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Math Help - Cauchy problem

  1. #1
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    Cool Cauchy problem

    Consider u_t + (v.grad)u = 0 , where grad = upside down triangle, and v is a vector, dotted with the grad symbol.

    where v = (-ax, ay) represents 2D stagnation point flow and u = u(x,y,t). Obtain a solution of the Cauchy problem given that u = u0(x,y) at t = 0

    (u0 is read as "u naught")
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  2. #2
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    Quote Originally Posted by Waikato View Post
    Consider u_t + (v.grad)u = 0 , where grad = upside down triangle, and v is a vector, dotted with the grad symbol.

    where v = (-ax, ay) represents 2D stagnation point flow and u = u(x,y,t). Obtain a solution of the Cauchy problem given that u = u0(x,y) at t = 0

    (u0 is read as "u naught")
    Your PDE is

    u_t - a x u_x + a y u_y = 0

    MofC are:

    \frac{dt}{1} = \frac{dx}{-ax} = \frac{dy}{ay} ; du = 0

    Now pick in pairs

     <br />
1)\;\;\;\frac{dt}{1} = \frac{dx}{-ax}\;\;\; \text{so}\;\;\; I_1 = x e^{at}<br />

     <br />
2)\;\;\;\frac{dt}{1} = \frac{dy}{ay}<br />
\;\;\; \text{so}\;\;\; I_2 = y e^{-at}<br />

     <br />
3)\;\;\;du = 0\;\;\; \text{so}\;\;\; I_3 = u<br />

    Solution  I_3 = f(I_1,I_2) \;\; \text{or}\;\; u = f\left(x e^{at},y e^{-at}\right)
    Last edited by Jester; May 28th 2010 at 12:04 PM. Reason: fixed typo
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