相关论文: An Exploration of the Approximation of Derivative …

A unified explicit form for difference formulas to approximate the fractional and classical derivatives is presented. The formula gives finite difference approximations for any classical derivatives with a desired order of accuracy at nodal…

Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the…

The calculus of finite differences is a solid foundation for the development of operations such as the derivative and the integral for infinite sequences. Here we showed a way to extend it for finite sequences. We could then define…

Approximate group analysis technique, that is, the technique combining the methodology of group analysis and theory of small perturbations, is applied to finite-difference equations approximating ordinary differential equations.…

Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…

In this paper we focus on the linear functionals defining an approximate version of the gradient of a function. These functionals are often used when dealing with optimization problems where the computation of the gradient of the objective…

Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…

Finite difference schemes are here solved by means of a linear matrix equation. The theoretical study of the related algebraic system is exposed, and enables us to minimize the error due to a finite difference approximation.

We approximate functionals depending on the gradient of $u$ and on the behaviour of $u$ near the discontinuity points, by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise…

In the given paper we consider finite difference approximations to systems of polynomially-nonlinear partial differential equations whose coefficients are rational functions over rationals in the independent variables. The notion of strong…

This paper provides a new approach to derive various arbitrary high order finite difference formulae for the numerical differentiation of analytic functions. In this approach, various first and second order formulae for the numerical…

Let $z_{1},z_{2},...,z_{N}$ be a sequence of distinct grid points. A finite difference formula approximates the $m$-th derivative $f^{(m)}(0)$ as $\sum w_{k}f(z_{k})$, with $w_{k}$ being the weights. We derive an algorithm for finding the…

Two approximations of the integral of a class of sinusoidal composite functions, for which an explicit form does not exist, are derived. Numerical experiments show that the proposed approximations yield an error that does not depend on the…

Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…

This work investigates finite differences and the use of interpolation models to obtain approximations to the first and second derivatives of a function. Here, it is shown that if a particular set of points is used in the interpolation…

The goal of this paper is to investigate an approach for derivative-free optimization that has not received sufficient attention in the literature and is yet one of the simplest to implement and parallelize. It consists of computing…

Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…

Many mathematical models utilize limit processes. Continuous functions and the calculus, differential equations and topology, all are based on limits and continuity. However, when we perform measurements and computations, we can achieve…

In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise…

Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…