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Method of Characteristics

  1. Mar 22, 2017 #1

    joshmccraney

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    Gold Member

    1. The problem statement, all variables and given/known data
    $$\frac{\partial u}{\partial t} + x^2 + t+\left(\frac{\partial u}{\partial x}\right)^2 = 0\\
    u(x,0)=0$$

    2. Relevant equations
    $$
    \dot{x} = 2 u_x ;\,\,t=0,\,\,x=\xi\\
    \dot{u}=2(u_x)^2+u_t ;\,\,t=0,\,\,u=0\\
    \dot{u_x}=-2x ;\,\,t=0,\,\,p=0\\
    \dot{u_t}=-1 ;\,\,t=0,\,\,u_t=-\xi^2.
    $$
    where ##\dot{f}## is the total derivative of ##f## with respect to ##t##, or ##\dot{f} \equiv \frac{df}{dt}## where ##x## is a function of ##t##.
    3. The attempt at a solution
    Write equation 1 as ##\ddot{x} = 2 \dot{u_x}##. Next substitute equation 3 in to arrive at $$\ddot{x}=-4x^2 \implies\\ x = A \sin(2t) + B\cos(2t)$$ The first BC associated with equation 1 implies ##B = \xi##, but now I'm stuck. Any ideas how to proceed?
     
  2. jcsd
  3. Mar 28, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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