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-1 528 "Times" 1 14 0 0 1 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle56" -1 267 1 {CSTYLE " " -1 -1 "Times" 1 14 0 0 1 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle76" -1 529 "Times" 1 14 0 0 1 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle57" -1 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle 77" -1 530 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle58 " -1 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {PARA 240 "" 0 "" {TEXT 494 5 " " }{TEXT 495 38 "Heat Cond uction in 1 and 2 Dimensions:" }{TEXT 496 0 "" }}{PARA 241 "" 0 "" {TEXT 497 7 " " }{TEXT 497 32 "An Application of Fourier Series" }{TEXT 497 0 "" }}{PARA 242 "" 0 "" {TEXT 498 0 "" }}{PARA 242 "" 0 " " {TEXT 498 0 "" }}{PARA 243 "" 0 "" {TEXT 499 438 "In this worksheet \+ we introduce three Maple procedures for studying the heat equation in \+ dimension 1 (along a thin rod of finite length) and in dimension 2 (ac ross a thin rectangular plate), with homogeneous boundary conditions. \+ These procedures will allow you to study the time evolution of the tem perature symbolically, numerically and graphically. Examine the follow ing procedures and examples, then go ahead with your own experiments. \+ " }{TEXT 499 0 "" }}{PARA 244 "" 0 "" {TEXT 500 2 " " }{TEXT 500 0 "" }}{SECT 1 {PARA 245 "" 0 "" {TEXT 501 29 " 0. Summary of the Procedur es" }{TEXT 501 0 "" }}{PARA 246 "" 0 "" {TEXT 502 11 "Section 1: " } {TEXT 503 21 "heat1d_sin( f, a, N )" }{TEXT 502 31 " computes the sum \+ of the first " }{TEXT 503 2 " N" }{TEXT 502 96 " terms of the series \+ expansion of the solution to the heat equation along a thin rod of len gth " }{TEXT 503 1 "a" }{TEXT 503 1 " " }{TEXT 502 25 "with initial te mperature " }{TEXT 503 4 "f(x)" }{TEXT 502 66 ", provided that the tem perature at the boundary points is kept at " }{TEXT 503 1 "0" }{TEXT 502 2 ". " }{TEXT 502 0 "" }}{PARA 242 "" 0 "" {TEXT 498 0 "" }}{PARA 246 "" 0 "" {TEXT 502 11 "Section 1: " }{TEXT 503 21 "heat1d_cos( f, a , N )" }{TEXT 502 31 " computes the sum of the first " }{TEXT 503 2 " \+ N" }{TEXT 502 96 " terms of the series expansion of the solution to t he heat equation along a thin rod of length " }{TEXT 503 1 "a" }{TEXT 503 1 " " }{TEXT 502 25 "with initial temperature " }{TEXT 503 4 "f(x) " }{TEXT 502 61 ", provided that the ends of the rod are thermally ins ulated. " }{TEXT 502 0 "" }}{PARA 242 "" 0 "" {TEXT 498 0 "" }}{PARA 246 "" 0 "" {TEXT 502 11 "Section 2: " }{TEXT 503 20 "heat2d( f, a, b, N )" }{TEXT 502 31 " computes the sum of the first " }{TEXT 503 2 " N " }{TEXT 502 118 " terms of the double series expansion of the soluti on to the heat equation in a thin rectangular plate of dimensions " } {TEXT 503 5 "a , b" }{TEXT 502 27 " with initial temperature " } {TEXT 503 6 "f(x,y)" }{TEXT 502 61 ", provided that the temperature al ong the boundary is kept at" }{TEXT 503 2 " 0" }{TEXT 502 2 ". " } {TEXT 502 0 "" }}}{SECT 1 {PARA 247 "" 0 "" {TEXT 504 33 " 1. Heat Con duction in a thin rod" }{TEXT 