On a nonlocal extension of differentiation

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.