On a nonlocal extension of differentiation

TL;DR

This paper introduces a nonlocal integral operator extending anti-differentiation, demonstrating its convergence to the classical derivative as the nonlocality diminishes, with explicit solutions and convergence properties analyzed.

Contribution

It formulates a nonlocal extension of differentiation, analyzes its properties, and proves convergence to the classical derivative in the limit of vanishing nonlocality.

Findings

Nonlocal operator converges weakly to classical derivative as nonlocality vanishes.

General solution involves an infinite set of functions plus a constant.

Special kernels ensure weak convergence to classical antiderivative.

Abstract

We study an integral equation that extends the problem of anti-differentiation. We formulate this equation by replacing the classical derivative with a known nonlocal operator similar to those applied in fracture mechanics and nonlocal diffusion. We show that this operator converges weakly to the classical derivative as a nonlocality parameter vanishes. Using Fourier transforms, we find the general solution to the integral equation. We show that the nonlocal antiderivative involves an infinite dimensional set of functions in addition to an arbitrary constant. However, these functions converge weakly to zero as the nonlocality parameter vanishes. For special types of integral kernels, we show that the nonlocal antiderivative weakly converges to its classical counterpart as the nonlocality parameter vanishes.