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(1)延遲時間的特徵值表示法 (2)以擴散過程延遲時間及平均首度經過時間探討成核誘導期及膠體穩定性

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國 立 交 通 大 學

應用化學系碩士班

碩士論文

(1) 延遲時間的特徵值表示法

Representations of Time Lag by Eigenvalues

(2) 以擴散過程延遲時間及平均首度經過時間探討成核誘導

期及膠體穩定性

Study of Induction Time in Nucleation and Stability in

Colloids via Time Lag and Mean First Passage Time in

Diffusion

研 究 生:葉于榮

指導教授:陳振興 博士

(2)

(1) 延遲時間的特徵值表示法

Representations of Time Lag by Eigenvalues

(2) 以擴散過程延遲時間及平均首度經過時間探討成核誘導期及膠

體穩定性

Study of Induction Time in Nucleation and Stability in Colloids via

Time Lag and Mean First Passage Time in Diffusion

研究生:葉于榮 Student:Yu-Jung Yeh

指導教授:陳振興 博士 Advisor:Dr. Jenn-Shing Chen

國立交通大學

應用化學系碩士班

碩士論文

A Thesis

Submitted to M. S. Program,

Department of Applied Chemistry

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of Master

in

Applied Chemistry

July 2011

(3)

(1)

(2)

(4)
(5)

(1) Representations of Time Lag by Eigenvalues

(2) Study of Induction Time in Nucleation and Stability in Colloids

via Time Lag and Mean First Passage Time in Diffusion

Student Yu-Rung Yeh Advisor Dr. Jenn-Shing Chen

M. S. Program, Department of Applied Chemistry

National Chiao Tung University

Abstract

In this thesis, the eigenvalue and Green’s function representations for

the time lag of first and second moments were formulated. The Green’s

function mentioned above is the one subject to the boundary conditions

on both ends being absorbing. The homogeneous nucleation and the

coagulation of colloids were discussed with the help of diffusion. Time

lag and mean first passage time were employed to interpret the induction

time in nucleation and the stability of colloids.

The time lag of the first and of second moments will decrease as a

result of the properties of Sturm-Liouville operator. We have derived the

kinetic equations of homogeneous nucleation in the discrete number of

particle coordinate, followed by solving in the Laplace domain. In this

way, time lag, mean first passage time, and their corresponding second

moments can be obtained. The formulas were tested in the problem of

condensing water vapor. The results show that induction time for vapor

condensation decreases with increasing vapor pressure.

The stability of colloids is commonly expressed by stability ratio. We

attempted to interpret the stability with the viewpoint of diffusion via the

parameters, relative time lag and relative mean first passage time. It is

indicated that relative mean first passage time matches with stability ratio

quite well. The relation between the barrier height and the stability ratio

is also discussed by applying the method of steepest descent, to obtain an

approximate formula. Furthermore, a linear equation was proposed to

calculate the critical coagulation concentration from known parameters.

(6)

(7)

...I ...III ...IV ...V ...VIII ...XI 1.1 ...1 1.2 – ...3 1.3 ...5 1.4 ...9 1.5 ρ(x, t) Jout(t) ...11 1.6 ...13 1.7 ...16 1.8 ...19 1.9 ...28 ...37 A1.1 x t t t t ( , )d 0 2 2

ρ

∞ ∂ ∂ t t x t t ( , )d 0 3 3 2

ρ

∞ ∂ ∂ ...40 A1.2 f y y x x K x D h x h d d ) ( ) ( ) ( 1 0

...42

(8)

A1.4 K x K y G x y G y x x y y x h h d d ) , , ( ) , ( ) ( ) ( 2 0 0

∫ ∫

       < ...49 2.1. ...54 2.2. ...56 2.3. ...60 2.4. ...65 2.5. ...69 2.6. ...71 2.7. ...74 2.8. ...81 ...88 A 2.1 (2-14) ...90 A 2.2 (2-41) ...92 A2.3 ...94 3.1. ...100 3.2. ...102 3.3. ...105 3.4. ...108 3.5. ...112 3.6. ...120

(9)

3.8. ...142 3.9. ...155 ...159 A3.1 (3-41) T22(0)=1 ...162 A3.2 ...163 A3.3 ...167 ...175 ...183

(10)

1-1 ...1 1-2 h2 / 6D ± h2 / √90D ...36 2-1 G n n* ...57 2-2 n=1 n=5 ...58 2-3 n=1 n=m ...63 2-4 n = 1 n=4 ...69 2-5 n = 1 n = m ...73 2-6 n* n = m ...81 2-7 P/Psat ...87 3-1 ...109 3-2 ...112 3-3 1-1 ...122 3-4 1-1 h = r/a r a ...123 3-5 2-2 ...124 3-6 2-2 h = r/a r a ...125 3-7 3-3 ...126

(11)

3-9 Hamaker A = 0.7×10-20 J 1-1 ...128 3-10 Hamaker A = 1.2×10-20 J 1-1 ...129 3-11 Hamaker A = 1.7×10-20 J 1-1 ...130 3-12 Hamaker A = 2.2×10-20 J 1-1 ...131 3-13 A = 2×10-20 J ...134 3-14 1-1 Hamaker ...135 3-15 logW C ...136 3-16 ...140

3-17 logWapproximation kT h V( 0) (3-46) ion approximat W (3-44) ...141 3-18 c.c.c. log(C/molL-1) = -2.219 h = r/a r a ...142 3-19 (3-80) (3-81) ...146 3-20 (3-84) (3-85) ...148 3-21 3-13 ...154

(12)

2-1 P/Psat = 5 ...86 3-1 3-3 3-8 ...121 3-2 Hamaker ...121 3-3 (3-85) ...151 3-4 ...152

(13)

1.1

(time lag)

(absorbing)

Q(t)

1-1

1-1 1 ( n = 200)

(14)

1-1

J

ss

(steady state)

Q(t)

J(t)

1-1

J

ss

t

L 2 ss L) ( lim ) ( limQ t t t J t t→∞ = →∞ −

(1-1)

(1-1)

Daynes

2

Daynes

(permeability)

(time lag)

(diffusion coefficient)

(absorption coefficient)

Barrer

3,4

Goodknight

5

(porous

media)

(porosity)

(tortuosity)

Frisch

6-7

(solubility)

(15)

Leypoldt

Gough

10

Siegel

11-12

Chen

13

Siegel

11

(Dirac delta function)

14

(breakthrough curve)

15 16

(higher mome-

nts)

ss

ρ

(Green’s function)

1.2

Q(t)

(1-1)

(1-1)

Daynes

2

(16)

(1-1)

Q(t)

J

out

(t)

′ ′ = t t t J t Q 0 out( )d ) (

(1-2)

Q(t)

) (t Q

′ ′ ′ ∂ ∂ ′ − ′ ′ = t t t t J t t t J t 0 out 0 out( ) ( )d

(1-3)

(1-1)

t

L

(1-4)

(1-3)

Q(t)

) ) ( ( lim ss L J t Q t t t − = ∞ →

(1-4)

) d ) ( ) ( ( lim ss 0 out out J t t J t t t tJ t t t

′ ′ ′ ∂ ∂ ′ − − = ∞ → ss 0 out ss out d ) ( ) ( lim J t t J t t J t J t t t

∞ ∞ → ′ ∂ ∂ +       − = ss 0 out( )d J t t J t t

∞ ∂ ∂ =

(1-5)

(1-5)

(1-1)

(1-5)

(17)

function)

t = 0

t → ∞

(first moment)

(1-5)

(2) L t ss 0 out 2 (2) L d ) ( J t t J t t t

∞ ∂ ∂ =

(1-6)

(variance)

17

( )

2 L (2) L out ) ( ) ( t t J t J t Variance ss − = ∂ ∂

(1-7)

σ

Variance =

σ

(1-8)

(second moment)

1.3

1.4

(18)

(advection)

D(x) K(x)

ρ 8 ) , ( ) ( ) , ( ) ( ) ( x t t x K t x x x K x D x

ρ

ρ

∂ ∂ = ∂ ∂ ∂ ∂

(1-9)

x

t

x

0 ) ( ) 0 , (x =

δ

xx0

ρ

δ

(xx0)

(Dirac delta function)

(1-9)

(Laplace transform)

) ( ) , , ( ˆ ) ( ) , , ( ˆ ) ( ) ( 0 s x x0 s x x0 x K s x x x x K x D x ∂ = − − ∂ ∂ ∂ ρ ρ δ

(1-10)

^

ρˆ(x,x0,s)=L{ρ(x,x0,t)}

L

(activity)

) ( ) , ( ) , ( x K t x t x a = ρ

(1-11)

) ( ) , ( ˆ ) , ( ˆ x K s x s x a = ρ

(1-12)

(1-10)

ρˆ(x,x0,s) aˆ(x,x0,s) ) ( ) , , ( ˆ ) ( ) , , ( ˆ ) ( ) ( a x x0 s K x sa x x0 s x x0 x x K x D x ∂ = − − ∂ ∂ ∂ δ

(1-13)

(19)

∞ → t 0 ) , , ( lim ) , , ( ˆ lim 0 0 0 = →∞ = → sa x x s t a x x t s

(1-13)

s → 0

) ( ) 0 , , ( ˆ ) ( ) ( a x x0 x x0 x x K x D x ∂ =− − ∂ ∂ ∂ δ

(1-14)

(Green’s function)

18 aˆ(x,x0,0) x x K x D x ∂ ∂ ∂ ∂ − ( ) ( )

G(x,y)

(1-14)

) ( ) , ( ) ( ) ( G x x0 x x0 x x K x D x ∂ =− − ∂ ∂ ∂ δ

(1-15)

0 ) , ( ) , 0 ( x0 =G h x0 = G

(1-16)

(1-15)

δ

(

x

x

0

)

(unit step

function)

( ) ( 0) 0 0 dx u x x x x x − = −

δ

(1-15)

0

x

) ( ) , ( ) 0 ( ) 0 ( ) , ( ) ( ) ( 0 0 0 0 G x x u x x x K D x x G x x K x D x − − = ∂ ∂ − ∂ ∂ =

(1-17)

0 0 0 1 0 ) ( x x x x x x u > <    = −

(1-18)

(20)

) ( ) , ( ) 0 ( ) 0 ( 0 0 0 A x x x G x K D x = ∂ ∂ = ) ( ) ( ) ( ) ( ) ( ) ( ) , ( 0 0 0 x K x D x x u x K x D x A x x G x − − = ∂ ∂

(1-19)

(1-19)

x = 0

x = h

(1-16)

(1-18)

A

(x

0

)

− = − h h x x K x D x x u x x K x D x A x G x h G 0 0 0 0 0 0 d ) ( ) ( ) ( d ) ( ) ( ) ( ) , 0 ( ) , (

− = h x h x x K x D x x K x D x A 0 d ) ( ) ( 1 d ) ( ) ( 1 ) ( 0 0 0

(1-20)

A

(x

0

)

= h h x x x K x D x x K x D x A 0 0 d ) ( ) ( 1 d ) ( ) ( 1 ) ( 0

(1-21)

A

(x

0

)

(1-19)

G(x,x0)

(1-19)

0

x

G(x,x0)

′ ′ ′ − − ′ ′ ′ = x x x x K x D x x u x x K x D x A x x G 0 0 0 0 0 d ) ( ) ( ) ( d ) ( ) ( 1 ) ( ) , (

(1-22)

(1-18)

u(x, x

0

)

G(x, x

0

)

         ′ ′ ′ − ′ ′ ′ ′ ′ ′ =

x x x x x x K x D x x K x D x A x x K x D x A x x G d ) ( ) ( 1 d ) ( ) ( 1 ) ( d ) ( ) ( 1 ) ( ) , ( 0 0 0 0 0 0 0 x x x x > <

(1-23)

(21)

                 ′                 ′         =

− − h x x h h x x h x DK x DK x DK x DK x DK x DK x x G d 1 d 1 d 1 d 1 d 1 d 1 ) , ( 0 0 0 1 0 0 1 0 0 0 0 x x x x > <

(1-24)

1.4

8 ) (x ss

ρ

8 t t x x K t x x x K x D x ∂ ∂ = ∂ ∂ ∂ ∂ ( , ) ) ( ) , ( ) ( ) (

ρ

ρ

(1-25)

