- #1
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.
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.