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Math Help - Partial Differential Equation Problem

  1. #1
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    Partial Differential Equation Problem

    Solve the PDE

    \frac {\partial u}{\partial t} + 3t \frac {\partial u}{\partial x} = u and u(x,0) = \exp^(-x^2)

    Plot characteristc and solution at t = 0 and 1
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  2. #2
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    Quote Originally Posted by r2dee6 View Post
    Solve the PDE

    \frac {\partial u}{\partial t} + 3t \frac {\partial u}{\partial x} = u and u(x,0) = \exp^(-x^2)

    Plot characteristc and solution at t = 0 and 1
    If

    \frac{du}{dt} = \frac{\partial u}{\partial t} + \frac{dx}{dt} \frac{\partial u}{\partial x} then your characteristic curve is given by

    \frac{dx}{dt} = 3t.

    Using the method of characteristics

    \frac{dt}{1} = \frac{dx}{3t} = \frac{du}{u}

    which we solve in pairs giving

    dx = 3 t \, dt\;\;\; c_1 = x - \frac{3}{2} t^2 (you can plot this)

    du = dt\;\;\; c_2 = u e^{-t}

    The solution is

    u = e^t f(x - \frac{3}{2} t^2)

    Imposing the initial condition gives

    f(x) = e^{-x^2}

    and the final solution

    u = e^{t -\left( x - \frac{3}{2} t^2 \right)^2}
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  3. #3
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    hey thanks but how do i get teh characteristic equations?i mean how do i plot them ?
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