1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Very evil diffusion equation

  1. Sep 6, 2016 #1
    1. The problem statement, all variables and given/known data
    $$\frac{\partial U}{\partial t}=\nu \frac{\partial^{2} U}{\partial y^{2}}$$
    $$U(0,t)=U_0 \quad for \quad t>0$$
    $$U(y,0)=0 \quad for \quad y>0$$
    $$U(y,t) \rightarrow {0} \quad \forall t \quad and \quad y \rightarrow \infty$$
    2. Relevant equations
    This is a diffusion problem on fluid mechanics, but it's more of a math problem so i posted it here.

    3. The attempt at a solution
    I'm trying to solve this via separation of variables (the textbook uses a "similarity" method i've never seen before, and concludes the function U must be erf) is it even possible to reach an analytic result via SV?
    The first boundary condition is what gets me, I tried
    $$U_{0} e^{{k^{2}t}} e^{{-\frac{k}{\sqrt{\nu}}y}}$$
    But it clearly doesn't work for any boundary condition except the last.
    I don't think sinusoidal is the answer here either because it must eventually converge to zero, for every t.
    Is there really no analytic answer?
  2. jcsd
  3. Sep 7, 2016 #1


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member
    2017 Award

    Please show your separation of variables and the resulting differential equations.
  4. jcsd
  5. Sep 7, 2016 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Use Laplace transforms with respect to ##t##. Let
    $$W(y,s) = \int_0^{\infty} e^{-st} U(y,t) \, dt $$
    be the Laplace transform. Then, using standard properties of Laplace transforms, we get the DE
    $$\nu W_{yy}(y,s) = s W(y,s) - U(y,0) = s W(y,s),$$
    where ##W_{yy} = \partial^2 W / \partial y^2##.
    Also: ##U(0,t) = U_0## implies that
    $$W(0,s) = \frac{U_0}{s} $$
    Finally, the initial value theorem requires that ##\lim_{s \to \infty} s W(y,s) = 0## for ##y > 0##.

    These are enough to determine ##W(y,s)##. Then it is just a matter of taking the inverse Laplace transform of ##W(y,s)## to get ##U(y,t)##.

    Separation of variables will never work in this example, simply because it leads to the wrong kind of function.
    Last edited: Sep 7, 2016
  6. Sep 7, 2016 #3

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted