Abstract
We study the discrete approximation to solutions of first-order system arising from applying the trapezoidal rule to asecond-order scalar ordinary differential equation. In the trapezoidal rule the finite difference approximation are Dyk =(zk + zk−1)/2, Dzk = ( fk + fk−1)/2, for k = 1, 2, .., n, and tk = kh for k = 0, ..., n, 0 = G(y0, yn), (z0 + z1)/2, (zn−1 + zn)/2),where fi ≡ f (ti, yi, zi) and G = (g0, g1) are continuous and fully nonlinear. We assume there exist strict discrete lowerand strict discrete upper solutions and impose additional conditions on fk and G which are known to yield a priori boundson, and to guarantee the existence of solutions of the discrete approximation for sufficiently small grid size. We usethe homotopy to compute the solutions of the discrete approximations. In this paper we study the first-order system ofdifference equations that arise when one applies the trapezoidal rule to approximate solutions of the second-order scalarordinary differential equation.
Highlights
A solution of the discrete approximation, (5) and (6) is a solution of the second order difference equation
Henderson and Thompson (2001) showed that under these assumptions α = (α(t0), · · ·, α(tn)) and β = (β(t0), · · ·, β(tn)) are strict discrete lower and strict discrete upper solutions, respectively, for (9) provided the step size h = ti − ti−1 = 1/n is sufficiently small. Using these strict discrete lower and strict discrete upper solutions and the Nagumo growth bound they established a priori bounds on difference quotients of solutions independently of step size provided the step size is sufficiently small
They introduced the central notion of very strong discrete compatibility of the boundary conditions G = 0 with the strict discrete lower and strict discrete upper solutions, α and β, respectively
Summary
A solution of the discrete approximation, (5) and (6) is a solution of the second order difference equation. Following Henderson and Thompson (2001) we assume that there are strict lower and strict upper solutions for (1), α and β, respectively, which are very strongly compatible with the very general nonlinear boundary conditions given by G = 0. Let α ≤ y ≤ β be strict lower and strict upper solutions for (1).
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