How do we go from BVP to IVP in determining the Green's function?

[member 428835]

Hi PF!

I'm reading a text where the authors construct a Green's function for a given BVP by variation of parameters. The authors construct the Green's function by finding first the fundamental solutions (let's call these $v_1$ and $v_2$) to the homogenous BVP. However, the authors determine $v_1$ and $v_2$ from initial conditions (not given anywhere in the physical setup) rather than boundary conditions.

Specifically, the boundary conditions initially presented are $u'(s_0)+\mu u(s_0) = -u'(-s_0)+\mu u(-s_0) = 0:s\in[-s_0,s_0]$, where $\mu$ is a constant. The authors state that solving the BVP for the fundamental solutions is equivalent to solving the homogenous equation for the fundamental solutions $v_1$ and $v_2$ subject to $v_1(0)=0,v_1'(0)=1$ and $v_2(0)=1,v_2'(0)=0$.

Can anyone help me understand how the went from the BVP to the IVP? I should say the governing differential equation (not shown here) does not change from the BVP to the IVP.

 

[hilbert2]

I'm not sure what kind of differential equation is being solved here, but consider a BVP where the values of an unknown function $f(x,y)$ are given on a circular curve in the xy-plane, and then you make the circle larger and larger to make the shape of the boundary approach a straight line. Wouldn't this kind of a construction convert a boundary value problem to an initial value one?