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{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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 71 "Section \+
7.2: Two Dimensional Diffusion with Neumann Boundary Conditions" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 274 30 "
Maple Packages for Section 7.2" }}{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 65 "In Lecture 32, we conside
red boundary conditions that are called " }{TEXT 258 29 "Dirichlet Bou
ndary Conditions" }{TEXT -1 36 ". Here, we will work a problem with " 
}{TEXT 259 18 "Neumann Conditions" }{TEXT -1 64 " corresponding to an \+
insulated surface in the diffusion of heat." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We illustrate this type problem
 with a simple example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 257 36 "Consider the heat conduction problem" }{TEXT -1 
100 " in a rectangle where there is insulation on all boundaries and t
he initial condition is as follows:" }}{PARA 0 "" 0 "" {TEXT -1 48 "  \+
                                              " }{XPPEDIT 18 0 "du/dt \+
= Delta(u);" "6#/*&%#duG\"\"\"%#dtG!\"\"-%&DeltaG6#%\"uG" }{TEXT -1 1 
"," }}{PARA 0 "" 0 "" {TEXT -1 26 "Insulated boundaries:     " }
{XPPEDIT 18 0 "du/dy;" "6#*&%#duG\"\"\"%#dyG!\"\"" }{TEXT -1 10 "(t,x,
0) = " }{XPPEDIT 18 0 "du/dy;" "6#*&%#duG\"\"\"%#dyG!\"\"" }{TEXT -1 
15 "(t,x,1)= 0,    " }{XPPEDIT 18 0 "du/dx;" "6#*&%#duG\"\"\"%#dxG!\"
\"" }{TEXT -1 10 "(t,0,y) = " }{XPPEDIT 18 0 "du/dx;" "6#*&%#duG\"\"\"
%#dxG!\"\"" }{TEXT -1 11 "(t,1,y)= 0." }}{PARA 0 "" 0 "" {TEXT -1 65 "
Initial condition:              u(0, x, y) = 100 x (1-x) y (1-y)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Here are \+
the tasks we will accomplish:" }}{PARA 0 "" 0 "" {TEXT -1 109 "(A) Giv
e three ODE's, together with boundary conditions, whose solutions lead
 to a solution for this problem." }}{PARA 0 "" 0 "" {TEXT -1 42 "(B) G
ive the solution for the three ODE's." }}{PARA 0 "" 0 "" {TEXT -1 42 "
(C) Give the general solution for the PDE." }}{PARA 0 "" 0 "" {TEXT 
-1 78 "(D) Approximate the particular solution that goes with this ini
tial condition." }}{PARA 0 "" 0 "" {TEXT -1 36 "(E) Animate a graph of
 the solution." }}{PARA 0 "" 0 "" {TEXT -1 55 "(F) Compute the total h
eat for the system for all time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "(A.) Here
 are three ODE's and boundary conditions:" }}{PARA 0 "" 0 "" {TEXT -1 
17 "          X '' + " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#
" }{TEXT -1 29 " X = 0,  X '(0) = X '(1) = 0." }}{PARA 0 "" 0 "" 
{TEXT -1 16 "         Y '' + " }{XPPEDIT 18 0 "mu^2;" "6#*$%#muG\"\"#
" }{TEXT -1 29 " Y = 0,  Y '(0) = Y '(1) = 0." }}{PARA 0 "" 0 "" 
{TEXT -1 18 "          T ' + ( " }{XPPEDIT 18 0 "lambda^2+mu^2;" "6#,&
*$%'lambdaG\"\"#\"\"\"*$%#muGF&F'" }{TEXT -1 9 " ) T = 0." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "(B.) Solutions for
 these equations are" }}{PARA 0 "" 0 "" {TEXT -1 22 "        X(x) = co
s( n " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 19 " x),  Y(y) = cos(m
 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 21 " y),  T(t) = exp( - (
" }{XPPEDIT 18 0 "n^2+m^2;" "6#,&*$%\"nG\"\"#\"\"\"*$%\"mGF&F'" }
{TEXT -1 3 " ) " }{XPPEDIT 18 0 "Pi^2;" "6#*$%#PiG\"\"#" }{TEXT -1 4 "
 t )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "(
C.) The general solution for the pde is" }}{PARA 0 "" 0 "" {TEXT -1 
17 "      u(t,x,y) = " }{XPPEDIT 18 0 "sum(sum(A[mn]*cos(n*Pi*x)*cos(m
*Pi*y)*exp(-(n^2+m^2)*Pi^2*t),n),m);" "6#-%$sumG6$-F$6$**&%\"AG6#%#mnG
\"\"\"-%$cosG6#*(%\"nGF-%#PiGF-%\"xGF-F--F/6#*(%\"mGF-F3F-%\"yGF-F--%$
expG6#,$*(,&*$F2\"\"#F-*$F8FAF-F-*$F3FAF-%\"tGF-!\"\"F-F2F8" }{TEXT 
-1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 62 "This is so important I will c
heck an approximation of the sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 15 "Here defines u." }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 100 "u:=(t,x,y)->sum(sum(A[m,n]*cos(n*Pi*x)*cos(m*Pi*y
