We study the discrete approximation to solutions of first-order system arising from applying the trapezoidal rule to a second-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,...

The approximation of integral type functionals is studied for discrete observations a continuous Itô semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved L 2 -Sobolev functions with fractional smoothness. An explicit ( P ) -lower bound shows that already lower order quadrature rules, such as trapezoidal rule and classical Riemann estimator, rate...

Let f(z) be an analytic or meromorphic function in the closed unit disk sampled at the nth roots of unity. Based on these data, how can we approximately evaluate f(z) or f (m)(z) at a point z in the disk? How can we calculate the zeros or poles of f in the disk? These questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integ...

8. First-Order Equations: Numerical Methods 8.1. Numerical Approximations 2 8.2. Explicit and Implicit Euler Methods 3 8.3. Explicit One-Step Methods Based on Taylor Approximation 4 8.3.1. Explicit Euler Method Revisited 4 8.3.2. Local and Global Errors 4 8.3.3. Higher-Order Taylor-Based Methods (not covered) 5 8.4. Explicit One-Step Methods Based on Quadrature 6 8.4.1. Explicit Euler Method Re...

In an earlier paper by the author a detailed convergence and stability analysis of a generalization of the Richardson extrapolation process was given under certain conditions. In the present work these conditions are modified and relaxed considerably, and results are obtained on convergence and stability under the new conditions. As the previous ones, these new results are asymptotic in nature,...

Given a function f ∈ C[0, 1] and some q ∈ (0, 1), we look at the approximation for the Hadamard finite-part integral = ∫ 1 0 x−q−1f(x)dx based on a piecewise linear interpolant for f at n equispaced nodes (i.e., the product trapezoidal rule). The main purpose of this paper is to give sufficient conditions for the sequence of approximations to converge against the correct value of the integral i...

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euter Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the develo...

Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n≥ 0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A trapezoidal word is closed if its longest repeated prefix has exactly two occurrences in the word, the second one being a suffix of the word. Otherwise it is open...

This paper considers the steady-state heat transfer process in a fin with Robin boundary condition at base (instead of usual Dirichlet base). models effect thermal resistance between and surface on which is placed. work presents an equivalent minimum principle, represented by convex coercive functional, ensuring solution’s existence uniqueness. In order to illustrate use proposed functional for...