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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 33 "The One Dimensional Heat
 Equation" }{TEXT 259 0 "" }{TEXT 260 0 "" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 36 "The one dimensional heat conduction " }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 274 "We want to consider the problem of heat conductin
g in a medium (without currents or radiation) in the one dimensional c
ase. We are given a wire which has a given distribution of temperature
 at time t=0. The question is how the heat is conducted through the bo
dy of the wire." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 46 "The situation can be modeled by the so called " }{TEXT 
261 13 "heat equation" }{TEXT -1 144 ". It is an example of a partial \+
differential equation (PDE). If u(x,t) is the temperature at position \+
x and time t the equation looks like this:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(PDEtools):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "heat := diff(u(x,t),t)=kappa*diff(u
(x,t),x,x);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%heatG/-%%diffG6$-%\"uG6$%\"xG%\"tGF-*&%&kappaG\"\"\"
-F'6$F)-%\"$G6$F,\"\"#F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The c
onstant " }{XPPEDIT 18 0 "kappa" "6#%&kappaG" }{TEXT -1 49 " depends o
n the heat conductivity of the medium. " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 237 "The heat equation also shows what a
 PDE is in general: We are given some kind of relationship between a f
unction u(x,t) and/or its partial derivatives. In the one dimensional \+
case this can be partial derivatives in time t or in space x. " }}
{PARA 0 "" 0 "" {TEXT -1 172 "The problem is to find functions u(x,t) \+
which satisfy the given relationship and sometimes satisfy additional \+
conditions (initital value conditions or boundary conditions)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 270 "PDEs lik
e the heat equation (but usually much more complex) very often arise i
n modeling problems in science, engineering and financial engineering.
 Sometimes we are able to find exact solutions using analytic methods.
 Maple is also capable of solving some PDEs via the " }{TEXT 262 10 "p
dsolve( )" }{TEXT -1 16 " command (Click " }{HYPERLNK 17 "here" 2 "pds
olve" "" }{TEXT -1 199 " for details of this command). However, often \+
a PDE cannot be solved analytically and our only chance is a numerical
 (and therefore approximate) solution (we will discuss that later in t
his lecture)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 22 "Solution by separation" }}{PARA 0 "" 0 "" {TEXT -1 
296 "PDEs like the heat equation (but usually much more complex) very \+
often arise in modeling problems in science, engineering and financial
 engineering. Sometimes we are able to find exact solutions using anal
ytic methods. Maple is also capable of solving some PDEs via the pdsol
ve( ) command (Click " }{HYPERLNK 17 "here" 2 "pdsolve" "" }{TEXT -1 
199 " for details of this command). However, often a PDE cannot be sol
ved analytically and our only chance is a numerical (and therefore app
roximate) solution (we will discuss that later in this lecture)." }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "A typical technique to solve this \+
type of equations is " }{TEXT 263 13 "by separation" }{TEXT -1 155 ". \+
This means we assume that our solution u(x,t) is of the form u(x,t)=X(
x)T(t). From the following output, we can see that Maple also follow t
his strategy:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ansatz := pdsolve(
