Results 1 to 2 of 2

Math Help - shocks and strict inequalities

  1. #1
    Senior Member
    Joined
    Dec 2007
    From
    Melbourne
    Posts
    428

    shocks and strict inequalities

    I'm just doing some practice exam questions and the question asks me to solve
    <br />
\frac{\partial u}{\partial t}+u^2\frac{\partial u}{\partial x} = 0

    with
    u(x,0) = \left\{ \begin{array}{cc} 1, &x<0\\0, &x \geq 0 \end{array} \right.

    I am expected to use the method of characteristics. The calculation was very straightforward and I ended up with
    <br />
u(x,t) = \left\{ \begin{array}{ll} 1,&x<\frac13 t\\0,&x>\frac13 t \end{array} \right.

    What I'm not sure about is the value of u on the shock: does the fact that the initial conditions used \geq instead of > mean that u=0 on the shock? or do I need to add to my answer that u(0,0)=0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,572
    Thanks
    1412
    Quote Originally Posted by badgerigar View Post
    I'm just doing some practice exam questions and the question asks me to solve
    <br />
\frac{\partial u}{\partial t}+u^2\frac{\partial u}{\partial x} = 0

    with
    u(x,0) = \left\{ \begin{array}{cc} 1, &x<0\\0, &x \geq 0 \end{array} \right.

    I am expected to use the method of characteristics. The calculation was very straightforward and I ended up with
    <br />
u(x,t) = \left\{ \begin{array}{ll} 1,&x<\frac13 t\\0,&x>\frac13 t \end{array} \right.

    What I'm not sure about is the value of u on the shock: does the fact that the initial conditions used \geq instead of > mean that u=0 on the shock? or do I need to add to my answer that u(0,0)=0
    Yes, it does. Your solution must satisfy your initial condition and to do that you must have:
    <br />
u(x,t) = \left\{ \begin{array}{ll} 1,&x<\frac13 t\\0,&x\ge\frac13 t \end{array} \right.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. inequalities
    Posted in the Algebra Forum
    Replies: 11
    Last Post: August 25th 2010, 03:18 PM
  2. validity of convergent sequences with strict inequalities
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 16th 2010, 06:34 PM
  3. Strict
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: March 27th 2009, 07:41 AM
  4. inequalities
    Posted in the Algebra Forum
    Replies: 3
    Last Post: September 10th 2008, 07:57 PM

Search Tags


/mathhelpforum @mathhelpforum