504 1 " " }{TEXT 504 0 "" }}{PARA 248 "" 0 "" {TEXT 505 53 "Consider a thin metal rod placed along the interva l " }{TEXT 506 7 "00" }{TEXT 505 60 " is t he time. It satisfies the partial differential equation" }{TEXT 507 1 " " }{TEXT 507 0 "" }}{PARA 249 "" 0 "" {XPPEDIT 18 0 "diff(u(x,t),t) \+ = k*diff(u(x,t),`$`(x,2));" "6#/-%%diffG6$-%\"uG6$%\"xG%\"tGF+*&%\"kG \"\"\"-F%6$F'-%\"$G6$F*\"\"#F." }{TEXT 508 3 " " }{TEXT 509 0 "" }} {PARA 242 "" 0 "" {TEXT 498 0 "" }}{PARA 246 "" 0 "" {TEXT 502 5 "with " }{TEXT 503 5 "k > 0" }{TEXT 502 64 " being a constant. Here we make the simplifying assumption that " }{TEXT 503 5 "k = 1" }{TEXT 502 26 ". The initial temperature " }{TEXT 503 6 "u(x,0)" }{TEXT 502 48 " is \+ given by a reasonably well-behaved function " }{TEXT 503 4 "f(x)" } {TEXT 502 59 ". There are two basic cases that we will be interested i n: " }{TEXT 502 0 "" }}{PARA 242 "" 0 "" {TEXT 498 0 "" }}{PARA 246 "" 0 "" {TEXT 503 6 "Case 1" }{TEXT 502 38 ". The boundary temperature i s kept at " }{TEXT 503 1 "0" }{TEXT 502 8 " so that" }{TEXT 502 1 " " }{TEXT 503 16 "u(0,t)=u(a,t)=0 " }{TEXT 502 8 "for all " }{TEXT 503 1 "t" }{TEXT 502 2 ". " }{TEXT 502 0 "" }}{PARA 246 "" 0 "" {TEXT 502 76 "In this case, the method of separation of variables shows that the function " }{TEXT 503 1 "u" }{TEXT 502 33 " can be represented by the series" }{TEXT 502 0 "" }}{PARA 250 "" 0 "" {XPPEDIT 18 0 "u(x,t) = S um(B(n)*exp(-n^2*Pi^2*t/(a^2))*sin(n*Pi*x/a),n = 1 .. infinity);" "6#/ -%\"uG6$%\"xG%\"tG-%$SumG6$*(-%\"BG6#%\"nG\"\"\"-%$expG6#,$**F0\"\"#%# PiGF7F(F1*$%\"aGF7!\"\"F;F1-%$sinG6#**F0F1F8F1F'F1F:F;F1/F0;F1%)infini tyG" }{TEXT 510 0 "" }}{PARA 246 "" 0 "" {TEXT 502 23 "where the coeff icients " }{TEXT 503 4 "B(n)" }{TEXT 502 14 " are given by " }{TEXT 502 0 "" }}{PARA 250 "" 0 "" {XPPEDIT 18 0 "B(n) = 2/a;" "6#/-%\"BG6#% \"nG*&\"\"#\"\"\"%\"aG!\"\"" }{XPPEDIT 18 0 "Int(f(x)*sin(n*Pi*x/a),x \+ = 0 .. a);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#**%\"nGF+%#PiGF+ F*F+%\"aG!\"\"F+/F*;\"\"!F2" }{TEXT 510 0 "" }}{PARA 246 "" 0 "" {TEXT 503 8 "Case 2. " }{TEXT 502 52 "The boundary points are thermall y insulated so that " }{TEXT 503 20 "u_x(0,t)=u_x(a,t)=0 " }{TEXT 502 8 "for all " }{TEXT 503 1 "t" }{TEXT 502 2 ". " }{TEXT 502 0 "" }} {PARA 246 "" 0 "" {TEXT 502 39 "In this case, the temperature function " }{TEXT 503 1 "u" }{TEXT 502 33 " can be represented by the series" }{TEXT 502 0 "" }}{PARA 250 "" 0 "" {XPPEDIT 18 0 "u(x,t) = A(0)/2+Sum (A(n)*exp(-n^2*Pi^2*t/(a^2))*cos(n*Pi*x/a),n = 1 .. infinity);" "6#/-% \"uG6$%\"xG%\"tG,&*&-%\"AG6#\"\"!\"\"\"\"\"#!\"\"F/-%$SumG6$*(-F,6#%\" nGF/-%$expG6#,$**F8F0%#PiGF0F(F/*$%\"aGF0F1F1F/-%$cosG6#**F8F/F>F/F'F/ F@F1F//F8;F/%)infinityGF/" }{TEXT 510 0 "" }}{PARA 246 "" 0 "" {TEXT 502 23 "where the coefficients " }{TEXT 503 4 "A(n)" }{TEXT 502 14 " a re given by " }{TEXT 502 0 "" }}{PARA 250 "" 0 "" {XPPEDIT 18 0 "A(n) \+ = 2/a;" "6#/-%\"AG6#%\"nG*&\"\"#\"\"\"%\"aG!\"\"" }{XPPEDIT 18 0 "Int( f(x)*cos(n*Pi*x/a),x = 0 .. a);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$c osG6#**%\"nGF+%#PiGF+F*F+%\"aG!