I am solving two the first order ODEs ('=d/dz, all other variables are known constants):
p0 p0'=-32 beta/R^4
p0p1'=-((2-sigma_v)/sigma_v)*8*p0'/R
I have next conditions
p0(z=0)=p0i (I can choose value)
p0(z=1)=1
p1(z=0)=0
p1(z=1)=0
It is necessary to find p0'(z=0) and p1'(z=0) with shooting method (literature says like that), according to already mentioned p0(z=1)=1 and p1(z=1)=0. *How to connect this two conditions* and shoot p0'(z=0) for already known p1'(z=0)?
*Are that conditions p0'(z=0) and p1'(z=0) necessary*, because these are the first order equations, is there only one initial condition enough?
Instead of missing conditions, I also need to solve numerically two ODEs, from the beginning of text, with Runge Kutta method. How to connect that solving with shooting? Is it possible to find missing condition with bvp4c, as shooting method, and after that solve equation with ode45?
Best Answer
If you know $p_0|_{z=1}=1$ and $p_1|_{z=1}=0$ and you want to know $p_0'|_{z=0}$ and $p_1'|_{z=0}$, you don't need a shooting method.
Just use ODE45 as you already did and define tspan=[1 0]. This way, you integrate back in z-direction from z=1 to z=0. Once you have reached z=0,
$p_0'|_{z=0}=-\dfrac{32 \beta}{R^4}/p_0$and
$p_1'|_{z=0}=(-\dfrac{2-\sigma_v}{\sigma_v}\dfrac{8}{R}p_0'-p_0'*p_1)/p_0$Best wishes
Torsten.