Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Writing PDEs as differential equations on Hilbert space

  1. Oct 4, 2011 #1
    Hi,

    I was reading a paper on control of the 1-D heat equation with boundary control, the equation being
    [itex] \frac{\partial u(x,t)}{\partial x}[/itex]= [itex] \frac{\partial^2 u(x,t)}{\partial x^2} [/itex] with boundary conditions:

    [itex] u(0,t)=0[/itex] and [itex] u_x(1,t)=w(t)[/itex], where [itex] w(t)[/itex] is the control input.

    The authors then proceed to write the equation as:

    [itex] v(t)=Av(t)+\delta_1(x)w(t) [/itex] where [itex] A [/itex] is the double differential operator, [itex] v(t)=u(\cdot,t) [/itex] is the state with [itex] L_2(0,1) [/itex] being the state space and [itex] \delta_1[/itex] being the dirac delta distribution centered at [itex] x=1 [/itex].

    I understand the idea behind putting [itex] A [/itex] but I do not understand how the two equations are same.

    Any help is much appreciated.
    Thanks.
     
  2. jcsd
  3. Oct 5, 2011 #2
    I will be following this thread. :smile:

    For a diffusion equation, the first derivative is wrt to time.
    [tex] \frac{\partial u(x,t)}{\partial t}= \frac{\partial^2 u(x,t)}{\partial x^2} [/tex]
     
  4. Oct 6, 2011 #3
    Sorry, was a typo. Fixed. Thank you.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...