{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 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 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE " " -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 55 "Module 28: Laplace's equation with insulated boundaries" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 29 "We work the following problem" }}{PARA 0 "" 0 "" {TEXT -1 42 " " }{XPPEDIT 18 0 "0 = diff(u,`$`(x,2))+diff(u,`$`(y,2));" "6#/\"\"!,&-%%diffG6$%\"uG-%\"$G6 $%\"xG\"\"#\"\"\"-F'6$F)-F+6$%\"yGF.F/" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "Boundary conditions: " } {XPPEDIT 18 0 "u(0,y) = sin(Pi*y);" "6#/-%\"uG6$\"\"!%\"yG-%$sinG6#*&% #PiG\"\"\"F(F." }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[x](1,y)=0" "6#/- &%\"uG6#%\"xG6$\"\"\"%\"yG\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 45 " \+ " }{XPPEDIT 18 0 "u[y](x,0)=0" "6#/-&%\"uG6#%\"yG6$%\"xG\"\"!F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " u[y](x,1)=0" "6#/-&%\"uG6#%\"yG6$%\"xG\"\"\"\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "spacecurve(\{[x,0,0],[x,1,0],[1,x, 0],[0,x,sin(Pi*x)]\},x=0..1,color=BLACK,\n axes=NORMAL,orientation =[-25.,50]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 213 "We will not need to break this into a set of t wo PDE's with boundary conditions for one of them will have only the z ero solution. The other one will break into two differential equation s with boundary conditions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " X '' - " }{XPPEDIT 18 0 "lambda ^2;" "6#*$%'lambdaG\"\"#" }{TEXT -1 20 " X = 0, and Y '' + " } {XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#" }{TEXT -1 28 " Y = 0, Y '(0) = 0 = Y '(1)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 29 "These two will have solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " " } {XPPEDIT 18 0 "x[0](x)" "6#-&%\"xG6#\"\"!6#F%" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "A[0]*x+B[0]" "6#,&*&&%\"AG6#\"\"!\"\"\"%\"xGF)F)&%\"BG6 #F(F)" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "X[n](x)" "6#-&%\"XG6#%\"nG6# %\"xG" }{TEXT -1 10 " = sinh(n " }{XPPEDIT 18 0 "pi;" "6#%#piG" } {TEXT -1 16 " x) or sinh(n " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 13 " (1-x)) " }{XPPEDIT 18 0 "Y[n](y)=cos(n*Pi*y)" "6#/-&%\"YG 6#%\"nG6#%\"yG-%$cosG6#*(F(\"\"\"%#PiGF/F*F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus," }}{PARA 0 "" 0 "" {TEXT -1 32 " u(x,y) = " }{XPPEDIT 18 0 "A[0]*x+B[0]+s um((A[n]*sinh(n*Pi*x)+B[n]*sinh(n*Pi*(1-x)))*cos(n*Pi*y),n=1..infinity )" "6#,(*&&%\"AG6#\"\"!\"\"\"%\"xGF)F)&%\"BG6#F(F)-%$sumG6$*&,&*&&F&6# %\"nGF)-%%sinhG6#*(F6F)%#PiGF)F*F)F)F)*&&F,6#F6F)-F86#*(F6F)F;F),&F)F) F*!\"\"F)F)F)F)-%$cosG6#*(F6F)F;F)%\"yGF)F)/F6;F)%)infinityGF)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We use the remaining boundary conditions to com pute coefficients: " }}{PARA 0 "" 0 "" {TEXT -1 23 "Because \+ sin( " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 6 " y) = " }{XPPEDIT 18 0 "B[0]+sum(B[n]*sinh(n*Pi)*cos(n*Pi*y),n=1..infinity" "6#,&&%\"BG6 #\"\"!\"\"\"-%$sumG6$*(&F%6#%\"nGF(-%%sinhG6#*&F/F(%#PiGF(F(-%$cosG6#* (F/F(F4F(%\"yGF(F(/F/;F(%)infinityGF(" }}{PARA 0 "" 0 "" {TEXT -1 25 " We can compute the B's: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 18 " " }{XPPEDIT 18 0 "B[0];" "6#&%\" BG6#\"\"!" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(sin(Pi*y)*1,y = 0 .. \+ 1)/int(1^2,x = 0 .. 1);" "6#*&-%$intG6$*&-%$sinG6#*&%#PiG\"\"\"%\"yGF- F-F-F-/F.;\"\"!F-F--F%6$*$F-\"\"#/%\"xG;F1F-!\"\"" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 18 " " }{XPPEDIT 18 0 "B[n]* sinh(n*Pi) = int(sin(Pi*y)*cos(n*Pi*y),y = 0 .. 1)/int(cos(n*Pi*y)^2,y = 0 .. 1);" "6#/*&&%\"BG6#%\"nG\"\"\"-%%sinhG6#*&F(F)%#PiGF)F)*&-%$in tG6$*&-%$sinG6#*&F.F)%\"yGF)F)-%$cosG6#*(F(F)F.F)F8F)F)/F8;\"\"!F)F)-F 16$*$-F:6#*(F(F)F.F)F8F)\"\"#/F8;F?F)!\"\"" }{TEXT -1 14 " for n > \+ 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "bec ause 0 = " }{XPPEDIT 18 0 "u[x](1,y)" "6#-&%\"uG6#%\"xG6$\" \"\"%\"yG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "A[0]" "6#&%\"AG6#\"\"!" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(n*Pi*(cosh(n*Pi)*A[n]-B[n])*cos(n *Pi*y),n=1..infinity);" "6#-%$sumG6$**%\"nG\"\"\"%#PiGF(,&*&-%%coshG6# *&F'F(F)F(F(&%\"AG6#F'F(F(&%\"BG6#F'!