D

K

x = 0

ρ

(0,t)=

ρ

0

x = h

ρ(h,t)=0

t = 0

ρ(x,0)=0

(1-25)

8

= h ss x x K x D J 0 0 d ) ( ) ( 1

ρ

(1-26)

8

(22)

= h h x ss dx x K x D dy y K y D x K x 0 0 ) ( ) ( 1 ) ( ) ( 1 ) ( ) (

ρ

ρ

(1-27)

(1-24)

G(x, x

0

)

x

0

x

G(x, x)

′ ′ = h h x x x DK x DK x DK x x G 0 0 d 1 d 1 d 1 ) , (

(1-28)

(1-27)

(1-28)

ρ

ss(x)

G(x, x)

= x ss dy y K y D x x G x K x 0 0 ) ( ) ( 1 ) , ( ) ( ) (

ρ

ρ

(1-29)

(1-29)

G(x, x)

(23)

1.5

ρ( tx, ) Jout(t)

) , ( tx ρ

(1-15)

) ( ) , ( ) ( ) ( G x x0 x x0 x x K x D x ∂ =− − ∂ ∂ ∂ δ

(1-15)

0 ) , ( ) , 0 ( x0 =G h x0 = G

(1-16)

) , ( ) ( ) , ( ) ( ) ( a x t t x K t x a x x K x D x ∂ ∂ = ∂ ∂ ∂ ∂

(1-30)

) 0 ( ) , 0 ( t 0 K a =

ρ

, a(h,t)=0

(1-31)

A 1.3

(A 1-31)

a(x, t)

0 0 0 ) , ( ) 0 ( d ) , ( ) , ( ) , ( =       ∂ ∂ + ∂ ∂ − =

y h y x G y D y t y y y x G t x a

ρ

ρ

(1-32)

) , ( tx

ρ

(24)

0 0 0 ) , ( ) 0 ( ) ( d ) , ( ) , ( ) ( ) , ( =       ∂ ∂ + ∂ ∂ − =

y h y x G y D x K y t y y y x G x K t x

ρ

ρ

ρ

(1-33)

A 1.3

-J

out

(t)

(1-9)

x

h

= ∂ ∂ − ∂ ∂ → h x h x t y t y x K t x x x K x D x K t x x x K x D ( , )d ) ( ) , ( ) ( ) ( ) ( ) , ( ) ( ) ( lim

ρ

ρ

ρ

= ∂ ∂ − − h x y t t y x K t x x x K x D t J ( , )d ) ( ) , ( ) ( ) ( ) ( out

ρ

ρ

(1-34)

K t x x x K x D t x J( , ) ( ) ( ) ρ( , ) ∂ ∂ − = Jout(t)

x = h

(1-34)

D(x) K(x)

D

K

+ = ∂ ∂ − h x y t t y DK DK t J K t x x d ) , ( 1 1 ) ( ) , ( out ρ ρ

(1-35)

x

h

+ ∂ = + − h x h y h x dy z t t z DK y DK t J x K t x h K h d ) , ( 1 d 1 ) ( ) ( ) , ( ) ( ) ( out

ρ

ρ

ρ

(1-36)

0 ) (h = ρ

K(x)

(25)

+ ∂ = h x h y h x dy z t t z DK x K y DK x K t J t x, ) ( ) ( ) 1 d ( ) 1 ( , )d ( out

ρ

ρ

(1-37)

(1-37)

x = 0

ρ

(0,t)=

ρ

0

+ ∂ = h h y h dy z t t z DK K y DK K t J 0 0 out 0 d ) , ( 1 ) 0 ( d 1 ) 0 ( ) (

ρ

ρ

(1-38)

) ( out t J

∂ ∂ − = h h h y h y DK K dy z t t z DK K y DK K t J 0 0 0 0 out d 1 ) 0 ( d ) , ( 1 ) 0 ( d 1 ) 0 ( ) (

ρ

ρ

(1-39)

h y DK K 0 0 d 1 ) 0 (

ρ

(1-26)

J

ss 8 ) ( out t J

− = h h h y y DK dy z t t z DK J t J 0 0 ss out d 1 d ) , ( 1 ) (

ρ

(1-40)

(1-40)

1.6

(1-44)

(1-47)

(1-5)

J

out

(t)

(1-40)

(26)

ss 0 out L d ) ( J t t J t t t

∞ ∂ ∂ =

(1-5)

∞               ∂ ∂ − ∂ ∂ = 0 0 0 ss ss d d 1 d ) , ( 1 1 t y DK dx y t t y DK J t t J h h h x

ρ

∫ ∫

∞ ∂ ∂ − = h h x h t t ydx t y t DK y DK J 0 0 2 2 0 ss d d ) , ( 1 d 1 1 1

ρ

(1-41)

(A 1-5)

( , )d ss( ) 0 2 2 y t t t y t

ρ

=−

ρ

∂ ∂

(1-26)

J

ss

(1-41)

− = h h x ss h h dx y y DK y DK x DK t 0 0 0 0 L - ( )d 1 d 1 1 d 1

ρ

ρ

= h h x x y y DK 0 ss 0 d )d ( 1 1

ρ

ρ

(1-42)

A 1.2

(A 1-10)

(

)

∫ ∫

       = h x y y x DK t 0 ss 0 0 L d ( )d 1 1

ρ

ρ

(1-43)

) ( ss x

ρ

(1-27)

(27)

       = h x x x y DK x x G x K y DK t 0 0 0 0 0 L d d 1 ) , ( ) ( d 1 1

ρ

ρ

= hK x G x x x 0 d ) , ( ) (

(1-44)

19

= i i i i x v y w y x G

λ

) ( ) ( ) , (

(1-45)

wi

(eigenfunction)

vi

(adjoint

operator)

wi

vi

(orthogonal

function)

wi

vi

20    = ≠ =

x w x v x x nn mm h m n 1 0 d ) ( ) ( ) ( 0

ψ

(1-46)

) (x ψ

(weighting function)

K(x)

L t =

h i i i i x x v x w x K 0 d ) ( ) ( ) ( λ

= i λi 1

(1-47)

(1-44)

(1-47)

(28)

1.7

1.6

(1-53)

(1-54)

J

out

(t)

(1-40)

J

out

(t)

