# Cauchy problem

• May 18th 2010, 09:27 PM
Waikato
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")
• May 19th 2010, 02:32 PM
Jester
Quote:

Originally Posted by Waikato
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")

$u_t - a x u_x + a y u_y = 0$

MofC are:

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

Now pick in pairs

$
1)\;\;\;\frac{dt}{1} = \frac{dx}{-ax}\;\;\; \text{so}\;\;\; I_1 = x e^{at}
$

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

$
3)\;\;\;du = 0\;\;\; \text{so}\;\;\; I_3 = u
$

Solution $I_3 = f(I_1,I_2) \;\; \text{or}\;\; u = f\left(x e^{at},y e^{-at}\right)$