\"\"F(-%$cosG6#*(F'F(F)F(%\"yGF(F (/F';F(%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "then " }{XPPEDIT 18 0 "A[0]=0" "6#/&%\"AG6 #\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "A[n]=B[n]/cosh(n*Pi)" "6 #/&%\"AG6#%\"nG*&&%\"BG6#F'\"\"\"-%%coshG6#*&F'F,%#PiGF,!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Here are the details." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Details" }}{PARA 0 "" 0 "" {TEXT -1 50 "We proceed to compute coefficients: first the B's" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "B[0]:=int(sin(Pi*y)*1,y=0..1)/int(1^2,y=0..1); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "for i from 1 to 6 do\n B[i]:=int(sin(Pi*y)*cos(i*Pi*y),y=0..1)/\n int(cos(i*Pi*y )^2,y=0..1)/sinh(i*Pi);\nod;" }}}{PARA 0 "" 0 "" {TEXT -1 11 "Now the \+ A's" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "for i from 1 to 6 do \n A[i]:=B[i]/cosh(i*Pi);\nod;" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Her e is a candidate for a solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "u:=(x,y)->B[0]+sum((A[n]*sinh(n*Pi*x)+B[n]*sinh(n*Pi*(1-x)))*c os(n*Pi*y),\n n=1..6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "D[1](u)(1,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 56 "When x = 0, this u should approximate the sine funct ion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot([sin(Pi*y),u(0, y)],y=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 68 "Here's checking the boun dary condition, and THAT U SATISFIES THE PDE" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "D[1](u)(1,y);\nD[2](u)(x,0);\nD[2](u)(x,1);\nsimpl ify(D[1,1](u)(x,y)+D[2,2](u)(x,y));" }}}{PARA 0 "" 0 "" {TEXT -1 75 "F inally, draw the graph. Guess what it should look like before you draw it." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot3d(u(x,y),x=0..1 ,y=0..1,axes=NORMAL,orientation=[-135,45]);" }}}{PARA 0 "" 0 "" {TEXT -1 47 "Can you guess what the contour lines look like?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plots[contourplot](u(x,y),x=0..1,y= 0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 74 "Would you guess that for each \+ x, the total \"heat\" is the same as at x = 0?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "int(u(x,y),y=0..1);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 28 "We work this second problem:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 " \+ " }{XPPEDIT 18 0 "0 = diff(u,`$`(x,2))+diff(u,` $`(y,2));" "6#/\"\"!,&-%%diffG6$%\"uG-%\"$G6$%\"xG\"\"#\"\"\"-F'6$F)-F +6$%\"yGF.F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Boundary conditions: " }{XPPEDIT 18 0 "u[x](0,y) = 0; " "6#/-&%\"uG6#%\"xG6$\"\"!%\"yGF*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[x](1,y)=0" "6#/-&%\"uG6#%\"xG6$\"\"\"%\"yG\"\"!" }}{PARA 0 "" 0 " " {TEXT -1 45 " " } {XPPEDIT 18 0 "u[y](x,0) = 1-cos(2*Pi*x);" "6#/-&%\"uG6#%\"yG6$%\"xG\" \"!,&\"\"\"F--%$cosG6#*(\"\"#F-%#PiGF-F*F-!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "u[y](x,1) = 1;" "6#/-&%\"uG6#%\"yG6$%\"xG\"\"\"F+" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 456 "Here is a way to think of this problem: first, it is not changing in time. And, second, material is flowing into the bottom of the rectangle and flowing out the top. Since nothing crosses the left or right side, there had better be as much going in the bottom as is \+ going out the top. The total going out the top is 1. We compute how mu ch is going in the bottom, confident I have made the problem well. Als o, we draw the graph of the input from the bottom." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 54 "int(1-cos(2*Pi*y),y=0..1);\nplot(1-cos(2*Pi* y),y=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "We know how to se parate variables. The differential equations will be" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " X '' + " } {XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#" }{TEXT -1 44 " X = 0, X '(0) = 0 = X '(1), and Y '' = " }{XPPEDIT 18 0 "lambda^2;" "6#*$% 'lambdaG\"\"#" }{TEXT -1 3 " Y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "Solutions for this will be X(x) = 1 or cos(n " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 13 " x) if n > 0 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 0 "" 0 "" {TEXT -1 69 " \+ Y(y) = 1 and y or cosh(n" }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 14 " y) and cosh(n" }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 17 " ( 1-y)) if n > 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "We can construct a general solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " U(x, y) = " } {XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "b[0];" "6#&%\"bG6#\"\"!" }{TEXT -1 5 " y + " }{XPPEDIT 18 0 "sum (a[n]*cosh(n*Pi*y)+b[n]*cosh(n*Pi*(1-y))*cos(n*Pi*x),n);" "6#-%$sumG6$ ,&*&&%\"aG6#%\"nG\"\"\"-%%coshG6#*(F+F,%#PiGF,%\"yGF,F,F,*(&%\"bG6#F+F ,-F.6#*(F+F,F1F,,&F,F,F2!\"\"F,F,-%$cosG6#*(F+F,F1F,%\"xGF,F,F,F+" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We have only to find the " }{TEXT 256 1 "a" }{TEXT -1 8 " 's and " }{TEXT 257 2 "b " }{TEXT -1 4 " 's." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 " Coming in from the bott om is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " f(x) = " }{XPPEDIT 18 0 "1-cos(2*Pi*x)" "6#,&\"\"\"F$-%$cosG6#*(\"\" #F$%#PiGF$%\"xGF$!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "diff(u,y);" "6#-%%diffG6$%\"uG%\"yG" }{TEXT -1 9 "(x, 0) = " }{XPPEDIT 18 0 "b[0]; " "6#&%\"bG6#\"\"!" }{TEXT -1 4 " - " }{XPPEDIT 18 0 "sum(b[n]*n*Pi*s inh(n*Pi)*cos(n*Pi*x),n);" "6#-%$sumG6$*,&%\"bG6#%\"nG\"\"\"F*F+%#PiGF +-%%sinhG6#*&F*F+F,F+F+-%$cosG6#*(F*F+F,F+%\"xGF+F+F*" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Going out the top is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " 1 = " }{XPPEDIT 18 0 "diff(u,y);" "6#-%%diffG6$%\"uG% \"yG" }{TEXT -1 10 "(x, 1) = " }{XPPEDIT 18 0 "b[0];" "6#&%\"bG6#\"\" !" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(n*Pi*a[n]*sinh(n*Pi)*cos(n*Pi *x),n);" "6#-%$sumG6$*,%\"nG\"\"\"%#PiGF(&%\"aG6#F'F(-%%sinhG6#*&F'F(F )F(F(-%$cosG6#*(F'F(F)F(%\"xGF(F(F'" }{TEXT -1 2 " ." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 172 "Finding the coefficien ts is now a job for Fourier Series. For this example, we can see what \+ the coefficients are by inspection. Is it not clear from the last equa tion that " }{XPPEDIT 18 0 "b[0];" "6#&%\"bG6#\"\"!" }{TEXT -1 17 " = \+ 1 and all the " }{TEXT 258 1 "a" }{TEXT -1 77 " 's are zero? From the \+ next to last equation, we can see by inspection that " }{XPPEDIT 18 0 "-b[2]*2*Pi*sinh(n*Pi) = -1;" "6#/,$**&%\"bG6#\"\"#\"\"\"F)F*%#PiGF* -%%sinhG6#*&%\"nGF*F+F*F*!\"\",$F*F1" }{TEXT -1 85 " and all the other b 's are zero. We can now make the u that satisfies this equation." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "D etails" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Here we define u, check that it satisfies the equation, and draw a gr aph." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "u:=(x,y)->y+cosh(2*P i*(1-y))*cos(2*Pi*x)/(2*Pi*sinh(2*Pi));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "diff(u(x,y),x,x)+diff(u(x,y),y,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "D[1](u)(0,y);\nD[1](u)(1,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "D[2](u)(x,0);\nD[2](u)(x,1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " We draw a graph of the sol ution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot3d(u(x,y),x=0. .1,y=0..1,axes=NORMAL,orientation=[-100,100]);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 29 "Assignment: Solve the problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 " \+ " }{XPPEDIT 18 0 "0 = diff(u,`$`(x,2))+diff(u,`$`(y,2 ));" "6#/\"\"!,&-%%diffG6$%\"uG-%\"$G6$%\"xG\"\"#\"\"\"-F'6$F)-F+6$%\" yGF.F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Boundary conditions: " }{XPPEDIT 18 0 "u(0,y) = sin(Pi*y); " "6#/-%\"uG6$\"\"!%\"yG-%$sinG6#*&%#PiG\"\"\"F(F." }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "u[x](1,y)=0" "6#/-&%\"uG6#%\"xG6$\"\"\"%\"yG\"\"!" } }{PARA 0 "" 0 "" {TEXT -1 45 " \+ " }{XPPEDIT 18 0 "u(x,0) = sin(Pi*x);" "6#/-%\"uG6$%\"xG\"\"!-%$s inG6#*&%#PiG\"\"\"F'F." }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[y](x,1)= 0" "6#/-&%\"uG6#%\"yG6$%\"xG\"\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }