lecture4 - Types of Solutions to Ficks Second Law Error...

Types of Solutions to Fick’s Second Law
Error function solutions – short-time or infinite systems
Trigonometric function solutions – long-time or finite systems
( )
Dt l >> size system
( )
Dt l ~ size system
Methods of solution: (1) Fourier transforms; (2) separation of
variables, typically employed to obtain long-time solutions; (3)
Laplace transforms, the most general and powerful method.
Definition of Error Function
Gaussian function:
( )





=
+∞ →
−∞ →
→ − =
0 at 1
as 0
as 0
exp
2
x
x
x
x y
( )
π = −
∫
∞ +
∞ −
dx x
2
exp
( )
2 exp
0
2
π = −
∫
∞ +
dx x
Error function:
( )
( ) ( )
∫ ∫
∞
− − =
0
2
0
2
exp exp du u du u x erf
x
( )
( )
∫
− =
x
du u x erf
0
2
exp
2
π

Properties of Error Function
0 ) 0 ( = erf 1 ) ( = ∞ erf
( )
x erf x erf − = − ) (
( )
x erf x erfc − = 1 ) (
Complimentary error function:
( )
( )
2
exp
2
x
dx
x derf
− =
π
( )
x
x
x erf small at
2
π
≈
Method of Fourier Transform and Superposition
Fourier transform gets rid of the space-derivative, and thus transforms the