(1-6)

ss 0 out 2 ) 2 ( L d ) ( J t t J t t t

∞ ∂ ∂ =

(1-5)

∞               ∂ ∂ − ∂ ∂ = 0 0 0 ss 2 ss d d 1 d ) , ( 1 1 t y DK dx y t t y DK J t t J h h h x

ρ

∫ ∫

∞ ∂ − = h h x dx y t t t y t DK 0 0 2 2 2 0 d d ) , ( 1 1

ρ

ρ

(1-48)

∞ ∂ ∂ 0 2 2 2 d ) , (y t t t t

ρ

∞ - ∞

t

(A 1-8)

(1-33)

t

0 0 0 ) , ( ) 0 ( ) ( d ) , ( ) , ( ) ( ) , ( =       ∂ ∂ + ∂ ∂ − =

z h z y G z D y K z t z z z y G y K t y

ρ

ρ

ρ

(1-33)

(1-33)

(1-48)

(1-33)

t

(29)

) 2 ( L t z t t z y x t z y G y K t t DK h h x h d d d d ) , ( ) , ( ) ( 1 1 0 0 0 2 2 2 0

∫ ∫ ∫

           ∂ ∂ − ∂ ∂ − = ∞

ρ

ρ

x y x t t z t t z y G y K DK h h x h d d d d ) , ( ) , ( ) ( 1 1 0 0 0 0 3 3 2 0

∫ ∫

       ∂ ∂ = ∞

ρ

ρ

(1-49)

(1-49)

t

(A 1-8)

∞ = ∂ ∂ 0 ss 3 3 2 ) ( 2 d ) , (z t t z t t

ρ

ρ

(1-49)

) 2 ( L t K y G y z z z y x DK h h x h d d )d ( ) , ( ) ( 1 2 0 0 ss 0

∫ ∫

        =

ρ

ρ

(1-50)

(1-50)

(A 1-10)

) 2 ( L t z K x G x y y y x DK h x h d )d ( ) , ( ) ( d 1 2 0 0 ss 0 0

                =

ρ

ρ

(1-51)

) ( ss y

ρ

(1-27)

= h h x ss dx x K x D dy y K y D x K x 0 0 ) ( ) ( 1 ) ( ) ( 1 ) ( ) (

ρ

ρ

(1-27)

) ( ss y

ρ

(1-27)

(1-51)

      

x z DK 0 d 1 ) 2 ( L t y x dz DK dz DK y K y x G x K z DK h h h h y x d d 1 1 ) ( ) , ( ) ( d 1 2 0 0 0 0 0 0

                      =

ρ

ρ

(30)

x y z DK z DK z DK y x G y K x K h h h h y x d d d 1 d 1 d 1 ) , ( ) ( ) ( 2 0 0 0 0

∫ ∫

       = x y y x x y G y x G y K x K h h d d ) , , ( ) , ( ) ( ) ( 2 0 0

∫ ∫

       < =

(1-52)

(1-52)

K x K y G x y G y x x y y x h h d d ) , , ( ) , ( ) ( ) ( 2 0 0

∫ ∫

       <

(A 1-58)

x y y x x y G y x G y K x K h h d d ) , , ( ) , ( ) ( ) ( 2 0 0

∫ ∫

       < x y x y G y x G y K x K x y y y G x x G y K x K h h h h d d ) , ( ) , ( ) ( ) ( d d ) , ( ) , ( ) ( ) ( 0 0 0 0

∫ ∫

∫ ∫

       +         =

(A 1-58)

(A 1-58)

A 1.4

(A 1-58)

(1-52)

= ) 2 ( L t K x K y G x x G y y y x h h d d ) , ( ) , ( ) ( ) ( 0 0

∫ ∫

       x y x y G y x G y K x K h h d d ) , ( ) , ( ) ( ) ( 0 0

∫ ∫

       +

(1-53)

x y y v y w x v x w y K x K h h i i i i i i i i d d ) ( ) ( ) ( ) ( ) ( ) ( 0 0

∫ ∫

       =

λ

λ

x y x v y w y v x w y K x K h h i i i i i i i i d d ) ( ) ( ) ( ) ( ) ( ) ( 0 0

∫ ∫

       +

λ

λ

+ = 1 2 1

(31)

(1-53)

(1-54)

1.8

(1-9)

(first order reaction)

-) , ( ) , ( ) ( ) ( ) , ( ) ( ) ( x t t t x x R x K t x x x K x D x

ρ

ρ

ρ

∂ ∂ = − ∂ ∂ ∂ ∂

(1-55)

R(x)

(rate coefficient)

0 ) 0 , (x = ρ ρ(0,t)=ρ0 ρ(h,t)=0

(activity)

) ( ) , ( ) , ( x K t x t x a = ρ

(1-56)

(1-55)

) , ( ) ( ) , ( ) ( ) ( ) , ( ) ( ) ( a x t t x K t x a x K x R t x a x x K x D x ∂ ∂ = − ∂ ∂ ∂ ∂

(1-57)

(1-57)

(Laplace transform)

) , ( ˆ ) ( ) , ( ˆ ) ( ) ( ) , ( ˆ ) ( ) ( a x s R x K x a x s K x sa x s x x K x D x ∂ − = ∂ ∂ ∂

(1-58)

(32)

s K s a 0 ) 0 ( 1 ) , 0 ( ˆ = ρ

,

aˆ(h,s)=0

(1-59)

s

Lx

) ( ) ( d d ) ( ) ( d d x K x R x x K x D x Lx =− +

(1-60)

(1-58)

) , ( ˆ ) ( ) , ( ˆ x s K x sa x s a Lx =−

(1-61)

) , ( ˆ x s a s ) , ( ) , ( ˆ x s P x s a s =

(1-62)

(1-61)

s

saˆ(x,s)= P(x,s)

[

ˆ( , )

]

) ( ) , ( ˆ x s K x s sa x s a s Lx =− ) , ( ) ( ) , (x s K x sP x s P Lx =−

(1-63)

(1-61)

(1-63)

aˆ(x,s) ) , (x s P P(x,s) ) , ( ˆ x s a

P(x,s)

s = 0

L + ′′ + ′ + = ( ,0) 2 ) 0 , ( ) 0 , ( ) , ( 2 x P s x P s x P s x P

(1-64)

(33)