\"\"F+/F*;\"\"!F2" }{TEXT 510 0 "" }} {PARA 251 "" 0 "" {TEXT 511 45 "Below are two procedures with call seq uences " }{TEXT 511 0 "" }}{PARA 252 "" 0 "" {TEXT 502 3 " " }{TEXT 503 48 "heat1d_sin ( f, a, N ), heat1d_cos ( f, a, N )" }{TEXT 502 0 "" }}{PARA 246 "" 0 "" {TEXT 502 35 "which compute the sum of the fi rst " }{TEXT 503 2 " N" }{TEXT 502 52 " terms of above series for the initial temperature " }{TEXT 503 4 "f(x)" }{TEXT 502 21 " and a rod o f length " }{TEXT 503 3 "a. " }{TEXT 502 0 "" }}{PARA 253 "" 0 "" {TEXT 512 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 8 "restart; " }{MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 23 "heat1d_sin:=proc(f,a,N)" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 " " {MPLTEXT 1 513 25 "local n; global Bn, u, x;" }{MPLTEXT 1 513 0 "" } }{PARA 254 "> " 0 "" {MPLTEXT 1 513 20 "for n from 1 to N do" } {MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 50 " Bn(n) := (2/a)*int(f(x)*sin(n*Pi*x/a), x=0..a):" }{MPLTEXT 1 513 0 "" }} {PARA 254 "> " 0 "" {MPLTEXT 1 513 3 "od:" }{MPLTEXT 1 513 0 "" }} {PARA 254 "> " 0 "" {MPLTEXT 1 513 86 "u:= t-> convert( [ seq ( exp(-( n^2/a^2)*Pi^2*t)*Bn(n)*sin(n*Pi*x/a), n=1..N) ] , `+`):" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 22 "print(`u(x,t)` =u(t) ):" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 4 "end:" }{MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 23 "heat1d_cos:=proc(f,a,N)" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 29 "local n; global An, A0, u, x;" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 20 "for n from 1 to N do" } {MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 49 " An(n) :=(2/a)*int(f(x)*cos(n*Pi*x/a), x=0..a):" }{MPLTEXT 1 513 0 "" }} {PARA 254 "> " 0 "" {MPLTEXT 1 513 3 "od:" }{MPLTEXT 1 513 0 "" }} {PARA 254 "> " 0 "" {MPLTEXT 1 513 28 "A0:=(1/a)*int(f(x),x=0..a): " } {MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 91 "u:= t-> A 0 + convert( [ seq ( exp(-(n^2/a^2)*Pi^2*t)*An(n)*cos(n*Pi*x/a), n=1.. N) ] , `+`):" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 22 "print(`u(x,t)` =u(t)):" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 4 "end:" }{MPLTEXT 1 513 0 "" }}}{PARA 255 "" 0 "" {TEXT 514 70 "Let us put these two procedures to the test in the foll owing examples:" }{TEXT 514 0 "" }}{SECT 1 {PARA 256 "" 0 "" {TEXT 507 1 " " }{TEXT 515 9 "Example 1" }{TEXT 507 0 "" }}{PARA 246 "" 0 "" {TEXT 502 55 "We begin by the simple example of a thin rod of length \+ " }{TEXT 503 1 "2" }{TEXT 502 33 " when the initial temperature is " } {TEXT 503 1 "0" }{TEXT 502 17 " on the interval " }{TEXT 503 5 "0 " 0 "" {MPLTEXT 1 513 21 "f:=x->H eaviside(x-1):" }{MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 19 "heat1d_sin(f,2,30);" }{MPLTEXT 1 513 0 "" }}} {EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 20 "with(plots,animate):" } {MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 53 "a nimate(u(t),x=0..2,t=0..2,numpoints=150,frames=150);" }{MPLTEXT 1 513 0 "" }}}{PARA 257 "" 0 "" {TEXT 516 184 "As you see the system is diss ipative: it eventually loses all the energy and the temperature approa ches zero. This can also be seen from the series expansion of the solu tion, or by " }{TEXT 516 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 12 "u(infinity);" }{MPLTEXT 1 513 0 "" }}}{PARA 258 "" 0 "" {TEXT 517 37 "Now consider the insulated ends case:" }{TEXT 517 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 19 "heat1d_cos(f,2,30 );" }{MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 55 "animate(u(t),x=0..2,t=0..2.5,numpoints=150,frames=200);" } {MPLTEXT 1 513 0 "" }}}{PARA 255 "" 0 "" {TEXT 514 157 "In this case t he energy is not lost (intuitively, the heat has nowhere to escape). I n the long run, the temperature just distributes uniformly along the r od:" }{TEXT 514 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 12 "u (infinity);" }{MPLTEXT 1 513 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT 507 1 " " }{TEXT 515 9 "Example 2" }{TEXT 507 0 "" }}{PARA 246 "" 0 "" {TEXT 502 28 "Now let the rod have length " }{TEXT 503 1 "1" }{TEXT 502 63 " but the initial temperature be concentrated near the midpoint " }{TEXT 503 6 "x=0.5 " }{TEXT 502 164 "in the form of a unit impulse (think of touching the rod at its midpoint by an enormously hot needl e). First, we consider the rod with zero temperature at the ends:" } {TEXT 502 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 19 "f:=x->D irac(x-0.5):" }{MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 19 "heat1d_sin(f,1,30);" }{MPLTEXT 1 513 0 "" }}}{PARA 246 "" 0 "" {TEXT 502 119 "Because of the exponential terms in this so lution, this temperature tends to change extremely fast for small valu es of " }{TEXT 503 3 "t. " }{TEXT 502 28 "For this reason, we animate \+ " }{TEXT 503 8 "u(x,t^3)" }{TEXT 502 12 " instead of " }{TEXT 503 6 "u (x,t)" }{TEXT 502 59 ". This trick will allow us to slow down the fast change of " }{TEXT 503 1 "u" }{TEXT 502 21 " for small values of " } {TEXT 503 1 "t" }{TEXT 502 1 ":" }{TEXT 502 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 57 "animate(u(t^3),x=0..1,t=0..0.8,numpoints=15 0,frames=300);" }{MPLTEXT 1 513 0 "" }}}{PARA 255 "" 0 "" {TEXT 514 64 "Again, the temperature tends to zero as time tends to infinity. " }}{PARA 255 "" 0 "" {TEXT 514 45 "Now the case of the rod with insulat ed ends: " }{TEXT 514 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 19 "heat1d_cos(f,1,30);" }{MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 57 "animate(u(t^3),x=0..1,t=0..0.5,numpoints= 150,frames=200);" }{MPLTEXT 1 513 0 "" }}}{PARA 259 "" 0 "" {TEXT 508 90 "Unlike the first case, you see that the temperature tends to the s teady state temperature " }{TEXT 518 3 "u=1" }{TEXT 508 42 ", as can b e checked in the above formula. " }{TEXT 509 1 " " }{TEXT 509 0 "" }}} }{SECT 1 {PARA 260 "" 0 "" {TEXT 502 1 " " }{TEXT 519 41 "2. Heat Cond uction in a Rectangular Plate" }{TEXT 502 0 "" }}{PARA 246 "" 0 "" {TEXT 502 161 "The procedure in this section is a straightforward gene