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
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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

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- Spring '02
- CHEN
- Partial differential equation, Error function, plane source
-
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