# Thread: Runge Kutta shooting with partial differential equation

1. ## Runge Kutta shooting with partial differential equation

I need to solve system of PDE-s:

$\dfrac{2}{r} \dfrac{\partial (r V_0 p_0)}{\partial r} + \dfrac{\partial (u_0 p_0)}{\partial z} =0$

$u_0 \dfrac{\partial u_0}{\partial z}=\dfrac{4}{3} \beta \dfrac{\partial^2 u_0}{\partial r^2} -\dfrac{\partial p_0}{\partial z} +\dfrac{4 \beta}{r}\dfrac{\partial u_0}{\partial r}$

$\dfrac{\partial p_0}{\partial r}=0$

$u_0 |_{r=1}=0$, $p_0|_{z=0}=p_{0i}$, $p_0|_{z=1} = 1$

where u_0, V_0, p_0 are dimensionless velocity in z axis, dimensionless radial velocity, and dimensionless pressure. z=0 is inlet and z=1 is outlet of this circular microtube with linear change of geometry ( $r(z)=r_i-z(r_1-1)$ ), where r is radial coordinate.

I need to solve this system to get velocities and pressure, but I dont know how.
With Runge Kutta shooting methods I have different differentials, and couple of them in equations, so how to calculate these values for unknown boundary conditions?

And how to solve these eqautions later?