) 0 , (x P ) ( ) , ( lim ) , ( ˆ lim ) 0 , ( ss 0sa x s a x t a x x P t s = = = ∞ → →

(1-65)

(1-65)

(1-64)

L + ′′ + ′ + = ( ,0) 2 ) 0 , ( ) ( ) , ( 2 ss P x s x P s x a s x P

(1-66)

) , (x s P

(1-66)

[

( )

]

0 lim ) , ( lim 0 0 = → = → sP x s s sass x s

(1-67)

    ∂ ∂ + = ∂ ∂ → → ( , ) lim ( , ) ( , ) lim 0 0 ssP x s s P x s s sP x s s =lims→0P(x,s)=ass(x)

(1-68)

[

( , )

]

lim 2 ( , ) ( , ) 2 ( ,0) lim 2 2 0 2 2 0 s sP x s s sP x s s s P x s P x s= ′     ∂ ∂ + ∂ ∂ = ∂ ∂ → →

(1-69)

(1-66)

(1-69)

(1-63)

LxP(x,s)=−K(x)sP(x,s)

1.

(1-63)

s

0

(1-66)

(1-67)

    − =     → → ( , ) ( ) lim ( , ) lim 0 0P x s K x sP x s L s s x 0 ) ( ss x = a Lx

(1-70)

(34)

0 ) ( , ) 0 ( ) 0 ( 0 ss ss = a h = K a ρ

(1-71)

(1-70)

(1-71)

(A 1-40)

0 0 ss( ) (0) ( , ) =       ∂ ∂ = y y x G y D x a

ρ

(1-72)

(1-70)

(1-72)

A 1.3

2.

(1-63)

s

s

0

(1-66)

(1-68)

    ∂ ∂ − =     ∂ ∂ → → ( , ) ( ) lim ( , ) lim 0 0 s P x s K x ssP x s L s s x ) ( ) ( ) 0 , (x K x ass x P Lx ′ =−

(1-73)

0 ) 0 , ( , 0 ) 0 , 0 ( = ′ = ′ P h P

(1-74)

(1-72)

ass(x)

− = ′ x hG x y K y ass y y P 0 d ) ( ) ( ) , ( ) 0 , (

(

)

0 0 ( , ) ( ) ( , ) d ) 0 ( =         ∂ ∂ − =

hG x y K y G y z y z D

ρ

(1-75)

(35)

3.

(1-63)

s

s

0

(1-66)

(1-69)

      ∂ ∂ − =       ∂ ∂ → → ( , ) ( ) lim ( , ) lim 2 2 0 2 2 0 s P x s K x s sP x s L s s x ) 0 , ( ) ( 2 ) 0 , (x K x P x P Lx ′′ =− ′

(1-76)

0 ) 0 , ( , 0 ) 0 , 0 ( = ′′ = ′′ P h P

(1-77)

(1-76)

(1-75)

P(x,0)

′ − = ′′ x hG x y K y P y y P 0 d ) 0 , ( ) ( ) , ( 2 ) 0 , (

(

)(

)

0 0 0 0 ( , ) ( ) ( , ) ( ) ( , ) d d ) 0 ( 2 =         ∂ ∂ =

∫ ∫

z h h y z z G z K z y G y K y x G D

α

α

ρ

(1-78)

) ( ss x a P(x,0) P′′(x,0) ) ( out t J h x t x a x h K h D t J =     ∂ ∂ − = ( ) ( ) ( , ) ) ( out

(1-79)

(Laplace domain)

(36)

h x s x a x h K h D s J =     ∂ ∂ − = ( ) ( ) ˆ( , ) ) ( ˆ out

(1-80)

(1-5)

(1-6)

ss 0 out L d ) ( d d J t t J t t t

∞ =

(1-5)

ss 0 out( )d lim J t t J t t t t

′ ′ ′ ∂ ∂ ′ = ∞ →               ′ ′ ′ ∂ ∂ ′ =

ss 0 out 0 d ) ( lim J t t J t t s t s L ) ( ˆ ) ( ˆ d d lim out out 0 sJ s s J s s s − = →

(1-81)

ss J

L

[ f(x) ]

f(x)

ss 0 out 2 (2) L d ) ( d d J t t J t t t

∞ =

(1-6)

) ( ˆ ) ( ˆ d d lim out out 2 2 0 sJ s s J s s s→ =

(1-82)

(1-80)

Jˆout(s)

(1-81)

ˆ ) ( ˆ d d lim out L s J s s t = − →

(37)

          ∂ ∂ −           ∂ ∂ − − = = = → h x h x s s x a x h K h D s s x a x h K h D s s ) , ( ˆ ) ( ) ( ) , ( ˆ ) ( ) ( d d lim 0 h x s h x s s x a s x s x a s s x = → = →           ∂ ∂             ∂ ∂ ∂ ∂ − = ) , ( ˆ lim ) , ( ˆ lim 0 0 h x s h x s s x P x s x P s x = → = →           ∂ ∂             ∂ ∂ ∂ ∂ − = ) , ( lim ) , ( lim 0 0 h x ss h x x a x x P x = =     ∂ ∂     ∂ ∂ − = ) ( ) 0 , (

(1-83)

(1-72)

ass(x)

(1-75)

P(x,0)

(

)

h x y h x z h y x G y x y z y G y K y x G z x t = = = =               ∂ ∂ ∂ ∂                 ∂ ∂ ∂ ∂ =

0 0 0 L ) , ( d ) , ( ) ( ) , (

(1-84)

(L’Hôspital’s rule)

f(a) = g(a) = 0

) ( ) ( lim ) ( ) (a g a f x g x f a x→ = ′ ′

(1-84)

G(0, y) = G(h, y) = G(x, 0) = G(x, h) = 0

(1-84)

) , ( d ) , ( ) ( ) , ( lim 0 0 L z x G y z y G y K y x G t h z h x

→→ =

(1-85)

22       = ) ( ) ( ) ( ) ( ) ( ) ( ) , ( 1 2 2 1 y W y f x f y W y f x f y x G

y x y x ≤ ≥

(1-86)

f

1

f

2

(homogeneous solution)

W

f

1

f

2

(Wronskian)

22

(38)

) ( d d ) ( d d ) ( ) ( ) ( 2 1 2 1 y f y y f y y f y f y W =

(1-87)

(1-86)

(1-85)

) ( ) ( ) ( d ) ( ) ( ) ( ) ( ) ( ) ( ) ( lim 2 1 0 2 1 2 1 0 L z W z f x f y z W z f y f y K y W y f x f t h z h x

→→ =

= h y y W y f y f y K 0 1 2 d ) ( ) ( ) ( ) (

= hK y G y y y 0 d ) , ( ) (

(1-88)

= h i i i i y y v y w y K 0 d ) ( ) ( ) (

λ

= i λi 1

(1-89)

(1-80)

Jˆout(s)

(1-82)

) ( ˆ ) ( ˆ d d lim out out 2 2 0 (2) L s J s s J s s t s − = →           ∂ ∂ −           ∂ ∂ − − = = = → h x h x s s x a x h K h D s s x a x h K h D s s ) , ( ˆ ) ( ) ( ) , ( ˆ ) ( ) ( d d lim 2 2 0 h x s s sa x s x = →  ∂                 ∂ ∂ ∂ ∂ − = ) , ( ˆ lim 2 2 0 h x s s P x s x = →   ∂                 ∂ ∂ ∂ ∂ − = ) , ( lim 2 2 0

(39)

h x ss h x x a x x P x = =     ∂ ∂     ′′ ∂ ∂ − = ) ( ) 0 , (

(1-90)

(1-72)

ass(x)

(1-78)

P′′(x,0)

(

)(

)

h x y h x z h h y x G y x y z z G z K z y G y K y x G x t = = = =               ∂ ∂ ∂ ∂                 ∂ ∂ ∂ ∂ =

∫ ∫

0 0 0 0 ) 2 ( L ) , ( d d ) , ( ) ( ) , ( ) ( ) , ( 2

α

α

(1-91)

) , ( d d ) , ( ) ( ) , ( ) ( ) , ( 2 lim 0 0 0 ) 2 ( L

α

α

α G x y z z G z K z y G y K y x G t h h h x

∫ ∫

→ → =

(1-92)

(1-86)

(1-92)

) ( ) ( ) ( d d ) ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) ( 2 lim 2 1 0 0 2 1 2 1 0 ) 2 ( L

α

α

α

α

α W f x f y z W f z f z K z y G y K y W y f x f t h h h x

∫ ∫

→ → =

∫ ∫

= h h K y K z G y z z y y W y f z f 0 0 2 1 ( ) ( ) ( , )d d ) ( ) ( ) ( 2

∫ ∫

> = h hK z G z y z y K y G y z z y 0 0 d d ) , ( ) ( ) , , ( ) ( 2

(1-93)

(1-93)

A 1.4

(A 1-58)

x y y x x y G y x G y K x K h h d d ) , , ( ) , ( ) ( ) ( 2 0 0

∫ ∫

       < x y x y G y x G y K x K x y y y G x x G y K x K h h h h d d ) , ( ) , ( ) ( ) ( d d ) , ( ) , ( ) ( ) ( 0 0 0 0

∫ ∫

∫ ∫

       +         =

(A 1-58)

(40)

A 1.4

(1-93)

= ) 2 ( L t K x K y G x x G y y y x h h d d ) , ( ) , ( ) ( ) ( 0 0

∫ ∫

       x y x y G y x G y K x K h h d d ) , ( ) , ( ) ( ) ( 0 0

∫ ∫

       +

(1-94)

x y y v y w x v x w y K x K h h i i i i i i i i d d ) ( ) ( ) ( ) ( ) ( ) ( 0 0

∫ ∫

       =

λ

λ

x y x v y w y v x w y K x K h h i i i i i i i i d d ) ( ) ( ) ( ) ( ) ( ) ( 0 0

∫ ∫

       +

λ

λ

+ = i i i i 2 2 1 ) 1 (

λ

λ

(1-95)

1.7

1.8

1.9

(first order reaction)

-) , ( ) , ( ) ( ) ( ) , ( ) ( ) ( x t t t x x R x K t x x x K x D x

ρ

ρ

ρ

∂ ∂ = − ∂ ∂ ∂ ∂

(1-96)

) , (x t

ρ

=

(41)

(1-96)

) , ( ) ( ) , ( ) ( ) ( ) , ( ) ( ) ( a x t t x K t x a x K x R t x a x x K x D x ∂ ∂ = − ∂ ∂ ∂ ∂

(1-98)

R(x)

(rate constant)

a(x, t)

(1-98)

t → ∞

0 ) ( ) ( ) ( ) ( d d ) ( ) ( d d ss ss xR x K x a x = a x x K x D x

(1-99)

L

x ) ( ) ( d d ) ( ) ( d d x R x K x x K x D x Lx =− +

(1-100)

) ( ) ( ) (x K x u x u Lx i =

λ

i i

(1-101)

K(x)

(weighting function)

ui

(0) = u

i

(h) = 0

λ

i

i

ui

λ

i

(1-101)

) ( ) ( ) ( ) ( ) ( ) ( d d ) ( ) ( d d x u x K x u x K x R x u x x K x D x i + ii i

(1-102)

(1-102)

Sturm-Liouville

[

( ) ( ) ( )

]

( ) 0 ) ( d d ) ( ) ( d d + =     x u x K x R x K x u x x K x D x i λi i

(1-103)

(42)

Sturm-Liouville

1.

D(x) > 0

K(x) > 0

R(x) > 0

ui

(0) = u

i

(h) = 0

(1-103)

Sturm-Liouville

23

(positive

definite operator)

24

2.

(1-103)

R(x)=0 i

λ

) rxn ( i

λ

Birkhoff

25

(1-103)

R(x)K(x)

i i

λ

λ

(rxn) >

R(x) = 0

R(x) > 0

R(x)K(x)

R(x)

1.9

L t =

i λi 1

(1-104)

) 2 ( L t =

+

i i i i 2 2 1 ) 1 (

λ

λ

(1-105)

i

λ

λ

i

(43)

R(x)

D

K

) , ( ) , ( 2 2 t x t t x x D

ρ

ρ

∂ ∂ = ∂ ∂

(1-106)

0 ) , 0 (

ρ

ρ

t = ρ(h,t)=0 ρ(x,0)=0

(1-107)

Crank

1

(1-106)

∞ = − − − = 1 2 2 2 0 0 sin exp( ) 2 ) 1 ( ) , ( n t h n D x h n n h x t x

π

π

π

ρ

ρ

ρ

(1-108)

2 2 2 h n D n

π

λ

=

(1-109)

(1-107)

Jout(t)

(44)

h x t x x D t J =       ∂ ∂ − = ( , ) ) ( out

ρ

∞ = − + = 1 2 2 2 0 0 2 (cos )exp( ) n t h n D n h D h D

π

π

ρ

ρ

(1-110)

h D Jss = ρ0

(1-111)

(1-110)

ss 0 out L d ) ( d d J t t J t t t

∞ = t t h n D h n D n h D t J 2 n (cos ) exp( ) d 1 0 1 2 2 2 2 2 2 0 ss

∞ ∞ =         −         − =

ρ

π

π

π

= ∞ −         − = 1 0 2 2 2 2 2 2 d ) exp( ) (cos 2 n t t h n D t h n D n

π

π

π

(1-112)

4 4 2 4 0 2 2 2 d ) exp(

π

π

n D h t t h n D t − =

(1-113)

(1-113)

(1-112)

∞ =         − = 1 4 4 2 4 2 2 2 L 2 (cos ) n D n h h n D n t

π

π

π

1 2

π

h

∞ − =

(45)

    + + = L 2 2 2 2 2 2 4 1 3 1 2 1 1 1 2

π

D h D h D h 6 12 2 2 2 2 2 =         =

π

π

(1-114)

12 ) 2 ( 1 ) 1 2 ( 1 2 1 2 2

π

=       − −

∞ = n n n 27 ss 0 out 2 (2) L d ) ( d d J t t J t t t

∞ = t t h n D h n D n h D t J 2 n (cos ) exp( ) d 1 0 1 2 2 2 2 2 2 0 2 ss

∞ ∞ =         −         − =

ρ

π

π

π

= ∞ −         − = 1 0 2 2 2 2 2 2 2 d ) exp( ) (cos 2 n t t h n D t h n D n

π

π

π

(1-115)

6 6 3 6 0 2 2 2 2 2 d ) exp(

π

π

n D h t t h n D t − =

(1-116)

(1-116)

(1-115)

∞ =                 − = 1 6 6 3 6 2 2 2 (2) L 2 (cos ) 2 n D n h h n D n t

π

π

π

) (cos 1 4 1 4 4 2 4

π

π

n n D h n

∞ = − =     + + = L 4 4 4 4 4 2 4 4 1 3 1 2 1 1 1 4

π

D h

(46)

2 4 4 4 2 4 180 7 720 7 4 D h D h =         =

π

π

(1-117)

27 720 7 ) 2 ( 1 ) 1 2 ( 1 4 1 4 4

π

=       − −

∞ = n n n D h t 6 2 L = 2 4 ) 2 ( L 180 7 D h t = 2 2 2 h n D n

π

λ

= D h n D h Dn h t n n n n 6 1 1 2 1 2 2 2 1 2 2 2 1 L =

=

=

= ∞ = ∞ = ∞ =

λ

π

π

(1-118)

        +       =

∞ = ∞ = 1 2 2 1 (2) L 1 1 n n n n t

λ

λ

        +         =

∞ = ∞ = 2 4 1 4 4 2 1 2 2 2 1 1 n n D n h n D h

π

π

2 4 2 4 2 4 180 7 90 36 D h D h D h + = =

(1-119)

(1-106)

tL =h2 6D tL(2) =7h2 180D2 ) ( 1 out ss t J t J ∂ ∂

(47)

) ( 1 ) ( out ss t J t J t P ∂ ∂ =

(1-120)

1 d ) ( 0 =

t t P

(1-5)

t

> =< =∞

tP t t t t 0 L ( )d

(1-121)

> <t

t

(1-6)

> =< =

∞ 2 0 2 (2) L t P(t)dt t t

(1-122)

(variance)

(standard deviation)

µ

σ

(

)

∞ > < − = 0 2 d ) (t t P t t

µ

(

)

∞ > < + > < − = 0 2 2 d ) ( 2t t t P t t t

∞ ∞ ∞ > < + > < − = 0 2 0 0 2 d ) ( d ) ( 2 d ) (t t t tP t t t P t t P t 2 2 ) 2 ( L −2< > +< > =t t t

(48)

2 L ) 2 ( L (t ) t − =

(1-123)

2 L ) 2 ( L (t ) t − = =

µ

σ

(1-124)

D h tL = 2 6 tL(2) =7h4 180D2 D h2 90 1 =

σ

(1-125)

D h 6 2 D h2 90 1 =

σ

1-2

1-2 h2 / 6D ± h2 / √90D D h t 6 2 >= < D h2 90 1 =

σ

(49)

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8.

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9.

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10.

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11.

R. A. Siegel, J. Phys. Chem. 95, 2556 (1991).

12.

W. Liang and R. A. Siegel, J. Chem. Phys. 125, 044707 (2006).

13.

J. S. Chen and W. Y. Chang, J. Chem. Phys. 106, 8022 (1997).

14.

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15.

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16.

X. Ye, L. Lv, X. S. Zhao, and K. Wang, J. Membr. Sci. 283, 425

(50)

17.

D. C. Montgomery and G. C. Runger, Applied Statistics and

Probability for Engineers (John Wiley & Sons, 1994), p. B-9.

18.

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ed. (McGraw-Hill, New York, 1995), p. 173.

19.

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2

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ed. (John Wiley & Sons, New York, 1989), p. 447.

21.

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ed. (McGraw-Hill, New York, 1995), p. 181.

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Press, New York, 2002), p. 318.

23.

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San Diego, 1989), p.59.

24.

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and Methods, 2

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ed. (Van Nostrand Reinhold, New York, 1990),

p.319.

25.

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(John Wiley & Sons, Singapore, 1989), p. 320.

(51)

26.

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publications, New York, 1995), pp. 296-297.

