1. Not finding help here? Sign up for a free 30min 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!

Existance of unique solutions for differential equation

  1. Oct 7, 2006 #1
    Hi guys,

    x'=A(t)x(t)+B(x,t)u(t) (1)

    x, t belongs to R^n and R+ respectively. u(t) is a function of t.

    I am required to find the sufficient conditions for A(t), f(x), B(x,t) and u(t)
    for the (1) to have a existence of a unique solution.

    In the lecture notes, the prof lists 4 conditions to be sufficient conditions for the
    existence of a unique solution for x'=f(x,t):

    1. for a fixed t, f(x) is continous.
    2. There exists a set S contained in R+ containing at most a finite
    number of points per unit interval. S will depend on f(x,t) and will denote
    possible discontinuity points. With this set, for x is fixed, f(t) will be discontinuous only when t belongs to S.

    3. If x(t) is continous, then f(x,t) is piecewise continous in t

    4.satisifies a global Lipschitz condition.


    Thus my solution is as follows:

    1. for a fixed t: B(x,t) are continuous.

    2. for a fixed x, A(t), u(t), B(x,t) are discontinuous only in S

    3. A(t)x(t)+B(x,t)u(t) is picewise continuous in t.

    4. I derive the global Lipschitz condition:


    ||A(x1-x2)+U(B(x1)-B(x2))||<=|A|*||(x1-x2)||+|u|*||B(x1)-B(x2)||

    so if ||B(x1)-B(x2)||<=K(t)||x1-x2|| then ||A(x1-x2)+U(B(x1)-B(x2))||<=K1||x1-x2||

    with k1=max(|A|, |k*u|).

    So the fourth condition will be B(x,t) satisfies the Lipschitz condition.


    I am not sure if the conditions can be narrower? can you please verify my solution.

    Thanks
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?



Similar Discussions: Existance of unique solutions for differential equation
  1. 1D wave equation PDE (Replies: 0)

  2. Laplace's eqn solution (Replies: 0)

Loading...