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Homework Help: Help solving a PDE

  1. Mar 19, 2012 #1
    1. The problem statement, all variables and given/known data

    The Cauchy problem for the advection-diffusion equation is given by:

    u.sub.t + c u.sub.x = K u.sub.xx (−∞< x < ∞)

    u(x, 0) = Phi(x)

    where c and K are positive constants.

    The advection-diffusion equation essentially combines the effects of the
    transport equation and the heat equation, so that the concentration profile
    is carried with speed c as it diffuses. The purpose of this problem is to
    solve the advection-diffusion equation using the following three steps:

    (1) Let v(x, t) = u(x + ct, t) and show that v(x, t) satisfies the heat equation,

    (2) Determine the initial condition that v(x, t) must satisfy; then, solve the
    resulting Cauchy problem for v(x, t).

    (3) Use the formula for v(x, t) from Step 2 to find u(x, t), the solution of the
    Cauchy problem for the advection-diffusion equation.


    2. Relevant equations

    See above.

    3. The attempt at a solution

    See https://www.physicsforums.com/showthread.php?t=588387
  2. jcsd
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