Instantaneous Point-source Solution - IfH

Section 3: Similarity solution to the one-dimensional diffusion equation 13
Using the chain rule to compute ∂C/∂t we have:
∂C
= ∂ [ ] M
∂t ∂t A √ Dt f(η) (
M
=
A √ − 1 ) 1
Dt 2 t f(η) + M
A √ Dt
= − M (
2At √ f + η ∂f
Dt ∂η
∂f ∂η
∂η ∂t
Using the chain rule to compute ∂ 2 C/∂x 2 we have:
∂ 2 C
∂x 2 = ∂ [ ( ∂ M
∂x ∂x
= ∂
∂x
=
[ ∂η
∂x
M
ADt √ Dt
∂f
∂η
)
. (22)
A √ Dt f(η) )]
M
]
A √ Dt
∂ 2 f
∂η 2 . (23)
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