chapter 8 partial differential equation. 8.1 introduction independent variables formulation boundary...
TRANSCRIPT
Chapter 8
Partial Differential Equation
8.1 Introduction
• Independent variables
• Formulation
• Boundary conditions
• Compounding & Method of Image
• Separation of variable
• Laplace transform
8.3 Formulation
• Transport phenomena
8.4 Boundary Conditions
• First type function specified
• Second type derivative specified
• IV,259
• Third type Mixed Condition
0 ,0 TTt
2
22
1
11 r
Tk
r
Tk
0TThx
Tk
8.4 Boundary Conditions
• Integro-differential B.C.
dtr
c
V
DaNC
a
t
0
24
8.5 Particular Solution
• Skip
8.5.2 Superposition
Second order linear partial differential equation
06542
2
3
2
22
2
1
ufy
uf
x
uf
y
uf
yx
uf
x
uf
21 uuuu Is the solutions
213 uuu Is also the solution
Infinite number of particular solutions can be added together to give a further solution
8.6 Orthogonal functions
The method of separation of variables gives the solution as sum of an infinite function.
xx mn ,
Are said to be orthogonal with respect to xr
Over the interval from to ifa b
nmxxxxrb
a
mn 0d,
Example 1 1)( dsinsin 0
xrxmxnx
xmnxmnmxnx coscossin2sin
000
dcosdcosdsinsin xxmnxxmnxmxnx
00 sin
1sin
1xmn
mnxmn
mn
mn 0
mnx d0
Sturm-Liouville Equation
0d
d
d
d
yxrxqx
yxp
x
xy
xy
mm
nn
,
,
0d
d
d
d
nnn xrxqx
xpx
0d
d
d
d
mmm xrxqx
xpx
xm
xn
0d
d
d
d
mnnn
m xrxqx
xpx
0d
d
d
d
nmmm
n xrxqx
xpx
0d
d
d
d
d
d
d
d
mnmnm
nn
m xrx
xpxx
xpx
Integrate over a b
xx
xpx
xx
xpx
xxrb
a
nm
b
a
mn
b
a
mnmn dd
d
d
dd
d
d
d
dd
xx
xpx
xx
xpx
xxrb
a
nm
b
a
mn
b
a
mnmn dd
d
d
dd
d
d
d
dd
b
a
mnnm
b
a
nmmn
xxx
xpx
xp
xxx
xpx
xp
dd
d
d
d
d
d
dd
d
d
d
d
d
b
a
b
a
b
ad
d
d
dd
xxxpxxr n
mm
n
b
a
mnmn
If orthogonal
0d
d
d
dd
b
a
xxxpxxr n
mm
n
b
a
mnmn
0d
d
d
dd
b
a
xxxpxxr n
mm
n
b
a
mnmn
0
0 ,
mn
yax
i
ii
0d
d
d
d
0d
d ,
xx
x
yax
mn
iii
mm
nn
xx
yx
yax
d
d ,
d
d
d
d ,
8.7 Method of Separation of Variables
txFT ,
tgxfT
t
T
x
T
2
2
tgxft
T
tgxfx
T
2
2
tgxftgxf
tgxf
tgxf
tgxf
tgxf
tg
tg
xf
xf
0 xfxf
0 tgtg
0 xfxf
0 tgtg
xBxAf cossin
tCg exp
0
txBxAgfT expcossin
00 BxAT
0
0 ,
0 ,0
TLx
Tx
tBA exp0cos0sin0
tLA expsin0
LA sin0 nL L
n Eigen
t
L
n
L
xnAT n
2
22
expsin
Super position
12
22
expsinn
n tL
n
L
xnAT
Initial Condition xfTt 0 ,0
10 sin
nn L
xnAxf
10 sin
nn L
xnAxf
0 xfxf
0d
d
d
d
yxrxqx
yxp
x
0 1 xqxrxp
Orthogonal
1 00
0 dsinsindsinn
L
n
L
xL
xm
L
xnAx
L
xmxf
1 00
0 dsinsindsinn
L
n
L
xL
xm
L
xnAx
L
xmxf
L
m
L
xL
xmAx
L
xmxf
0
2
0
0 dsindsin
mnmn ,
L
m
L
LxmLxmx
AxL
xmxf
0
2
0
0 4
sin
2dsin
Lm
Lm
LLmLLmL
AxL
xmxf m
L
04
0sin
2
04
sin
2dsin
0
0
Lm
Lm
LLmLLmL
AxL
xmxf m
L
04
0sin
2
04
sin
2dsin
0
0
2
dsin0
0
LAx
L
xmxf m
L
L
m xL
xmxf
LA
0
0 dsin2
12
22
expsinn
n tL
n
L
xnAT
12
22
0
0 expsindsin2
n
L
tL
n
L
xnx
L
xmxf
LT
Heat conduction in cylinder
r
T
rr
T
t
T 12
2
0 0 TTt
1 TTar
finite 0 Tr01
1
TT
TT
t
T
TTt
01
1
t
T
tTT
01
r
T
TTr
01
12
2
012
2 1
r
T
TTr
2
2
2
2
01 r
T
rTT
r
T
rTT
01
rr
TT
rTT
tTT
012
2
0101
0 0 TTt 1
1 TTar
01
1
TT
TT
0
finite 0 Tr finite
rrrt
12
2
rrrt
12
2
tgrf
gfr
gfgf 1
fg
gf
rfg
gf
fg
gf
1
21
f
f
rf
f
g
g
02 gg
0222 frfrfr
tCg 2exp
rBYrAJf 00
finite 0 Tr finite
00Y
trAJ 20 exp
terBYrAJ 2
00
1 TTar 0
taAJ 20 exp0
00 aJ
taAJ 20 exp0
trAJ 20 exp
Super position
1
20 exp
nnn trJA n
00 aJ n Eigen
0 0 TTt 1
1
01n
nn rJA
1
01n
nn rJA
rrrJrJArrrJa
nmnn
a
m dd10 1
00
0
0
mnrrrJrJAa
nmnn
0d0 1
00
mnrrrJArrrJa
mn
a
m dd0
2
0
0 0
a
mmmma
mm rJrJrJr
ArrJm 0
1120
2
012 2
1
a
mmmma
mm rJrJrJr
ArrJm 0
1120
2
012 2
1
aJa
AaaJ mmmm
m
21
2
12 2
1
aJaA
mmm 1
2
1
2
1
0
01
1 exp2
n mm
n taJ
rJ
aTT
TTn
00 aJ n