;\n if type(g, equation) then var := lhs (g); d := rhs(g) fi;\n f := unapply(h, var);\n subs(\{'L' = lhs( d), 'D' = rhs(d) - lhs(d), 'F' = f\}, proc(\n x::algebraic)\n \+ local y;\n y := floor((x - L)/D); F(x - y*D)\n \+ end)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "Damped Oscillator Driven Periodically" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 " Consider the driven damped spring-mass system governed by the differen tial equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "q := diff(y(t),t,t) + 2*diff(y(t),t) + 10*y(t) = f(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "where the periodic forcing function " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 30 " is the periodic \+ extension of " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "g := piecewise(t<=Pi,t, t<=2*Pi,2*Pi-t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "g(t)" "6#-%\"gG6#%\"tG" } {TEXT -1 19 " on its domain of [" }{XPPEDIT 18 0 "0,2*Pi" "6$\"\"!*&\" \"#\"\"\"%#PiGF&" }{TEXT -1 13 "] is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "plot(g,t=0 ..2*Pi, labels=[t,`g `], labelfont=[TIMES,ITALIC,12], xtickmarks=6, yt ickmarks=3, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "A Maple repre sentation of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 22 " \+ is obtained with the " }{TEXT 256 2 "PX" }{TEXT -1 44 " command introd uced in Section 10.1, so that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := PX(g,t=0..2*Pi):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "with graph, Figure 10.24, given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "plot(f(t), t=0..6*Pi, color=black, scaling=constrained, ytickmarks=2, xtickmarks= [6,12,18], labels=[t,`f `],labelfont=[TIMES,ITALIC,12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "Exact Solution by Laplace Transform" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "A graph of the exact mot ion of this oscillator, started with the inert initial conditions" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 11 ", y'(0) = 0 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "can b e found via the Laplace transform. To this end, we obtain F(s), the t ransform of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 72 ", \+ with the formula for periodic functions from Section 6.7. Thus, with \+ " }{XPPEDIT 18 0 "p=2*Pi" "6#/%\"pG*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 16 ", the period of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 12 ". We obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(s)=1/(1-exp(-p*s))" "6#/-%\"FG6#%\"sG *&\"\"\"F),&F)F)-%$expG6#,$*&%\"pGF)F'F)!\"\"F1F1" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(f(t)*exp(-s*t),t=0..p)" "6#-%$IntG6$*&-%\"fG6#%\"tG \"\"\"-%$expG6#,$*&%\"sGF+F*F+!\"\"F+/F*;\"\"!%\"pG" }{TEXT -1 5 " = \+ " }{XPPEDIT 18 0 "(1-exp(-Pi*s))/s^2/(1+exp(-Pi*s))" "6#*(,&\"\"\"F%- %$expG6#,$*&%#PiGF%%\"sGF%!\"\"F-F%*$F,\"\"#F-,&F%F%-F'6#,$*&F+F%F,F%F -F%F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "a calculation executed in Maple via" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "F := simp lify(int(g*exp(-s*t),t=0..2*Pi)/(1-exp(-2*Pi*s)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Tra nsforming the differential equation and solving for " }{XPPEDIT 18 0 " Y" "6#%\"YG" }{TEXT -1 27 ", the Laplace transform of " }{XPPEDIT 18 0 "y(t)" "6#-%\"yG6#%\"tG" }{TEXT -1 8 ", we get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Y=1/s ^2/(s^2+2*s+10)" "6#/%\"YG*(\"\"\"F&*$%\"sG\"\"#!\"\",(*$F(F)F&*&F)F&F (F&F&\"#5F&F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-exp(-Pi*s))/(1+exp(- Pi*s))" "6#*&,&\"\"\"F%-%$expG6#,$*&%#PiGF%%\"sGF%!