ralization of the one we discussed in dimension 1. Consider a thin met al plate placed over the rectangle " }{TEXT 503 14 "00" }{TEXT 502 61 " is the time. It satis fies the partial differential equation " }{TEXT 502 0 "" }}{PARA 249 " " 0 "" {XPPEDIT 18 0 "diff(u(x,y,t),t) = k*(diff(u(x,y,t),`$`(x,2))+di ff(u(x,y,t),`$`(y,2)));" "6#/-%%diffG6$-%\"uG6%%\"xG%\"yG%\"tGF,*&%\"k G\"\"\",&-F%6$F'-%\"$G6$F*\"\"#F/-F%6$F'-F46$F+F6F/F/" }{TEXT 508 2 " \+ " }{TEXT 509 0 "" }}{PARA 246 "" 0 "" {TEXT 502 5 "with " }{TEXT 503 5 "k > 0" }{TEXT 502 65 " being a constant. Again we make the simplify ing assumption that " }{TEXT 503 6 "k = 1 " }{TEXT 502 40 "and the bou ndary temperature is kept at " }{TEXT 503 1 "0" }{TEXT 502 159 ". (As \+ an exercise, you can modify the following procedure to include the cas e where the boundary of the plate is thermally insulated.) The initial temperature " }{TEXT 503 8 "u(x,y,0)" }{TEXT 502 37 " is given by a w ell-behaved function " }{TEXT 503 6 "f(x,y)" }{TEXT 502 60 ". Using th e method of separation of variables, the function " }{TEXT 503 1 "u" } {TEXT 502 40 " can be represented by the double series" }{TEXT 502 0 " " }}{PARA 250 "" 0 "" {XPPEDIT 18 0 "u(x,y,t) = Sum(Sum(C(m,n)*exp(-(m ^2/(a^2)+n^2/(b^2))*Pi^2*t)*sin(m*Pi*x/a)*sin(n*Pi*y/b),n = 1 .. infin ity),m = 1 .. infinity);" "6#/-%\"uG6%%\"xG%\"yG%\"tG-%$SumG6$-F+6$**- %\"CG6$%\"mG%\"nG\"\"\"-%$expG6#,$*(,&*&F3\"\"#*$%\"aGF=!\"\"F5*&F4F=* $%\"bGF=F@F5F5*$%#PiGF=F5F)F5F@F5-%$sinG6#**F3F5FEF5F'F5F?F@F5-FG6#**F 4F5FEF5F(F5FCF@F5/F4;F5%)infinityG/F3FN" }{TEXT 510 0 "" }}{PARA 246 " " 0 "" {TEXT 502 31 "where the Fourier coefficients " }{TEXT 503 6 "C( m,n)" }{TEXT 502 33 " are given by the double integral" }{TEXT 502 0 " " }}{PARA 250 "" 0 "" {XPPEDIT 18 0 "C(m,n) = 4/(a*b);" "6#/-%\"CG6$% \"mG%\"nG*&\"\"%\"\"\"*&%\"aGF+%\"bGF+!\"\"" }{XPPEDIT 18 0 "Int(Int(f (x,y)*sin(m*Pi*x/a)*sin(n*Pi*y/b),x = 0 .. b),y = 0 .. a);" "6#-%$IntG 6$-F$6$*(-%\"fG6$%\"xG%\"yG\"\"\"-%$sinG6#**%\"mGF.%#PiGF.F,F.%\"aG!\" \"F.-F06#**%\"nGF.F4F.F-F.%\"bGF6F./F,;\"\"!F;/F-;F>F5" }{TEXT 510 0 " " }}{PARA 261 "" 0 "" {TEXT 520 73 "In this section, we will write a M aple procedure, with the call sequence " }{TEXT 520 0 "" }}{PARA 262 " " 0 "" {TEXT 521 21 "heat2d ( f, a, b, N )" }{TEXT 521 0 "" }}{PARA 246 "" 0 "" {TEXT 502 30 "Given the initial temperature " }{TEXT 503 1 "f" }{TEXT 502 20 " and the dimensions " }{TEXT 503 6 "a , b " } {TEXT 502 55 "of the plate, this procedure approximates the solution " }{TEXT 503 1 "u" }{TEXT 502 41 " by computing the above double sum up to " }{TEXT 503 12 "m = N, n = N" }{TEXT 502 118 ". Once we have the \+ truncated series, we can use various Maple utilities to plot the resul ting temperature function. " }{TEXT 502 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 21 "heat2d:=proc(f,a,b,N)" }{MPLTEXT 1 513 0 "" } }{PARA 254 "> " 0 "" {MPLTEXT 1 513 30 "local m, n; global C, u, x, y; " }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 20 "for m \+ from 1 to N do" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 22 " for n from 1 to N do" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 61 " C(m,n):= int(f(x,y)*sin(m*Pi*x/a)*sin(n*Pi *y/b), x=0..a):" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 28 " C(m,n):= int(%, y=0..b):" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 23 " C(m,n):= %*4/(a*b):" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 5 " od " }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 3 "od:" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 123 "u:= t-> convert( [ seq \+ ( seq ( exp(-(m^2/a^2+n^2/b^2)*Pi^2*t)*C(m,n)*sin(m*Pi*x/a)*sin(n*Pi*y /b),m=1..N), n=1..N) ] , `+`):" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 24 "print(`u(x,y,t)` =u(t)):" }{MPLTEXT 1 513 0 "" }}{PARA 254 "> " 0 "" {MPLTEXT 1 513 4 "end:" }{MPLTEXT 1 513 0 "" }} }{PARA 255 "" 0 "" {TEXT 514 50 "We test the above procedure on some s pecific data." }{TEXT 514 0 "" }}{SECT 1 {PARA 263 "" 0 "" {TEXT 522 1 " " }{TEXT 523 9 "Example 1" }{TEXT 522 0 "" }}{PARA 264 "" 0 "" {TEXT 524 9 "Consider " }{TEXT 525 9 "f(x,y)=x " }{TEXT 524 14 "as the initial" }{TEXT 525 1 " " }{TEXT 524 31 "temperature over the rectang le " }{TEXT 525 8 "a=2, b=1" }{TEXT 524 88 ". Let us see how the tempe rature evolves with time by plotting the truncated series for " } {TEXT 525 4 "N=4 " }{TEXT 524 5 "terms" }{TEXT 524 34 ". As before, we choose to animate " }{TEXT 525 10 "u(x,y,t^2)" }{TEXT 524 12 " instea d of " }{TEXT 525 8 "u(x,y,t)" }{TEXT 524 42 " in order to slow down t he fast change of " }{TEXT 525 1 "u" }{TEXT 524 11 " for small " } {TEXT 525 1 "t" }{TEXT 524 3 ". " }{TEXT 524 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 12 "f:=(x,y)->x:" }{MPLTEXT 1 513 0 "" }}} {EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 16 "heat2d(f,2,1,4);" } {MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 22 "w ith(plots,animate3d):" }{MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 66 "animate3d(u(t^2),x=0..2,y=0..1,t=0..0.5,numpoin ts=300,frames=200);" }{MPLTEXT 1 513 0 "" }}}{PARA 264 "" 0 "" {TEXT 524 153 "It pretty much looks like a melting iceberg! As can be seen, \+ the initial temperature distributes itself very quickly toward the ste ady-state temperature " }{TEXT 525 1 "0" }{TEXT 524 20 ". We can check this:" }{TEXT 524 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 12 "u(infinity);" }{MPLTEXT 1 513 0 "" }}}}{SECT 1 {PARA 263 "" 0 "" {TEXT 522 1 " " }{TEXT 523 9 "Example 2" }{TEXT 522 0 "" }}{PARA 264 " " 0 "" {TEXT 524 37 "Now consider the initial temperature " }{TEXT 525 24 "f(x,y) = (x-1)^2+(y-1)^2" }{TEXT 524 17 " over the square " } {TEXT 525 8 "a=2, b=2" }{TEXT 524 58 ". Here is the time evolution of the approximate solution " }{TEXT 524 4 "for " }{TEXT 525 4 " N=5" } {TEXT 524 7 " terms:" }{TEXT 524 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 26 "f:=(x,y)->(x-1)^2+(y-1)^2:" }{MPLTEXT 1 513 0 "" }} }{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 16 "heat2d(f,2,2,5);" } {MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 66 "a