# Diffusion equation

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```					             Diffusion equation

• One-dimensional diffusion equation is
∂u ∂ 2 u
−     = 0,         −∞ < x < ∞,         t > 0.
∂t   ∂x2
• Introduce unit heat source at time t = 0 at the origin x = 0.
• We wish to solve

ut − uxx = δ(x)δ(t),             −∞ < x < ∞,     0≤t

• Initial condition is
u(x, 0− ) = 0.

• Here δ is the Dirac delta function

APDEs (EMAT32110) 2007-08                                3.20                          Diffusion equations: similarity
Fundamental solution

• Observe that L.H.S. does NOT involve x and t
• Can translate the solution to ﬁnd the result for a source located at x = ξ and turned on at t = τ
• Suppose u = F (x, t) is the solution of ut − uxx = δ(x)δ(t).
• Consider
ut − uxx = δ(x − ξ)δ(t − τ ),             u(x, τ − ) = 0

• Its solution is then u = F (x − ξ, t − τ ).

APDEs (EMAT32110) 2007-08                                  3.21                                  Diffusion equations: similarity
Similarity

• Assume we have a solution u = F (x, t)
• Question: it is possible to ﬁnd a second solution u = G(x, t)?
¯       ¯
• Set x = βx, t = γt.
• Deﬁne G as G(x, t) ≡ αF (βx, γt)
• If G is a solution it must satisfy

Gt − Gxx = δ(x − ξ)δ(t − τ ),           u(x, τ − ) = 0

• Substituting the derivatives gives,
β2            β
Ft −
¯             Fxx =
¯            x ¯
δ(¯)δ(t),          F (¯, 0−) = 0
x
γ             α

APDEs (EMAT32110) 2007-08                                 3.22                                Diffusion equations: similarity
Similarity

• G(x, t) can be a solution only if β = α and γ = α2
• G(x, t) must be of the form
G(x, t) = αF (αx, α2 t)

• Have we found a new soluion ?
• NO! The solution of the diffusion equation is unique.
• We have discovered similarity structure of F , i.e.

αF (αx, α2 t) = F (x, t).

• This property implies that F must be of the form
1        x                 1       x2            1     x
F (x, t) = √ f      √     ,    or     √ g     √    ,   or     h   √     ,...
t        t                 t       t            x       t

APDEs (EMAT32110) 2007-08                                  3.23                                Diffusion equations: similarity
Reduction to an ODE

• Each choice of F (x, t) reduces the original PDE to an ODE !
• Solution of an ODE will give the same result for F .
• Let’s try
1                                   x
F (x, t) = √ f (ζ),                         ζ= √ .
t                                   t
• Substituting into PDE leads to
ζ     1
f ′′ + f ′ + f = 0
2     2
• The solution of the second order linear ODE is
ζ
−ζ 2 /4         2
/4               2
/4
f = Ae                 e−s        ds + Be−ζ            ,     A, B   − constants.

APDEs (EMAT32110) 2007-08                                       3.24                                           Diffusion equations: similarity
Finding the constants

• Constants A and B are determined by taking the total heat content H(t) in a bar.
• The total heat is just an integral of the temperature, i.e.
∞
H(t) =               F (x, t)dx
−∞
∞                         ∞
A              x           B            2
/4t
=     √          g   √      dx + √          e−x         dx.
t   −∞        t            t   −∞

• Integrating by parts gives
2
g(ζ) =        + O(ζ −3 ) as |ζ| → ∞
|ζ|
∞
1
• It implies that √          gdx is undounded.
t   −∞

• The total heat must be ﬁnite, so set A = 0.

APDEs (EMAT32110) 2007-08                                    3.25                                     Diffusion equations: similarity
Heaviside function

• Since A = 0, therefore,
B    2
F = √ e−x /4t ,                t > 0.
t
• Have to choose B to satisfy the homogeneous problem.
• Differentiate H w.r.t. t:
∞
dH
=              Ft (x, t)dx
dt             −∞
= Fx (∞, t) − Fx (−∞, t) + δ(t) = δ(t)

• H(t) is the Heaviside function, i.e.

1          if t   >0
H(t) =
0          if t   > 0.

APDEs (EMAT32110) 2007-08                                    3.26                      Diffusion equations: similarity
General solution

• Using properties of the Heaviside function, we obtain
∞
B    2
1=           √ e−x /4t dx
−∞     t
• Integrating gives
1
B= √ .
2 π
• The fundamental solution of a diffusion equation is
1 −x2 /4t
F (x, t) = √          e     .
4πt
• More generally, the solution is
1                  2
/4(t−τ )
F (x − ξ, t − τ ) =                    e−(x−ξ)                  .
4π(t − τ )

APDEs (EMAT32110) 2007-08                                 3.27                                         Diffusion equations: similarity

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