heat,u(x,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ansatzG-%'&whereG6
$/-%\"uG6$%\"xG%\"tG*&-%$_F1G6#F,\"\"\"-%$_F2G6#F-F27#<$/-%%diffG6$F3F
-*(%&kappaGF2&%#_cG6#F2F2F3F2/-F:6$F/-%\"$G6$F,\"\"#*&F>F2F/F2" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "Maple can even compute a general \+
solution from this ansatz, however it cannot include the boundary and \+
initial conditions." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "anSol := build(ansatz);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
anSolG/-%\"uG6$%\"xG%\"tG,&**-%$expG6#*&-%%sqrtG6#&%#_cG6#\"\"\"F7F)F7
F7%$_C3GF7-F.6#*(%&kappaGF7F4F7F*F7F7%$_C1GF7F7*&*(F8F7F9F7%$_C2GF7F7F
-!\"\"F7" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 56 "Solving the initial-
boundary value problem by separation" }}{PARA 0 "" 0 "" {TEXT -1 73 "W
hat we can do, however, is reproduce the solution procedure using Mapl
e:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eq := subs(u(x,t)=X(x)
*T(t),heat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/-%%diffG6$*&-%
\"XG6#%\"xG\"\"\"-%\"TG6#%\"tGF.F2*&%&kappaGF.-F'6$F)-%\"$G6$F-\"\"#F.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "As X(x) is independent of  t \+
and T(t) of x the partial derivatives become much easier." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "sep := eq/X(x)/T(t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$sepG/*&-%%diffG6$-%\"TG6#%\"tGF-\"\"\"F*!\"\"*&*&%&k
appaGF.-F(6$-%\"XG6#%\"xG-%\"$G6$F8\"\"#F.F.F5F/" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 290 "We now have \"separated\" the variables: each sid
e of the equation depends only on one variable. The only way this equa
tion can be satisfied is if both sides are constant. We only consider \+
the case of negative values (this is the case that is of physical inte
rest) and denote the constant by " }{XPPEDIT 18 0 "-omega^2;" "6#,$*$%
&omegaG\"\"#!\"\"" }{TEXT -1 53 ". So we have for the left hand side o
f the equation: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "lhs(sep)= -omeg
a^2;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%%diffG6$-%
\"TG6#%\"tGF+\"\"\"F(!\"\",$*$)%&omegaG\"\"#F,F-" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 138 "This is an ordinary differential equation (as T(t
) is a function of one variable). This equation can easily be solved. \+
We can use Maple's " }{TEXT 264 10 "dsolve( ) " }{TEXT -1 33 "command \+
and obtain the following:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Tsol :
= dsolve(%,T(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%TsolG/-%\"TG6#
%\"tG*&%$_C1G\"\"\"-%$expG6#,$*&)%&omegaG\"\"#F,F)F,!\"\"F," }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The same can be done for the right
 hand side of the equation:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rhs(
sep)= -omega^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Xsol := dsolve(%
,X(x));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&*&%&kappa
G\"\"\"-%%diffG6$-%\"XG6#%\"xG-%\"$G6$F.\"\"#F'F'F+!\"\",$*$)%&omegaGF
2F'F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%XsolG/-%\"XG6#%\"xG,&*&%$_
C1G\"\"\"-%$cosG6#*&*&%&omegaGF-F)F-F-*$-%%sqrtG6#%&kappaGF-!\"\"F-F-*
&%$_C2GF--%$sinGF0F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "We now \+
have solutions X(x) and T(t). In " }{XPPEDIT 18 0 "Tsol;" "6#%%TsolG" 
}{TEXT -1 43 " we have the constant (due to integration) " }{TEXT 268 
3 "_C1" }{TEXT -1 8 " and in " }{XPPEDIT 18 0 "Xsol;" "6#%%XsolG" }
{TEXT -1 15 " the constants " }{TEXT 269 3 "_C1" }{TEXT -1 6 "  and " 
}{TEXT 270 6 "_C2.  " }{TEXT -1 15 "The constants _" }{TEXT 265 2 "C1
" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "Tsol" "6#%%TsolG" }{TEXT -1 6 " a
nd _" }{TEXT 266 3 "C1 " }{TEXT -1 3 "in " }{XPPEDIT 18 0 "Xsol" "6#%%
XsolG" }{TEXT 267 1 " " }{TEXT -1 78 "are not the same of course, so w
e need to rename them (contrary, the constant " }{XPPEDIT 18 0 "omega;
" "6#%&omegaG" }{TEXT 271 2 "  " }{TEXT 272 3 "is " }{TEXT -1 56 " rea
lly the same constant due to our separation ansatz):" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "Xsol := subs(_C2=_C3,_C1=_C2,%);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%XsolG/-%\"XG6#%\"xG,&*&%$_C2G\"\"\"-%$cosG6#*&*
&%&omegaGF-F)F-F-*$-%%sqrtG6#%&kappaGF-!\"\"F-F-*&%$_C3GF--%$sinGF0F-F
-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The complete solution of our
 problem therefore is " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "usol:=rhs
(Tsol)*rhs(Xsol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%usolG*(%$_C1G
\"\"\"-%$expG6#,$*&)%&omegaG\"\"#F'%\"tGF'!\"\"F',&*&%$_C2GF'-%$cosG6#
*&*&F.F'%\"xGF'F'*$-%%sqrtG6#%&kappaGF'F1F'F'*&%$_C3GF'-%$sinGF7F'F'F'
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The constants " }{XPPEDIT 18 
0 "omega;" "6#%&omegaG" }{TEXT 273 15 ", _C1, _C2, _C3" }{TEXT -1 339 
" can be chosen arbitrarily. Usually we are given some initial value c
onditions e.g. the values u(x,t=0) which would in our case be the temp
erature distribution at t=0. Using this additional information we can \+
calculate the constants and obtain a unique solution. In this exercise
 we just choose the constants and the heat conductivity k by" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 61 "usol_par := subs(_C1=1/2,_C2=1,_C3=1/2,om
ega=1,kappa=4,usol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)usol_parG,$
*&-%$expG6#,$%\"tG!\"\"\"\"\",&-%$cosG6#,$*&-%%sqrtG6#\"\"%F-%\"xGF-#F
-F7F-*&#F-\"\"#F--%$sinGF1F-F-F-F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 29 "Now we can plot the solution." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 50 "plot3d(usol_par,x=-2*Pi..2*Pi,t=0..10,axes=boxed);" }{TEXT -1 0 
"" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%%GRIDG6%;
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{PARA 0 "" 0 "" {TEXT -1 185 "Maple offers a nice feature: We can anim
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tart." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 0 ">
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2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 14 "Fourier method" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "In \+
our homework, we have shown that the function " }{XPPEDIT 18 0 "u(x,t)
 = sum(c[k]*exp(-k^2*Pi^2*t)*sin(k*Pi*x),k = 1 .. infinity);" "6#/-%\"
uG6$%\"xG%\"tG-%$sumG6$*(&%\"cG6#%\"kG\"\"\"-%$expG6#,$*(F0\"\"#%#PiGF
7F(F1!\"\"F1-%$sinG6#*(F0F1F8F1F'F1F1/F0;F1%)infinityG" }}{PARA 0 "" 
0 "" {TEXT -1 109 "is a solution candidate for the 1D heat equation fo
r homogenous boundary conditions and the initial condition" }}{PARA 0 
"" 0 "" {XPPEDIT 18 0 "f(x) = sum(c[k]*sin(k*Pi*x),k=1..infinity);" "6
#/-%\"fG6#%\"xG-%$sumG6$*&&%\"cG6#%\"kG\"\"\"-%$sinG6#*(F/F0%#PiGF0F'F
0F0/F/;F0%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The coeff
icients " }{XPPEDIT 18 0 "c[k]" "6#&%\"cG6#%\"kG" }{TEXT -1 27 " are g
iven by the integral " }{XPPEDIT 18 0 "c[k] = 2*int( f(x)*sin(k*Pi*x),
x=0..1);" "6#/&%\"cG6#%\"kG*&\"\"#\"\"\"-%$intG6$*&-%\"fG6#%\"xGF*-%$s
inG6#*(F'F*%#PiGF*F2F*F*/F2;\"\"!F*F*" }{TEXT -1 37 " if an analytical
 representation for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT 
-1 10 " is known." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 
7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefin
ed\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f := x->x;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowG
F(9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "We will not be able
 to compute the inifite sums. Thus, we define a maximal value for k:" 
}{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "kmax := 10;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%kmaxG\"#5" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 82 "The coefficients of the fourier sine transform of the \+
right hand side are given by" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "c := seq( 2*int( f(x)*sin(k*Pi*x), x=0..1),\n        \+
  k = 1..kmax);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG6,,$*&\"\"\"F
(%#PiG!\"\"\"\"#,$F'F*,$F'#F+\"\"$,$F'#F*F+,$F'#F+\"\"&,$F'#F*F/,$F'#F
+\"\"(,$F'#F*\"\"%,$F'#F+\"\"*,$F'#F*F4" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 60 "The function approximating the right hand side is therefo
re:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "fappr := \+
unapply( sum( c[k]*sin(k*Pi*x), k=1..kmax), x);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%&fapprGR6#%\"xG6\"6$%)operatorG%&arrowGF(,6*&-%$sinG6
#*&%#PiG\"\"\"9$F3F3F2!\"\"\"\"#*&-F/6#,$F1F6F3F2F5F5*&*&#F6\"\"$F3-F/
6#,$F1F>F3F3F2F5F3*&#F3F6F3*&-F/6#,$F1\"\"%F3F2F5F3F5*&*&#F6\"\"&F3-F/
6#,$F1FLF3F3F2F5F3*&#F3F>F3*&-F/6#,$F1\"\"'F3F2F5F3F5*&*&#F6\"\"(F3-F/
6#,$F1FZF3F3F2F5F3*&#F3FHF3*&-F/6#,$F1\"\")F3F2F5F3F5*&*&#F6\"\"*F3-F/
6#,$F1FboF3F3F2F5F3*&#F3FLF3*&-F/6#,$F1\"#5F3F2F5F3F5F(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Let's plot the right hand side and
 its approximation:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot([f,fapp
r],0..1);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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3qz%fARGHO*F:7$$\"3sK$eRA5\\Z*F:$\"3\\+PpPJ<C))F:7$$\"3[mTg_9sF&*F:$\"
3'oVz^`4T?)F:7$F^t$\"30`(*)pxMi](F:7$$\"3B+]il(z5j*F:$\"3!)4w$>_\\)pnF
:7$$\"3C+++]oi\"o*F:$\"3DzF]5t#>(fF:7$$\"3B+]PMR<K(*F:$\"3%)H6K^Py=^F:
7$Fat$\"3=uk!yEHw@%F:7$$\"3;+D1k2/P)*F:$\"3g\">ws*H%[?$F:7$$\"37+]P40O
\"*)*F:$\"34OKv18hc@F:7$$\"3k]7.#Q?&=**F:$\"3/)e=LgbFi\"F:7$$\"31+voa-
oX**F:$\"3**H]vtnO%3\"F:7$$\"3Z\\PMF,%G(**F:$\"3-#Q6MaP%HaF-7$Fdt$\"3!
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(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu
rve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "The solution \+
of the respective (approximate) initial-boundary value problem is:" }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "uappr := sum(  \+
c[k]* exp(-(k*Pi)^2*t) * sin(k*Pi*x), k=1..kmax);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%&uapprG,6*&*&-%$expG6#,$*&)%#PiG\"\"#\"\"\"%\"tGF0!\"
\"F0-%$sinG6#*&F.F0%\"xGF0F0F0F.F2F/*&*&-F)6#,$F,!\"%F0-F46#,$F6F/F0F0
F.F2F2*&*(#F/\"\"$F0-F)6#,$F,!\"*F0-F46#,$F6FDF0F0F.F2F0*&#F0F/F0*&*&-
F)6#,$F,!#;F0-F46#,$F6\"\"%F0F0F.F2F0F2*&*(#F/\"\"&F0-F)6#,$F,!#DF0-F4
6#,$F6FenF0F0F.F2F0*&#F0FDF0*&*&-F)6#,$F,!#OF0-F46#,$F6\"\"'F0F0F.F2F0
F2*&*(#F/\"\"(F0-F)6#,$F,!#\\F0-F46#,$F6F\\pF0F0F.F2F0*&#F0FWF0*&*&-F)
6#,$F,!#kF0-F46#,$F6\"\")F0F0F.F2F0F2*&*(#F/\"\"*F0-F)6#,$F,!#\")F0-F4
6#,$F6FcqF0F0F.F2F0*&#F0FenF0*&*&-F)6#,$F,!$+\"F0-F46#,$F6\"#5F0F0F.F2
F0F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot3d(uappr, x=0..
1, t=0..0.3);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6
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