nimate3d(u(t^2),x=0..2,y=0..2,t=0..0.5,numpoints=300,frames=200);" } {MPLTEXT 1 513 0 "" }}}{PARA 265 "" 0 "" {TEXT 526 82 "Again, the temp erature tends to the steady-state temperature (= zero) very fast . " } {TEXT 526 0 "" }}}{SECT 1 {PARA 263 "" 0 "" {TEXT 522 1 " " }{TEXT 523 11 "Example 3 " }{TEXT 522 0 "" }}{PARA 264 "" 0 "" {TEXT 524 27 "As the third example, let " }{TEXT 525 11 "f(x,y) = y " }{TEXT 524 3 "if " }{TEXT 525 8 "y > 0.5 " }{TEXT 524 4 "and " }{TEXT 525 1 "0" } {TEXT 524 25 " otherwise on the square " }{TEXT 525 8 "a=1, b=1" } {TEXT 524 115 ". We can express this function in terms of Heaviside's \+ unit step function. Let us look at the approximate solution " }{TEXT 524 4 "for " }{TEXT 525 3 "N=5" }{TEXT 524 7 " terms:" }{TEXT 524 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 29 "f:=(x,y)->y*Heaviside (y-0.5):" }{MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 16 "heat2d(f,1,1,5);" }{MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 " > " 0 "" {MPLTEXT 1 513 67 "animate3d(u(t^2),x=0..1,y=0..1,t=0..0.5,nu mpoints=300,frames=200); " }{MPLTEXT 1 513 0 "" }}}}{SECT 1 {PARA 266 "" 0 "" {TEXT 527 12 " Example 4 " }{TEXT 527 0 "" }}{PARA 264 "" 0 " " {TEXT 524 48 "As a final example, let the initial temperature " } {TEXT 525 2 "f " }{TEXT 524 22 " be a unit impulse at " }{TEXT 525 12 "x=0.8, y=0.2" }{TEXT 524 15 " on the square " }{TEXT 525 10 "a=1, b=1 . " }{TEXT 524 43 "This time we will calculate the series for " } {TEXT 525 4 "N=6 " }{TEXT 524 47 "terms (the answer is going to be pre tty long, " }{TEXT 525 3 "36 " }{TEXT 524 19 "terms to be exact)." } {TEXT 524 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 36 "f:=(x,y )->Dirac(x-0.8)*Dirac(y-0.2):" }{MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 16 "heat2d(f,1,1,6);" }{MPLTEXT 1 513 0 " " }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 67 "animate3d(u(t^2),x= 0..1,y=0..1, t=0..0.2,numpoints=400,frames=200);" }{MPLTEXT 1 513 0 "" }}}{PARA 264 "" 0 "" {TEXT 524 115 "It follows from the series expres sion of the temperature that the thermal energy stored in the metal pl ate at time " }{TEXT 525 1 "t" }{TEXT 524 63 " is proportional to the \+ double sum of all the terms of the form" }{TEXT 524 0 "" }}{PARA 252 " " 0 "" {TEXT 528 1 " " }{XPPEDIT 18 0 "C(m,n)^2*exp(-2*Pi^2*t*(m^2+n^2 ));" "6#*&-%\"CG6$%\"mG%\"nG\"\"#-%$expG6#,$**F)\"\"\"*$%#PiGF)F/%\"tG F/,&*$F'F)F/*$F(F)F/F/!\"\"F/" }{TEXT 502 0 "" }}{PARA 267 "" 0 "" {TEXT 529 122 "It is easy to approximate this sum and then plot it as \+ a function of time to check that the system is in fact dissipative:" } {TEXT 529 0 "" }}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 95 "energy: = t->convert( [ seq ( seq ( exp(-(m^2+n^2)*2*Pi^2*t)*C(m,n)^2,m=1..6), n=1..6) ] , `+`):" }{MPLTEXT 1 513 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {MPLTEXT 1 513 25 "plot(energy(t),t=0..0.1);" }{MPLTEXT 1 513 0 "" }} }}}{PARA 242 "" 0 "" {TEXT 498 0 "" }}{PARA 242 "" 0 "" {TEXT 498 0 "" }}{PARA 268 "" 0 "" {TEXT 530 0 "" }}{PARA 269 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }