1. ## Problems on characteristics

Obtain a solution of

u_t + u_x = t^a, where a = alpha, and a > 0.

given that u = u_0(x) on t = t0

2. Originally Posted by Waikato
Obtain a solution of

u_t + u_x = t^a, where a = alpha, and a > 0.

given that u = u_0(x) on t = t0
Characteristic equation

$\frac{dt}{1} = \frac{dx}{1} = \frac{du}{t^\alpha}$

$I_1 = x- t,\;\;\;I_2 = u - \frac{t^{\alpha+1}}{\alpha+1}$

Solution

$u = \frac{t^{\alpha+1}}{\alpha+1} + f(x-t)$.

3. ## Initial Condition - DE

Thanks

4. Originally Posted by Waikato

Thanks
Sure

Originally Posted by Danny
Characteristic equation

$\frac{dt}{1} = \frac{dx}{1} = \frac{du}{t^\alpha}$

$I_1 = x- t,\;\;\;I_2 = u - \frac{t^{\alpha+1}}{\alpha+1}$

Solution

$u = \frac{t^{\alpha+1}}{\alpha+1} + f(x-t)$.

$
u(x,0) = \frac{0}{\alpha+1} + f(x) = u_0(x)
$

so $f(x) = u_0(x)$ and the final solution is

$u = \frac{t^{\alpha+1}}{\alpha+1} + u_0(x-t)$.

5. ## IC's and help

Thanks for the IC's help.

Can you check this question out? You might be able to help.

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")