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