Solve $$\eta_k\frac{d^2C_k}{dz}(z)=-e_k, k = 1,2,3$$
$$C_1(0)=0, C_2(0)=A, C_3(0)=0$$ $$C_1(L)=B, \frac{dC_2}{dz}(L)=0, \frac{dC_3}{dz}(L)=0$$ where $A,B,\eta_k$ some known constant. $e_k, k=1,2,3$ are some functions we do not know the exact form, but we know the values of $e_k$ at the discretization points.
The thing bothers me is the right boundary condition for $C_2,C_3$ is the Neumann type.
My question is is there any references for solving this kind of equations? Is there any way I can convert this to a Dirichlet boundary value condition problem? I am thinking about using a shooting type method to convert this into a Dirichlet boundary condition problem, but I do not know if this will work. I could not find any references for this kind of equations.
Any help is appreciated! Many thanks!
Assuming $\eta \ne 0$, the general solution of $\eta y'' = e(x)$ is $$y(x) = c_1 + c_2 x + \eta^{-1} \int_0^x (x-t) e(t)\; dt$$ Substitute in the boundary conditions and solve for $c_1$ and $c_2$. Of course the result will depend on some integrals involving your unspecified function $e$.