Consider the following boundary value problem $$ \begin{align} y''(x) = f(x)\,, \textrm{ for } 0 < x < 1\,, \tag{4.1} \end{align} $$ with boundary conditions $$ \begin{align} y(0) = \alpha\,,\quad y(1)=\beta\,. \tag{4.2} \end{align} $$
Our goal is to solve this problem using finite differences. First we discretize the independent variable \( x_i = i h \), with \( h = 1/(N+1) \). Having defined a grid, automatically defines our unknowns, i.e. \( y_0, y_1, \dots y_N, y_{N+1} \). From boundary conditions we know that \( y_0=\alpha \) and \( y_{N+1}=\beta \), so that the number of true unknowns is \( N \). Next we approximate \( y''(x) \) by a centered finite difference approximation $$ \begin{align} y''(x_i) \approx \frac{1}{h^2} (y_{i-1} -2 y_i + y_{i+1})\,, \tag{4.3} \end{align} $$ obtaining the following algebraic equations $$ \begin{align} \frac{1}{h^2} (y_{i-1} -2 y_i + y_{i+1}) = f(x_i) \tag{4.4} \end{align} $$ for \( i=1,2,\dots,N \). It is important to note that the first equation (\( i=1 \)) involves the boundary value \( y_0=\alpha \) and that the last equation (\( i=N \)) involves the value \( y_{N+1}=\beta \). It can be easily noted that the set of equations (4.4) is as a linear system of \( N \) equations for \( N \) unknowns, which can be written as $$ \begin{align} A Y = F\,, \tag{4.5} \end{align} $$ where \( Y \) is the vector of unknowns \( Y=[y_1,y_2,\dots,y_N]^T \) and $$ \begin{equation} \tag{4.6} A = \frac{1}{h^2}\begin{bmatrix} -2 & 1 & & & & & & & \\ 1 & -2 & 1 & & & & & & \\ & \cdot & \cdot & \cdot & & & & & \\ & & \cdot & \cdot & \cdot & & & & \\ & & & \cdot & \cdot & \cdot & & & \\ & & & & & & 1 & -2 & 1 \\ & & & & & & & 1 & -2 \end{bmatrix} \,, F= \begin{bmatrix} f(x_1)-\alpha/h^2 \\ f(x_2) \\ \cdot \\ \cdot \\ \cdot \\ f(x_{N-1}) \\ f(x_N)-\beta/h^2 \end{bmatrix} \end{equation} $$
This is a nonsingular tridiagonal system, that can be solved for \( Y \) from any right-hand side \( F \).