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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