\"\"F-F%,&F%F%-F'6# ,$*&F+F%F,F%F-F%F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 50 "calculations which we execute in Maple as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The transform of the differential equation then yields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "q1 \+ := laplace(lhs(q),t,s) = F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Our notation has been simp lified by aliasing the letter \"Y\" to the Laplace transform of " } {XPPEDIT 18 0 "y(t)" "6#-%\"yG6#%\"tG" }{TEXT -1 83 ". Further simpli fication arises from imposing the inert initial contitions. Thus," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q2 := subs(y(0)=0, D(y)(0)=0, q1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Solving for t he unknown transform Y, we obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "q3 := factor(solve(q2,Y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 105 "It is impossible to invert this transform with just a \+ finite number of elementary functions, so we expand" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1- exp(-Pi*s))/(1+exp(-Pi*s))" "6#*&,&\"\"\"F%-%$expG6#,$*&%#PiGF%%\"sGF% !\"\"F-F%,&F%F%-F'6#,$*&F+F%F,F%F-F%F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "in powers of " } {XPPEDIT 18 0 "exp(-Pi*s)" "6#-%$expG6#,$*&%#PiG\"\"\"%\"sGF)!\"\"" } {TEXT -1 11 ", obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-1+2" "6#,&\"\"\"!\"\"\"\"#F$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)*exp(-k*Pi*s),k=0..infinity)" "6#-%$SumG6$*&,$\"\"\"!\"\"F(-%$expG6#,$*(%\"kGF(%#PiGF(%\"sGF(F)F(/F/ ;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 "The inverse Laplace transform of the fact or " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "1/s^2/(s^2+2*s+10)" "6#*(\"\"\"F$*$%\"sG\"\"#!\"\",( *$F&F'F$*&F'F$F&F$F$\"#5F$F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "is the function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " h(t)=exp(-t)/25" "6#/-%\"hG6#%\"tG*&-%$expG6#,$F'!\"\"\"\"\"\"#DF-" } {TEXT -1 3 " ( " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos(3*t)-2/3" "6#,&-%$cosG6#*&\"\"$\"\"\"%\"t GF)F)*&\"\"#F)F(!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(3*t)" "6# -%$sinG6#*&\"\"$\"\"\"%\"tGF(" }{TEXT -1 4 ") + " }{XPPEDIT 18 0 "(5*t -1)/50" "6#*&,&*&\"\"&\"\"\"%\"tGF'F'F'!\"\"F'\"#]F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "so the \+ solution to the differential equation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(t)=-h(t)+2" "6#/-%\"yG6#%\"tG,&-%\"hG6#F'!\"\"\"\"#\"\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Sum((-1)^k*h(t-k*Pi)*Heaviside(t-k*Pi),k=0..infinity)" "6#-%$SumG6$*(),$\"\"\"!\"\"%\"kGF)-%\"hG6#,&%\"tGF)*&F+F)%#PiGF)F*F)- %*HeavisideG6#,&F0F)*&F+F)F2F)F*F)/F+;\"\"!%)infinityG" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "To ob tain this solution in Maple, begin by getting a series expansion of " }{XPPEDIT 18 0 "Y(s)" "6#-%\"YG6#%\"sG" }{TEXT -1 14 " in powers of " }{XPPEDIT 18 0 "exp(-Pi*s)" "6#-%$expG6#,$*&%#PiG\"\"\"%\"sGF)!\"\"" } {TEXT -1 16 ". For this, in " }{XPPEDIT 18 0 "Y(s)" "6#-%\"YG6#%\"sG " }{TEXT -1 10 ", replace " }{XPPEDIT 18 0 "exp(-Pi*s)" "6#-%$expG6#,$ *&%#PiG\"\"\"%\"sGF)!\"\"" }{TEXT -1 24 " with the single letter " } {XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 11 ", obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "if has(q 3,exp(-Pi*s)) then q4 := subs(exp(-Pi*s)=z,q3)\nelse q4 := subs(exp(Pi *s)=1/z,q3);fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 187 "The conditionals in the code for \+ making this replacement are required to guarantee that each time this \+ worksheet is executed, the result is the same. Maple can return the e xpression for " }{XPPEDIT 18 0 "Y(s)" "6#-%\"YG6#%\"sG" }{TEXT -1 33 " as a fraction containing either " }{XPPEDIT 18 0 "exp(-Pi*s)" "6#-%$e xpG6#,$*&%#PiG\"\"\"%\"sGF)!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "e xp(Pi*s)" "6#-%$expG6#*&%#PiG\"\"\"%\"sGF(" }{TEXT -1 52 ". The repla cement of the negative exponential with " }{XPPEDIT 18 0 "z" "6#%\"zG " }{TEXT -1 88 " is crucial for the next step, a Taylor expansion in p owers of the negative exponential." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 33 "Perform a Taylor expansion about " } {XPPEDIT 18 0 "z=0" "6#/%\"zG\"\"!" }{TEXT -1 10 " to obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "q 5 := convert(taylor(q4,z=0,7),polynom);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Change " } {XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 9 " back to " }{XPPEDIT 18 0 "ex p(-Pi*s)" "6#-%$expG6#,$*&%#PiG\"\"\"%\"sGF)!