)*\n       exp(-(n^2+m^2)*Pi^2*t),n=0..10),m=0..10);" }}}{PARA 0 "" 0 
"" {TEXT -1 36 "Here checks the boundary conditions." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 63 "D[3](u)(t,x,0);\nD[3](u)(t,x,1);\nD[2](u)
(t,0,y);\nD[2](u)(t,1,y);" }}}{PARA 0 "" 0 "" {TEXT -1 20 "Here checks
 the PDE." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "D[1](u)(t,x,y)-
D[2,2](u)(t,x,y)-D[3,3](u)(t,x,y);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 57 "(D.) We compute several terms of the par
ticular solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "100*x*(
1-x)*y*(1-y)=u(0,x,y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "
for n from 0 to 10 do\n  for m from 0 to 10 do\n    A[n,m]:=100*int(x*
(1-x)*cos(n*Pi*x),x=0..1) / \n            int(cos(n*Pi*x)^2,x=0..1)\n \+
           * int(y*(1-y) *cos(m*Pi*y),y=0..1)/int(cos(m*Pi*y)^2,y=0..1
);\n  end do: \nend do:\nn:='n': m:='m':" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 98 "In order to check my
 solution, I compare the graph of the initial value with the initial f
unction." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot3d(u(0,x,y),
x=0..1,y=0..1,axes=normal, shading=zhue);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 68 "plot3d(100*x*(1-x)*y*(1-y),x=0..1,y=0..1,axes=norma
l, shading=zhue);" }}}{PARA 0 "" 0 "" {TEXT -1 160 "Note that there is
 a small error along the boundaries. This error could be made smaller \+
by computing more than the 121 coefficients that we have computed abov
e." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot([u(0,x,0)],x=0..1
,y=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 42 "(E.) Here is an animation of the solution." }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 66 "animate3d(u(t,x,y),x=0..1,y=0..1,t=0..1/8,axe
s=normal, frames=30);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 88 "(F.) We compute the total heat and verify that it is
 the same as the initial total heat." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 87 "int(int(u(t,x,y),x=0..1),y=0..1);\nTotHeat:=int(int(1
00*x*(1-x)*y*(1-y),x=0..1),y=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 153 "Finally, we recall how to make changes in case the bound
ary conditions were not homogeneous. A quick review for working such a
 problem seems appropriate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 49 "Non homogeneous Neumann Type Boundary C
onditions." }}{PARA 0 "" 0 "" {TEXT -1 79 "Suppose the partial differe
ntial equation and boundary conditions have the form" }}{PARA 0 "" 0 "
" {TEXT -1 48 "                                                " }
{XPPEDIT 18 0 "du/dt = Delta(u);" "6#/*&%#duG\"\"\"%#dtG!\"\"-%&DeltaG
6#%\"uG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 25 "Boundary Condi
tions:     " }{XPPEDIT 18 0 "du/dy;" "6#*&%#duG\"\"\"%#dyG!\"\"" }
{TEXT -1 14 "(t,x,0) = 0,  " }{XPPEDIT 18 0 "du/dy;" "6#*&%#duG\"\"\"%
#dyG!\"\"" }{TEXT -1 16 "(t,x,1)= -1,    " }{XPPEDIT 18 0 "du/dx;" "6#
*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 14 "(t,0,y) = 0,  " }{XPPEDIT 18 0 
"du/dx;" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 11 "(t,1,y)= 1." }}
{PARA 0 "" 0 "" {TEXT -1 64 "Initial condition:              u(0, x, y
) = x (1-x) y (1-y) + (" }{XPPEDIT 18 0 "x^2-y^2;" "6#,&*$%\"xG\"\"#\"
\"\"*$%\"yGF&!\"\"" }{TEXT -1 6 ") / 2." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 260 12 "Note changes" }{TEXT -1 1 " " }
{TEXT 261 26 "from the previous problem:" }}{PARA 0 "" 0 "" {TEXT -1 
10 "      The " }{TEXT 262 19 "boundary conditions" }{TEXT -1 88 " hav
e been changed to model heat coming in at the top and going out the ri
ght side. The " }{TEXT 263 18 "initial conditions" }{TEXT -1 100 " hav
e been changed so that the results computed at the beginning of this w
orksheet can be used here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 148 "The pattern for working problems of this type ha
ve been set. First, we solve the steady state problem. The steady stat
e problem will have this form:" }}{PARA 0 "" 0 "" {TEXT -1 59 "PDE:   \+
                                                0 = " }{XPPEDIT 18 0 "
Delta;" "6#%&DeltaG" }{TEXT -1 4 " v, " }}{PARA 0 "" 0 "" {TEXT -1 25 
"Boundary Conditions:     " }{XPPEDIT 18 0 "dv/dy;" "6#*&%#dvG\"\"\"%#
dyG!\"\"" }{TEXT -1 14 "(t,x,0) = 0,  " }{XPPEDIT 18 0 "dv/dy;" "6#*&%
#dvG\"\"\"%#dyG!\"\"" }{TEXT -1 16 "(t,x,1)= -1,    " }{XPPEDIT 18 0 "
dv/dx;" "6#*&%#dvG\"\"\"%#dxG!\"\"" }{TEXT -1 14 "(t,0,y) = 0,  " }
{XPPEDIT 18 0 "dv/dx;" "6#*&%#dvG\"\"\"%#dxG!\"\"" }{TEXT -1 11 "(t,1,
y)= 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
106 "We have addressed this problem previously. We found the following
 solution which we define and check here." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "v:=(x,y)->(x^2-y^2)/2
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "D[2](v)(x,0);\nD[2](v)
(x,1);\nD[1](v)(0,y);\nD[1](v)(1,y);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "D[1,1](v)(t,x,y)+D[2,2](v)(t,x,y);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 50 "Next we \+
find the transient solution which we call " }{TEXT 264 1 "w" }{TEXT 
-1 151 ". To do this we use homogeneous boundary conditions and an ini
tial condition to reflect that the solution u for the original problem
 will be defined by" }}{PARA 0 "" 0 "" {TEXT -1 22 "                  \+
    " }{TEXT 265 1 "u" }{TEXT -1 12 "(t, x, y) = " }{TEXT 266 1 "w" }
{TEXT -1 12 "(t, x, y) + " }{TEXT 267 1 "v" }{TEXT -1 7 "(x, y)." }}
{PARA 0 "" 0 "" {TEXT -1 145 "The initial conditions of this non-homog
eneous problem have been created so that the transient solution is exa
ctly the problem we worked above:  " }{TEXT 273 1 "w" }{TEXT -1 12 "(t
, x, y) = " }{TEXT 271 1 "u" }{TEXT -1 12 "(t, x, y) - " }{TEXT 272 6 
"v(x,y)" }{TEXT -1 28 ". We use these results here." }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 98 "w:=(t,x,y)->sum(sum(A[m,n]*cos(n*Pi*x)*cos(
m*Pi*y)*\n       exp(-(n^2+m^2)*Pi^2*t),n=0..4),m=0..4);" }}}{PARA 0 "
" 0 "" {TEXT -1 70 "The solution for the problem in this section shoul
d now be the sum of " }{TEXT 268 1 "w" }{TEXT -1 5 " and " }{TEXT 269 
1 "v" }{TEXT -1 16 ". We check this." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "u:=unapply(w(t,x,y)+v(x,y),(t,x,y)):" }}}{PARA 0 "" 
0 "" {TEXT -1 33 "We check the boundary conditions." }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 63 "D[3](u)(t,x,0);\nD[3](u)(t,x,1);\nD[2](u)(t
,0,y);\nD[2](u)(t,1,y);" }}}{PARA 0 "" 0 "" {TEXT -1 17 "We check the \+
PDE." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "D[1](u)(t,x,y)-D[2,2
](u)(t,x,y)-D[3,3](u)(t,x,y);" }}}{PARA 0 "" 0 "" {TEXT -1 59 "We chec
k the initial conditions by comparing the graphs of " }{TEXT 270 1 "u
" }{TEXT -1 14 "(0, x, y) and " }}{PARA 0 "" 0 "" {TEXT -1 33 "       \+
  100* x (1-x) y (1-y) - (" }{XPPEDIT 18 0 "x^2-y^2;" "6#,&*$%\"xG\"\"
#\"\"\"*$%\"yGF&!\"\"" }{TEXT -1 95 ") / 2.\nIn order to see that ther
e are two graphs here, we displace the second by an amount 0.1." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "plot3d(\{u(0,x,y),100*x*(1-
x)*y*(1-y)+(x^2-y^2)/2+0.1\},\n         x=0..1,y=0..1,axes=normal, sha
ding=zhue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "plot3d(\{u(1
,x,y),TotHeat+(x^2-y^2)/2+0.1\},\n         x=0..1,y=0..1,axes=normal, \+
shading=zhue);" }}}{PARA 0 "" 0 "" {TEXT -1 132 "Finally, to give us i
ntuition, we draw the animation as the solution moves from the initial
 condition to the steady state condition." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "animate3d(u(t,x,y),x=0..1,y=0..1,t=0..1/20, axes=norm
al, frames=30, shading=zhue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 50 "EMAIL: herod@math.gatech.edu   or   jherod@tds.net" }}
{PARA 0 "" 0 "" {TEXT -1 38 "URL: http://www.math.gatech.edu/~herod" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 36 "Copyri
ght \251  2003  by James V. Herod" }}{PARA 256 "" 0 "" {TEXT -1 19 "Al
l rights reserved" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 1 1 2 33 1 1 }