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PDE analytical solution

  1. Feb 12, 2010 #1
    LaTex Code: \frac{\partial U}{\partial t} + ax\frac{\partial U}{\partial x} + b\frac{\partial^2 U}{\partial x^2} = 0

    Can someone please tell me how to solve this PDE?

  2. jcsd
  3. Feb 12, 2010 #2
    Assuming that

    [tex]U(t,x)=\int_{-\infty}^\infty F(x,\tau)e^{it\tau}\,d\tau[/tex]

    we come to

    [tex]-ib\frac{\partial^2 F(x,\tau)}{\partial x^2}-iax\frac{\partial F(x,\tau)}{\partial x}+\tau F(x,\tau)=0[/tex]

    which solution is as follows

    [tex]F(x,\tau) = x[H1(\tau)KummerM(-\frac{-2a+i\tau}{2a},\frac{3}{2},\frac{ax^2}{2b})+H2(\tau)KummerU(-\frac{-2a+i\tau}{2a},\frac{3}{2},\frac{ax^2}{2b})]e^{-\frac{ax^2}{2b}}[/tex]

    so the general solution to your PDE is

    [tex]U(t,x)=xe^{-\frac{ax^2}{2b}}\int_{-\infty}^\infty [H1(\tau)KummerM(-\frac{-2a+i\tau}{2a},\frac{3}{2},\frac{ax^2}{2b})+H2(\tau)KummerU(-\frac{-2a+i\tau}{2a},\frac{3}{2},\frac{ax^2}{2b})]e^{it\tau}\,d\tau[/tex]

    where H1 and H2 are arbitrary functions.
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