A.2. Notations

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A.3. Green's function for 2-nd order ODE

Consider two-point boundary-value problem 2-nd order linear ODE \begin{align} &Lu = a_0 u''+a_1 u'+ a_2u=f\qquad 0<x< l, \label{eq-A.3.1}\\ &\alpha_1 u'(0)+\beta_1 u(0)=k_1, \label{eq-A.3.2}\\ &\alpha_2 u'(l)+\beta_2 u(l)=k_2 \label{eq-A.3.3} \end{align} with $a_j=a_j(x)$, $f=f(x)$.

Let $u_1$ and $u_2$ be two linearly independent solutions to the same homogeneous equation, satisfying special data \begin{align} &Lu = a_0 u_k''+a_1 u_k'+ a_2u_k=0\qquad 0<x< l, \label{eq-A.3.4}\\ &\alpha_1 u_k'(0)+\beta_1 u_k(0)=\delta_{k1}, \label{eq-A.3.5}\\ &\alpha_2 u_k'(l)+\beta_2 u_k(l)=\delta_{k2} \label{eq-A.3.6} \end{align} where we assume that such functions exist. Solving (\ref{eq-A.3.1}) by the method of variations, \begin{equation} u= C_1u_1+C_2u_2, \label{eq-A.3.7} \end{equation} where \begin{align*} &C_1' u_1 + c_2'u_2= 0,\\ &C_1' u_1' + c_2'u_2'= g:= a_0^{-1}f, \end{align*} we conclude that \begin{align*} &C_1 = -\int^x \frac{u_2(s)g(s)}{W(s)}\,ds,\\ &C_2 = \ \int^x \frac{u_1(s)g(s)}{W(s)}\,ds, \end{align*} with the Wronskian \begin{equation} W= W[u_1,u_2]= u_1u_2'-u_1'u_2 \label{eq-A.3.8} \end{equation} we need to satisfy (\ref{eq-A.3.2}) and (\ref{eq-A.3.3}), which due to \begin{equation} u'= C_1u_1'+C_2u_2' \label{eq-A.3.9} \end{equation} and (\ref{eq-A.3.5}) and (\ref{eq-A.3.6}) are equivalent to $C_1(0)=k_1$, $C_2(l)=k_2$ and therefore \begin{align*} &C_1 = -\int_0^x \frac{u_2(s)g(s)}{W(s)}\,ds + k_1,\\ &C_2 = - \int_x^l \frac{u_1(s)g(s)}{W(s)}\,ds+k_2, \end{align*} and we arrive to \begin{equation} u(x)= \int_0^l G(x,s)f(s)\,ds + k_1u_1(x)+ k_2u_2(x) \label{eq-A.3.10} \end{equation} with \begin{equation} G(x,s)=-\left\{\begin{aligned} & \frac{u_2(s)u_1(x)}{a_0(s)W(s)} &&0<s<x,\\[4pt] &\frac{u_1(s)u_2(x)}{a_0(s)W(s)} &&x<s<l. \end{aligned}\right. \label{eq-A.3.11} \end{equation}

Definition 1. Function $G(x,s)$ is a Green's function for problem (\ref{eq-A.3.1})-(\ref{eq-A.3.3}).

Remark 1.

  1. Green's function satisfies \begin{equation} L_x G(x,s)u =\delta(x-s)\qquad 0<x< l, \label{eq-A.3.12} \end{equation} and (\ref{eq-A.3.2})-(\ref{eq-A.3.3}) (with respect to $x$; $s$ is considered as a parameter), where $\delta(x-s)$ is Dirac's $\delta$-function.
  2. Also $G(x,s)$ is continuous, but it's derivative has a jump $a_0(x)^{-1}$ at $x=s$.

$\Leftarrow$  $\Uparrow$