where b, c, d, α, and β are given constants, the interval [0,1] is first divided into equal subintervals of length h; h is called the stepsize (or mesh or grid size). This gives the mesh points (or grid points) xn, xn = nh, n = 0,1,…,N+1, h = 1/(N+1)
At interior mesh points the derivatives are now replaced by finite-difference approximations, e.g. y′(xn) ≊ (½h)[y(xn+1) – y(xn–1)] y(xn) ≊ (1/h2)[y(xn+1) – 2y(xn) + y(xn–1)]
When combined with the boundary conditions these approximations result in a system of equations for approximations to y(xn), n = 1, 2,…, N. Nonlinear differential equations yield a system of nonlinear equations.
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