\"\"" }{TEXT -1 9 ", gett ing" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "q6 := subs(z=exp(-Pi*s),q5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "then in vert this truncated series representation of Y(s) to obtain what shoul d be an approximation to " }{XPPEDIT 18 0 "y(t)" "6#-%\"yG6#%\"tG" } {TEXT -1 8 ". Thus," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "q7 := invlaplace(q6,s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "collect( q7, [seq( Heaviside(t-k*Pi ), k=1..6 ) ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "A graph of this version of the sol ution, namely" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^`^`=-h(t)+2" "6#/)%\"yG%\"^G,&-%\"hG6 #%\"tG!\"\"\"\"#\"\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)^k*h(t -k*Pi)*Heaviside(t-k*Pi),k=0..7)" "6#-%$SumG6$*(),$\"\"\"!\"\"%\"kGF)- %\"hG6#,&%\"tGF)*&F+F)%#PiGF)F*F)-%*HeavisideG6#,&F0F)*&F+F)F2F)F*F)/F +;\"\"!\"\"(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "is given in Figure 10.25, where, on the interva l " }{XPPEDIT 18 0 "[0,20]" "6#7$\"\"!\"#?" }{TEXT -1 31 ", the soluti on is exact because" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Heaviside(t-k*Pi),k*`>`*7" "6$-%*Hea visideG6#,&%\"tG\"\"\"*&%\"kGF(%#PiGF(!\"\"*(F*F(%\">GF(\"\"(F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "is zero, since " }{XPPEDIT 18 0 "8*Pi*`>`*20" "6#**\"\") \"\"\"%#PiGF%%\">GF%\"#?F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "p := plot(q7,t=0 ..20, color=black, xtickmarks=5, ytickmarks=4,labels=[t,``],labelfont= [TIMES,ITALIC,12], view=[0..20,0.. .33]):\np;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The firs t term dropped from the Taylor expansion of Y(s) was " }{XPPEDIT 18 0 "z^7 = exp(-7*Pi*s);" "6#/*$%\"zG\"\"(-%$expG6#,$*(F&\"\"\"%#PiGF,%\"s GF,!\"\"" }{TEXT -1 89 ". If that term had been retained, and inverte d, it would have contributed a multiple of " }{XPPEDIT 18 0 "Heaviside (t-7*Pi);" "6#-%*HeavisideG6#,&%\"tG\"\"\"*&\"\"(F(%#PiGF(!\"\"" } {TEXT -1 28 ", and this term is zero for " }{XPPEDIT 18 0 "t < 21;" "6 #2%\"tG\"#@" }{TEXT -1 87 ". Hence, on the interval [0,20], we have k ept enough terms for the graph to represent " }{XPPEDIT 18 0 " y(t)" " 6#-%\"yG6#%\"tG" }{TEXT -1 9 " exactly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "Approximate Solu tion by Fourier Series" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "We next obtain a sequence of approximate solutions which rapidly converge to the exact solution. In the diffe rential equation, the driving function " }{XPPEDIT 18 0 "f(t)" "6#-%\" fG6#%\"tG" }{TEXT -1 16 " is replaced by " }{XPPEDIT 18 0 "f[k](t)" "6 #-&%\"fG6#%\"kG6#%\"tG" }{TEXT -1 41 ", a partial sum of the Fourier s eries of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 33 ", and the corresponding solution " }{XPPEDIT 18 0 "y[k](t)" "6#-&%\"yG6#%\" kG6#%\"tG" }{TEXT -1 50 " computed. The sequence of approximate solut ions " }{XPPEDIT 18 0 "y[k](t)" "6#-&%\"yG6#%\"kG6#%\"tG" }{TEXT -1 41 " converges rapidly to the exact solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Using the general formalism of Section 10.1, we obtain a Fourie r series representation of " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" } {TEXT -1 28 ". Hence, with the interval " }{XPPEDIT 18 0 "[0,2*L]" "6 #7$\"\"!*&\"\"#\"\"\"%\"LGF'" }{TEXT -1 11 " taken as [" }{XPPEDIT 18 0 "0,2*Pi" "6$\"\"!*&\"\"#\"\"\"%#PiGF&" }{TEXT -1 4 "] so" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "L := Pi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "we obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 268 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[0]=1/Pi" "6#/&%\"aG 6#\"\"!*&\"\"\"F)%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t), t=0..2*Pi)=Pi" "6#/-%$IntG6$-%\"fG6#%\"tG/F*;\"\"!*&\"\"#\"\"\"%#PiGF0 F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "a[n]=1/Pi" "6#/&%\"aG6#%\"nG*&\"\"\"F )%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)*cos(n*t),t=0..2* Pi)=2" "6#/-%$IntG6$*&-%\"fG6#%\"tG\"\"\"-%$cosG6#*&%\"nGF,F+F,F,/F+; \"\"!*&\"\"#F,%#PiGF,F6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "((-1)^n-1)/n^ 2/Pi" "6#*(,&),$\"\"\"!\"\"%\"nGF'F'F(F'*$F)\"\"#F(%#PiGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[n]=1/Pi" "6#/&%\"bG6#%\"nG*&\"\"\"F)%#PiG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(t)*sin(n*t),t=0..2*Pi)=0" "6#/-%$ IntG6$*&-%\"fG6#%\"tG\"\"\"-%$sinG6#*&%\"nGF,F+F,F,/F+;\"\"!*&\"\"#F,% #PiGF,F4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 24 "by writing the integrals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "q0 := Int(g,t=0. .2*L)/L;\nqa := Int(g*cos(n*Pi*t/L),t=0..2*L)/L;\nqb := Int(g*sin(n*Pi *t/L),t=0..2*L)/L;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "and evaluating them to get the Fou rier coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 58 "a0 := value(q0);\na := simplify(value(qa)); \nb := value(qb);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "We note that " }{XPPEDIT 18 0 "b[n ]=0" "6#/&%\"bG6#%\"nG\"\"!" }{TEXT -1 9 " for all " }{XPPEDIT 18 0 "n " "6#%\"nG" }{TEXT -1 68 ", so the sine terms disappear from the serie s. Also, examining the " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" } {TEXT -1 13 " via the list" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "seq(subs(n=k,a),k=1..10);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "we see that just the cosine terms with an odd index \"su rvive.\" Hence, the first three distinct partial sums of the resultin g Fourier series are " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "for k from 1 by 2 to 5 do\nf||((k+1 )/2) := a0/2 + sum(a*cos(n*Pi*t/L),n=1..k);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Lettin g " }{XPPEDIT 18 0 "y[k](t),k=1,2,3" "6&-&%\"yG6#%\"kG6#%\"tG/F'\"\"\" \"\"#\"\"$" }{TEXT -1 44 ", be the solutions of the original IVP when \+ " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 16 " is replaced b y " }{XPPEDIT 18 0 "f[k](t)" "6#-&%\"fG6#%\"kG6#%\"tG" }{TEXT -1 52 ", we obtain Figure 10.26 wherein the approximations " }{XPPEDIT 18 0 "y [k]" "6#&%\"yG6#%\"kG" }{TEXT -1 21 " are plotted against " }{XPPEDIT 18 0 "y^`^`" "6#)%\"yG%\"^G" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "To obtain the solutions \+ " }{XPPEDIT 18 0 "y[k](t)" "6#-&%\"yG6#%\"kG6#%\"tG" }{TEXT -1 187 " i n Maple, and to obtain the graphs shown in Figure 10.26, we execute th e following loop which uses Maple's dsolve command to solve three IVPs whose right-hand sides are the partial sums " }{XPPEDIT 18 0 "f[k]" " 6#&%\"fG6#%\"kG" }{TEXT -1 138 ". Within the same loop, the graphs of the solutions are generated, and displayed against the graph of the e xact solution in Figure 10.25." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 112 "Also, we re-generate figure 10.25 with a thickened black line, and use it in Figure 10.26 to enhance visibilit y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "p := plot(q7,t=0..20, color=black, thickness=3):\nfo r k from 1 to 3 do\nQ||k := rhs(dsolve(\{lhs(q) = f||k, y(0)=0, D(y)(0 )=0\}, y(t),method=laplace));\np||k := plot(Q||k,t=0..20, color=red, y tickmarks=2);\nd||k := display([p,p||k],xtickmarks=5, ytickmarks=4, la bels=[t,``], labelfont=[TIMES,ITALIC,12], view=[0..20,0.. .33]);\nod: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Figure 10.26 is then" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display(array([d||(1..3 )]),scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The third partial sum approx imates " }{XPPEDIT 18 0 "f(t) " "6#-%\"fG6#%\"tG" }{TEXT -1 61 " well \+ enough for the graph of the resulting approximation to " }{XPPEDIT 18 0 "y(t)" "6#-%\"yG6#%\"tG" }{TEXT -1 185 " to be indistinguishable fro m the graph of the exact solution. In Chapter 14 we will study additi onal techniques for obtaining approximate analytic solutions of differ ential equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 152 "We close by separately displaying the three graphs compr ising Figure 10.26. The thick black curve is the exact solution, and \+ the approximate solutions " }{XPPEDIT 18 0 "y[k]" "6#&%\"yG6#%\"kG" } {TEXT -1 12 " are in red." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "d1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "d2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "d3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }