{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 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 1 12 0 0 0 0 0 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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 } {PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 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 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 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 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}} {PARA 256 "" 0 "" {TEXT 260 42 "Section 5.5: Different Boundary Condit ions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 280 30 "Maple Packages for Section 5.5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "We exami ne a variety of boundary conditions that can be associated with the wa ve equation. We will provide illustrations for how these boundary cond itions would look in a vibrating string. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 16 "Fixed endpoints." }{TEXT -1 188 " Fixed endpoints is the situation we have considered to this point in \+ these notes. Here is a graph of a function that has this boundary con dition. Mathematically, the conditions have been" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 266 1 "u" }{TEXT -1 1 "(" }{TEXT 265 1 "t" }{TEXT -1 13 ", 0) = 0 and " }{TEXT 264 1 "u" }{TEXT -1 1 "(" } {TEXT 263 1 "t" }{TEXT -1 2 ", " }{TEXT 262 1 "L" }{TEXT -1 6 ") = 0. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "L:=Pi;\nu:=(t,x)->2*sin( x)*cos(t);" }}}{PARA 0 "" 0 "" {TEXT -1 93 "We check that this functio n satisfies the wave equation and is zero at the boundaries: 0 and " } {TEXT 261 1 "L" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "diff(u(t,x),t,t)-diff(u(t,x),x,x);\nu(t,0);\nu(t,L);" }}} {PARA 0 "" 0 "" {TEXT -1 31 "We graph the initial condition." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(u(0,x),x=0..L);" }}} {PARA 0 "" 0 "" {TEXT -1 126 "Here is an animation of the solution. Re member that it is what happens at the boundaries that is our interest \+ in this Section." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "animate( u(t,x),x=0..L,t=0..8, frames=40);" }}}{PARA 0 "" 0 "" {TEXT -1 49 "Fin ally, here is a graph of the solution surface." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot3d(u(t,x),x=0..L,t=0..2*Pi,axes=NORMAL,orien tation=[-160,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 19 "Elastic \+ attachment." }{TEXT -1 287 " This situation assumes we have a spring, \+ or other elastic device attached to the string. Such an arrangement wo uld tend to bring the string back from any displacement. How hard the \+ elastic device pulls the string back depends on how far it is displace d. Such conditions could be written" }}{PARA 0 "" 0 "" {TEXT -1 10 " \+ " }{XPPEDIT 18 0 "diff(u,x);" "6#-%%diffG6$%\"uG%\"xG" }{TEXT -1 11 "(t, 0) = k " }{TEXT 267 1 "u" }{TEXT -1 1 "(" }{TEXT 268 1 "t" }{TEXT -1 13 ", 0) and " }{XPPEDIT 18 0 "diff(u,x);" "6#-%%diffG6$ %\"uG%\"xG" }{TEXT -1 13 "(t, L) = - k " }{TEXT 269 1 "u" }{TEXT -1 7 "(t, L)." }}{PARA 0 "" 0 "" {TEXT -1 141 "Such conditions are reminisc ent of radiation cooling for heat diffusion problems. Here is an illus tration for how such a system would behave." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "L:=Pi/2;\nu:=(t,x)->cos(t)*sin(x+Pi/4);" }}}{PARA 0 "" 0 "" {TEXT -1 19 "We check that this " }{TEXT 270 1 "u" }{TEXT -1 57 " satisfies the wave equation and the boundary conditions." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "diff(u(t,x),t,t)-diff(u(t,x) ,x,x);\nD[2](u)(t,0)-u(t,0);\nD[2](u)(t,Pi/2)+u(t,L);" }}}{PARA 0 "" 0 "" {TEXT -1 33 "Here is the initial displacement." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(u(0,x),x=0..L,y=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Watch the boundaries." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "animate(u(t,x),x=0..Pi/2,t=0..L, frames=40);" }}} {PARA 0 "" 0 "" {TEXT -1 30 "Here is the solutions surface." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot3d(u(t,x),x=0..L,t=0..2*Pi,axes =NORMAL,orientation=[-30,70]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 20 "Frictionless sleeve." }{TEXT -1 233 " This models having the strin g attached to carts which hold the string with a horizontal tangent at the ends, but allows the string to move up and down as the cart runs \+ along a frictionless track. This situation can be described with" }} {PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "diff(u,x);" " 6#-%%diffG6$%\"uG%\"xG" }{TEXT -1 15 "(t, 0) = 0 and " }{XPPEDIT 18 0 "diff(u,x);" "6#-%%diffG6$%\"uG%\"xG" }{TEXT -1 12 "(t, L) = 0. " }} {PARA 0 "" 0 "" {TEXT -1 87 "In this illustration, the right end is fi xed, the left end is on a frictionless sleeve." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "L:=Pi/2;\nu:=(t,x)->sin(t+Pi/2)*cos(x);" }}} {PARA 0 "" 0 "" {TEXT -1 45 "We check the PDE and the boundary conditi ons." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "diff(u(t,x),t,t)-dif f(u(t,x),x,x);\nD[2](u)(t,0);\nu(t,L);" }}}{PARA 0 "" 0 "" {TEXT -1 33 "Here is the initial distribution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(u(0,x),x=0..L);" }}}{PARA 0 "" 0 "" {TEXT -1 80 "The animation is a good way to get an understanding for the boundary \+ conditions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "animate(u(t,x ),x=0..L,t=0..2*Pi, frames=40);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot3d(u(t,x),x=0..L,t=0..2*Pi,axes=NORMAL,orientation=[-40,55 ]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 27 "Changing Boundary Condit ion" }{TEXT -1 377 ". We suppose we have a half infinite string in thi s model and we change the boundary condition with time. Take the initi al conditions to be zero, and take U(t, 0) = b(t). We expect to see a \+ signal move down the string. Here is an analysis of the problem. With \+ no appeal to initial conditions or boundary conditions, we found in Se ction 5.2 that solutions should have the form " }{XPPEDIT 18 0 "psi; " "6#%$psiG" }{TEXT -1 12 "(x + c t) + " }{XPPEDIT 18 0 "phi;" "6#%$ph iG" }{TEXT -1 98 "(x - c t). Using that the initial conditions are zer o for x > 0, recall that we can conclude that " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 12 "(x) = 0 and " }{XPPEDIT 18 0 "phi;" "6#%$phiG " }{TEXT -1 30 "(x) = 0 for x > 0. The term " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 69 "(x + c t) will be zero since c > 0 and t > 0. Thus, whatever happens," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 15 " u(t, x) = " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 10 "(x - c t)," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 59 "and this is zero as long as x > c t. We ask how t o extend " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 107 " to the neg ative numbers. The answer must lie in the boundary condition. Recall t hat it did earlier, also. " }}{PARA 0 "" 0 "" {TEXT -1 37 " We kno w that u(t, 0) = b(t), so " }}{PARA 0 "" 0 "" {TEXT -1 18 " \+ " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 27 "(0 - c t) = u( t, 0) = b(t)." }}{PARA 0 "" 0 "" {TEXT -1 36 "So, for negative numbers n, we have " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 110 "(n) = b( -n/c). Here is an example. Think of standing at the end of a long rope and moving the end up and down." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "b:=x->sin(x);" }}}{PARA 0 "" 0 "" {TEXT -1 90 "From what came \+ above, here would be the solution for the wave equation with this boun dary." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "u:=(t,x)->piecewise (x " 0 "" {MPLTEXT 1 0 2 "c; " }}}{PARA 0 "" 0 "" {TEXT -1 34 "Watch the wave move down the rope." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "animate(u(t,x),x=0..20, t= 0..20, frames=40);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot3 d(u(t,x),x=0..20,t=0..20,axes=NORMAL,orientation=[-115,55]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 425 "The intent of this Section was to re call that there can be a variety of boundary conditions that change th e character of the solutions for the wave equation. Some of these were illustrated above. Before the reader grows weary with the endless var iations possible, we move to a new chapter and step up the dimension. \+ We next consider partial differential equations associated with the st eady state equation in multidimensions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 279 16 "Unassisted Maple" }}{PARA 0 "" 0 "" {TEXT -1 125 "This moving boundary problem suggests ideas ab out sending signals. Suppose we know the speed of the wave along the s tring is " }{TEXT 271 1 "c" }{TEXT -1 86 " = 3. We ask, how long will \+ it take for the peak of the wave to first reach the point " }{TEXT 272 1 "x" }{TEXT -1 166 " = 10? This is a problem that Maple can solve unassisted. We ask Maple to solve an equation involving the unknown a s part of the argument of a trigonometric function." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "First we define " } {TEXT 273 2 "u." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "u:=(t,x)- >piecewise(x " 0 "" {MPLTEXT 1 0 25 "solve(sin((t-10)/c)=1,t);" }}}{PARA 0 "" 0 " " {TEXT -1 64 "To get a visualization of this answer, we choose a part icular c." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c:=3;" }}}{PARA 0 "" 0 "" {TEXT -1 12 "Having this " }{TEXT 277 1 "c" }{TEXT -1 36 ", \+ watch to see that at the value of " }{TEXT 276 1 "t" }{TEXT -1 38 " fo und above, the wave first peaks at " }{TEXT 278 1 "x" }{TEXT -1 7 " = \+ 10. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "animate(u(t,x),x=0.. 10,t=0..10+Pi*c/2, frames=40);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "EMAIL: herod@math.gatech.edu or jhe rod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 38 "URL: http://www.math.gatec h.edu/~herod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 36 "Copyright \251 2003 by James V. Herod" }}{PARA 257 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }