Deriving a bound on a PDE

  • Thread starter physmatics
  • Start date
  • #1
16
0

Homework Statement


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).


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!
 

Answers and Replies

  • #2
gneill
Mentor
20,925
2,866
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.
 
  • #3
16
0
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 :)
 

Related Threads on Deriving a bound on a PDE

  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
1
Views
1K
Replies
2
Views
463
Replies
8
Views
2K
  • Last Post
Replies
1
Views
7K
Replies
6
Views
6K
Replies
0
Views
2K
Top