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Deriving a bound on a PDE

  1. Sep 23, 2014 #1
    1. The problem statement, all variables and given/known data
    Hi!
    Not sure if I'm posting in the right section, this problem is from a course in scientific computing. Anyway, we're considering a set of PDEs:

    [tex] u_t + Au_x = 0 \quad 0<x<1, \ t>0 \\
    u(x,0) = f(x) \quad 0 \leq x \leq1 \\
    u_1(0,t) = 0 \quad t \geq 0 \\
    u_2(1,t) = 0 \quad t \geq 0 \\
    u =
    \begin{pmatrix}
    u_1 \\
    u_2
    \end{pmatrix},
    \quad A = \begin{pmatrix}
    0 & 1 \\
    1 & 0
    \end{pmatrix}
    [/tex]

    Now, I want to compute a bound ||u(*, )|| in terms of f. We have the standard inner product and the norm, where ||u||^2 = (u,u).


    3. The attempt at a solution
    First and foremost, my idea is that I use what's called the energy method, where we multiply with u, integrate in space and apply the BCs, to find ||u||. What I can't wrap my head around though is the fact that A is a matrix and in reality I have two equations, but coupled since they both contain u1 and u2. Should I introduce two variables to multiply the equations with? How can I deal with the matrix while integrating? Or should I just give up the energy method completely and take on another approach?

    Any help or small hints are much appreciated!
     
  2. jcsd
  3. Sep 25, 2014 #2

    gneill

    User Avatar

    Staff: Mentor

    Hi Physmatics. You might find a better audience for your question in one of the mathematics homework forums. Maybe try
    Calculus & Beyond Homework.

    If you wish I can move your thread there. Let me know.
     
  4. Oct 1, 2014 #3
    Thank you for replying!
    I figured out how to solve it on my own though, and if anyone is interested I can obviously describe the solution here :)
     
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