Nathan
Summary
and
different
of
equations
wave
conditions for the heat
boundary
⑨ Heat equation
Equation
8eI=k8¥
:
Separation
ODES
of
X 't XX
'
)
-
and Tct )
General
ucx.tl
-
-
boundary
for
cases
XCO )
this
XK )
-
XkT=0
-
recent )
:[
ht
an
e-
:
O
X>0
case
conditions
O
solution
n
:
T 't
O
UCO ,t)=uCL,t )
In
t> 0
,
two
Usually
(i
-
,
ucx.tk/Cx)TCt )
variables
Xcx )
for
Axel
,
,
X=Xn=(¥5 near
:
"Ttsin(n¥ )
,
.
.
.
.
The constants
condition
an
Cii )
Ux
re
an
(x
,
(at )
this
General
ulxit )
-
from the initial
=fCx )
:
-_&§fCx)sin(n¥)dx
-
Ux
X' CO
In
O)
come
-
-
an={
X 'll )
vk.tl
9.2
.
.
.
.
Egan
O
'
X
{
CIT
"
O
n
,
)
ht
x
"
.
Hoos ( NTI)
)dx
O
,
n
-
9.2
-
aid
n=0
,
:
e-
§fCx cost
aqjofcxdx
{
-
X >O
solution
ao
>
( Litt O
case
t
n
,
,
.
.
.
.
.
.
.
② Wave
Equation
equation
8¥ 58¥
=
:
Separation
ODES
X
"
+
Usually
Ci )
-
two
recoil )
and Tct )
Xcx )
O
-
re
t> 0
:
T "tXET=0
,
cases
,
ucx.tk/CxJTCt )
variables
of
for
XX
Axel
,
boundary
for
( Lit )
-
conditions
:
O
XCO )=XC 4=0
In
this
General
X>
case
solution
0,7=7*415
ul x.t )
,
a-
9,2
.
.
.
.
:
ulx.tl-%Cancost.IT/tbnsin(ntEtDsin(nII)
The
constants
conditions
an ,
UCX O)
,
bn
from the initial
we Cx O )
gcx )
come
(x )
,
,
=
{
( ii )
2of( "¥)&×
bn-ni-fgcxls.int#olx
an
re,
x
,
Cat )
-
rex
) sink
a. 2
n,
,
.
.
.
( Lit 1=0
X' ( 01--1144=0
In this
General
case
solution
in> O
,x=Xn=f¥o? 2%62
'
vk.tl
"
"
:
ulxit -aotbott%2fancoslntftltbnsint.t #tHcoslnTT)&SofCx
{ q§fc×)dx
)
an
"
cos
(m)dx
,
n'
O
,
n-
9,2
.
.
.
.
{
•"
§
To
9 G) cos
9- guide
4¥ )d×
,
n
,
n-
9,2
.
.
.
.