I've been given a second-order non-linear ODE:$$\frac{d^{2}\theta(s)}{ds^{2}} = sf_{g}\cos{\theta} + sf_{x}\cos{\phi}\sin{\theta}$$where $f_{g}, f_{x}$ and $\phi$ are constants.The boundary conditions for $[0,L]$ are:$\frac{d\theta}{ds}\Big|_{s=0} = 0 $$\theta (L) = \theta_L$I will be repeating this numerous times for different values of $L$, implementing the solution method as a function. The $\theta_L$ is just a constant assumed known each time.I am unsure how to proceed with these boundary conditions (are they Neumann, Dirichlet etc.? I don't think so). I struggle when I try to convert the BVP into an IVP for the shooting method and I've read the documentation for bvp4c and ode45 over and over with no progress on this problem. I just can't seem to get started here.Could anyone give me some pointers or help? Thank you very much in advance.
Numerical method for a BVP with mixed boundary conditions (MATLAB)
matlab;boundary conditions;numerical modelling
To start you could change it into a system of first-order ODEs$$Y(s) := \left(  \begin{split}    \theta'(s)\\     \theta(s)  \end{split}\right) =   \left(\begin{split}    Y_1(s)\\     Y_2(s)  \end{split}\right)\,,$$$$ Y'(s) = \left(  \begin{split}    s f_g \cos(Y_2(s)) + s f_x \cos \phi\sin(Y_2(s))\\    Y_1(s)  \end{split}\right)\,.$$$$Y(0) = \left(\begin{align} 0\\ t \end{align}\right)\,.$$Then the shooting method, parametrised by $t$ as initial conditions, that means use different values of $t$ as an initial condition. Approximate value of $t$ can be find in the style taught in this lecture. The easiest root finding method is the bisection method. For that you should know a range in which the value of $t$ lies $[t_a t_b]$. Substitute $t_a$, $t_b$ and $t_c$ (mid point of $t_a$ and $t_b$), as three different I.C and solve that using  ode45 then check weather you are getting closer to $Y_2(L)=\theta_L$, then reduce the search range in bisection algorithm algorithm until you get desired accuracy.