(52)

A1.1

x t t t t ( , )d 0 2 2

ρ

∞ ∂ ∂ t t x t t ( , )d 0 3 3 2

ρ

∞ ∂ ∂ dt t x t t n n n ) , ( 0 1 1

ρ

∞ + + ∂ ∂ ) , ( tx ρ

t = 0

ρ(x,t)=0

n = 1

n = 2

udv=uvvdu

(A 1-1)

dt t x t t ( , ) 0 2 2 ρ

∞ ∂ ∂

t

u

=

x

t

t

t

v

(

,

)

d

d

2 2

ρ

=

d

u

=

d

t

)

,

( t

x

t

v

ρ

=

x t dt t t ( , ) 0 2 2 ρ

∞ ∂ ∂

∞ ∞ ∞ ∂ ∂ − ∂ ∂ = ∂ ∂ 0 0 0 2 2 d ) , ( ) , ( ) , ( x t t t t x t t dt t x t t ρ ρ ρ

(A 1-2)

) , ( tx ρ ) ( ) ( ) , (x t a w x e ss x i t i i i

ρ

ρ

=

−λ +

(A 1-3)

(A 1-2)

= − = ∂

ρ

λ

−λt

(53)

(A 1-2)

∞ ∞ ∂ ∂ − = ∂ ∂ 0 0 2 2 d ) , ( ) , ( x t t t dt t x t t

ρ

ρ

) (x ss

ρ

− =

(A 1-5)

(A 1-5)

1.7

(1-40)

(A 1-5)

(A 1-3)

x t dt t t ( , ) 0 3 3 2

ρ

∞ ∂ ∂ 2

t

u

=

x

t

t

t

v

(

,

)

d

d

3 3

ρ

=

d

u

=

2

t

d

t

(

,

)

2 2

t

x

t

v

ρ

=

dt t x t t t x t t dt t x t t ( , ) ( , ) 2 ( , ) 0 2 2 0 2 2 2 0 3 3 2

ρ

ρ

ρ

∞ ∞ ∞ ∂ ∂ − ∂ ∂ = ∂ ∂

(A 1-6)

(A 1-3)

(A 1-6)

0 e ) ( lim ) , ( 2 2 0 2 2 2 = = ∂ ∂ − ∞ → ∞

t i i i i t i x w a t t x t t

ρ

λ

λ

(A 1-7)

(A 1-5)

(A 1-5)

(A 1-6)

) ( 2 ) , ( 0 3 3 2 x dt t x t t

ρ

=

ρ

ss ∂ ∂

(A 1-8)

(A 1-8)

(A 1-8)

(1-46)

(54)

A1.2

f y y x x K x D h x h d d ) ( ) ( ) ( 1 0

1.7

1.8

= h x y y f u ( )d y x K x D v d ) ( ) ( 1 d =

d

u

=

f(y)dy

= x y y K y D v 0 d ) ( ) ( 1 x y y f x K x D h x h d d ) ( ) ( ) ( 1 0

h x h x h x h y y K y D y y f x y y f x K x D 0 0 0 d ) ( ) ( 1 d ) ( d d ) ( ) ( ) ( 1

=

(

)

∫ ∫

       −h x y f x x y K y D 0 0 d ) ( -d ) ( ) ( 1

(A 1-9)

(A 1-9)

(A 1-9)

=

f y y x x K x D h x h d d ) ( ) ( ) ( 1 0

∫ ∫

       h x x x f y y K y D 0 0 d ) ( d ) ( ) ( 1

(A 1-10)

(55)

A 1.3

G(x,y)

a( tx, ) ass(x)26 ) , ( ) , ( ) ( ) ( ) , ( ) ( ) ( y t t t y y R y K t y y y K y D y

ρ

ρ

ρ

∂ ∂ = − ∂ ∂ ∂ ∂

(A 1-11)

) , (y t

ρ

D(y)

K(y)

R(y)

(rate coefficient)

ρ(0,t)=ρ0 ρ(h,t)=0 0 ) 0 , (y = ρ

(activity)

) ( ) , ( ) , ( y K t y t y a = ρ

(A 1-12)

(A 1-12)

) , ( ) ( ) , ( ) ( ) ( ) , ( ) ( ) ( a y t t y K t y a y K y R t y a y y K y D y ∂ ∂ = − ∂ ∂ ∂ ∂

(A 1-13)

L

y ) ( ) ( d d ) ( ) ( d d y R y K y y y K y D y L =− +

(A 1-14)

G(x,y)

x

(source point) y

(field point)

G(x,y)

) ( ) , ( ) ( ) ( ) , ( ) ( ) ( G x y R y K y G x y x y y y K y D y ∂ − =− − ∂ ∂ ∂ δ

(A 1-15)

(56)

0 ) , ( ) 0 , (x =G x h = G

(A 1-16)

(A 1-13)

(A 1-15)

a(y,t)

(A 1-13)

(A 1-15)

) , ( ) , ( ) , ( ) ( ) , ( 2 2 t y a y K t y RKa t y a y DK t y a y DK ∂ ∂ = − ∂ ∂ ′ + ∂ ∂

(A 1-17)

) ( ) , ( ) , ( ) ( ) , ( 2 2 y x y x RKG y x G y DK y x G y DK − =− − ∂ ∂ ′ + ∂ ∂

δ

(A 1-18)

D(y)

K(y)

R(y)

D

K

R

) (DK

DK

y

DK y DK d d ) ( ′= ) , (y t RKa RKG(x,y)

(A 1-17)

G(x,y)

(A 1-18)

) , (y t a ) , ( ) , ( ) , ( ) , ( ) ( ) , ( ) , ( 2 2 t y a y K y x G t y RKa t y a y DK t y a y DK y x G ∂ ∂ =       − ∂ ∂ ′ + ∂ ∂

(A 1-19)

) ( ) , ( ) , ( ) , ( ) ( ) , ( ) , ( 2 2 y x t y a y x RKG y x G y DK y x G y DK t y a =− −      − ∂ ∂ ′ + ∂ ∂

δ

(A 1-20)

      ∂ ∂ − ∂ ∂ ′ +       ∂ ∂ − ∂ ∂ ) , ( ) , ( ) , ( ) , ( ) ( ) , ( ) , ( ) , ( ) , ( 2 2 2 2 y x G y t y a t y a y y x G DK y x G y t y a t y a y y x G DK ) ( ) , ( ) , ( ) , ( a y t a y t x y y K y x G + − ∂ ∂ =

δ

(A 1-21)

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