Trapezoidal rule and its error analysis for the Grünwald-Letnikov operator

Derivation of the Trapezoidal Rule Error Estimate

and |g′′(x)| = M. We will show that |f(x)| ≤ g(x) on [0, 1]. The desired conclusion follows. Suppose however that this is false, that there is a number q ∈ [0, 1] for which f(q) > g(q). (The other case, f(q) < −g(q), is similar.) Our strategy will be to show that there are real numbers s and t with s < t such that f ′(s) > g′(s) and f ′(t) < g′(t). See figure. We will then apply the Mean Value ...

Efficient computation of the Grünwald-Letnikov fractional diffusion derivative using adaptive time step memory

Article history: Received 1 June 2014 Received in revised form 15 April 2015 Accepted 29 April 2015 Available online 5 May 2015

A trapezoidal rule error bound unifying the Euler–Maclaurin formula and geometric convergence for periodic functions

The error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic func...

The Exponentially Convergent Trapezoidal Rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special funct...

-Stable Derivative-Free Error-Corrected Trapezoidal Rule for Burgers' Equation with Inconsistent Initial and Boundary Conditions