{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 "" 1 18 0 0 255 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 11 "Lecture 11:" }}{PARA 256 "" 0 "" {TEXT 256 30 "Laplace's Equation in a Circle" }}{PARA 256 "" 0 "" {TEXT -1 33 "Mihaela B. Vajiac and Juan Tolosa" }}{PARA 257 " " 0 "" {TEXT -1 17 "PCMI, Summer 2003" }{TEXT 258 0 "" }{TEXT 259 0 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The solut ion of the problem" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Delta*u(r,theta) \+ = 0;" "6#/*&%&DeltaG\"\"\"-%\"uG6$%\"rG%&thetaGF&\"\"!" }{TEXT -1 4 " \+ if " }{XPPEDIT 18 0 "r < a;" "6#2%\"rG%\"aG" }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "u(a,theta) = f(theta);" "6#/-%\"uG6$%\"aG%&th etaG-%\"fG6#F(" }{TEXT -1 11 " for every " }{XPPEDIT 18 0 "theta;" "6# %&thetaG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "can be writte n as" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "u(r,theta) = A[0]+Sum((r/a)^n*( A[n]*cos(n*theta)+B[n]*sin(n*theta)),n = 1 .. infinity);" "6#/-%\"uG6$ %\"rG%&thetaG,&&%\"AG6#\"\"!\"\"\"-%$SumG6$*&)*&F'F.%\"aG!\"\"%\"nGF., &*&&F+6#F7F.-%$cosG6#*&F7F.F(F.F.F.*&&%\"BG6#F7F.-%$sinG6#*&F7F.F(F.F. F.F./F7;F.%)infinityGF." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " where" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "A[0] = 1/2/Pi*Int(f(phi),phi = -Pi .. Pi);" "6#/&%\"AG6#\"\"!**\"\"\"F)\"\"#!\"\"%#PiGF+-%$IntG6$-% \"fG6#%$phiG/F3;,$F,F+F,F)" }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "A[n] = 1/Pi*int(f(phi)*cos(n*phi),phi = -Pi .. Pi);" "6 #/&%\"AG6#%\"nG*(\"\"\"F)%#PiG!\"\"-%$intG6$*&-%\"fG6#%$phiGF)-%$cosG6 #*&F'F)F3F)F)/F3;,$F*F+F*F)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "B[n] = 1/Pi*int(f(phi)*sin (n*phi),phi = -Pi .. Pi);" "6#/&%\"BG6#%\"nG*(\"\"\"F)%#PiG!\"\"-%$int G6$*&-%\"fG6#%$phiGF)-%$sinG6#*&F'F)F3F)F)/F3;,$F*F+F*F)" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The solution can also be wri tten as" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "u(r,theta) = 1/(2*Pi)*int((a ^2-r^2)/(a^2-2*a*r*cos(theta-phi)+r^2)*f(phi),phi = -Pi .. Pi);" "6#/- %\"uG6$%\"rG%&thetaG*(\"\"\"F**&\"\"#F*%#PiGF*!\"\"-%$intG6$*(,&*$%\"a GF,F**$F'F,F.F*,(*$F5F,F***F,F*F5F*F'F*-%$cosG6#,&F(F*%$phiGF.F*F.*$F' F,F*F.-%\"fG6#F>F*/F>;,$F-F.F-F*" }{TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "Separating the variables" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "First we compute the coefficients of the series" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assume(n, posint):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A0 := 1/2/Pi*int(f(theta), t heta = -Pi..Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "A := n -> 1/Pi*int(f(theta)*cos(n*theta), theta = -Pi..Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "B := n -> 1/Pi*int(f(theta)*sin(n*t heta), theta = -Pi..Pi):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Next \+ we write down the " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 12 "th har monic:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "u := (n, r, theta ) -> A0 + (r/a)^n*(A(n)*cos(n*theta)+B(n)*sin(n*theta)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 43 "Example 1: discontinuous boundary condition" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 64 "We plot the initial condition for the case when th e radius is 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f:= theta -> piecewise(theta<=0, -1, 1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(subs(a=1,f(theta)), theta = -Pi..Pi);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(theta);" "6#-%\"fG6#%&thetaG" }{TEXT -1 30 " is odd in this case, all the " } {XPPEDIT 18 0 "A[n];" "6#&%\"AG6#%\"nG" }{TEXT -1 12 "'s are zero:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "A0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A(n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B(n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{TEXT 260 1 "n " }{TEXT -1 11 " even, the " }{XPPEDIT 18 0 "B[n];" "6#&%\"BG6#%\"nG" }{TEXT -1 12 "'s are zero:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "B(2*n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "This is the n-th ei genfunction for the problem:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "u(n,r,theta);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "This is an approximation of the solution, choosing first 10 (nonzero) terms of t he Fourier series:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "solu \+ := (r, theta) -> sum(u(2*n-1,r,theta), n=1..10);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 66 "This is how the approximation looks when the radiu s is equal to 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(a= 1,solu(r,theta));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "This is a sh ortcut, in case we want just to show the plot and skip the math:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "shortcutsol:= (r, theta) -> 4*r/Pi*sin(theta)+4/3*r^3/Pi*sin(3*theta)+4/5*r^5/Pi*sin(5*theta)+4/7 *r^7/Pi*sin(7*theta)+4/9*r^9/Pi*sin(9*theta)+4/11*r^11/Pi*sin(11*theta )+4/13*r^13/Pi*sin(13*theta)+4/15*r^15/Pi*sin(15*theta)+4/17*r^17/Pi*s in(17*theta)+4/19*r^19/Pi*sin(19*theta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "addcoords(z_cylindrical,[z,r,theta],\n[r*cos(theta),r *sin(theta),z]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "plot3d( shortcutsol(r,theta),r=0..1,theta=-Pi..Pi,\ncoords=z_cylindrical,axes= BOXED, grid=[30,30]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "This is an animation of a one-parameter family of harmonic functions that sta rt with the zero boundary condition and end with our discontinuous bou ndary condition:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "animat e3d(t*shortcutsol(r,theta),r=0..1,theta=-Pi..Pi,t=0..1,\ncoords=z_cyli ndrical,axes=BOXED, grid = [30,30],frames = 15);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 89 "This is a plot of the function on a circle very cl ose to the boundary (with radius 0.95):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(shortcutsol(.95, theta), theta=-Pi..Pi);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "And this is an animation of the s olution on concentric circles, starting at 0 and ending at the boundar y r = 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "animate(shortcu tsol(r, theta), theta=-Pi..Pi, r=0..1,frames=60);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 "Ex ample 2: polynomial boundary condition" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "This is the given boundary condition (the restriction of \+ this polynomial to the unit circle):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "v:= (x, y) -> x^3-y^3:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "This is the polynomial harmonic function" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "u := (x, y) -> (3/4)*(x-y) + (1/4)* (x^3-3*x*y^2+ 3*x^2*y-y^3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "We check that u is indeed harmonic:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "uxx := diff(u(x,y),x$2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "uyy:=diff(u(x,y), y$2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "uxx + uyy;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 " We check that u satisfies the boundary condition;" }}{PARA 0 "" 0 "" {TEXT -1 62 "that is, we check that u=v on the boundary of the unit ci rcle:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "u(x,y) - v(x,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subs(\{x = cos(theta), y \+ = sin(theta)\}, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simp lify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "This is a plot of the solution on the square [-1, 1]x[-1, 1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot3d(u(x,y), x=-1..1, y=-1..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This is a restriction of the previous plot to the unit circle:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot3d([r* cos(t), r*sin(t), u(r*cos(t), r*sin(t))],\nr=0..1, t=0..2*Pi, axes=BOX ED);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "This is an animation of \+ a one-parameter family of harmonic functions which starts at minus the given polynomial boundary condition and ends at the given boundary co ndition:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "animate3d([r*c os(t), r*sin(t), s*u(r*cos(t), r*sin(t))],\nr=0..1, t=0..2*Pi, s=-1..1 , axes=BOXED, style = PATCHNOGRID, frames=30);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "The \+ Poisson Kernel" }}{PARA 0 "" 0 "" {TEXT -1 63 "We plot the Poisson ker nel to show it is positive and tends to " }{XPPEDIT 18 0 "infinity;" " 6#%)infinityG" }{TEXT -1 10 " at (0,1)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Po isson := (r, s) -> (a^2-r^2)/(a^2-2*a*r*cos(s)+r^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "addcoords(z_cylindrical,[z,r,theta],\n[r* cos(theta),r*sin(theta),z]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "T his was shamelessly copied from Jon Jake's web site (only his looks mu ch better):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "plot3d(Pois son(r,theta),r=0..1,theta=-Pi..Pi,\ncoords=z_cylindrical,axes=BOXED, g rid=[50,50], view=0..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "This one is also much better in Jon's web site:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "densityplot(Poisson(sqrt(x^2+y^2),arctan(y,x)),x= -1..0.9,y=-1..1, axes=none, style= patchnogrid,colorstyle=HUE, grid=[5 0,90],scaling=CONSTRAINED);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Th is was done with Andy's help; it looks nice:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "densityplot([r*cos(t),r*sin(t),Poisson(r,t)],r= 0..0.9,t=0..2*Pi, axes=none, style= patchnogrid,colorstyle=HUE, grid=[ 40,72],scaling=CONSTRAINED);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "5 18 0 0" 126 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }