On the one-dimensional SPH approximation of fractional-order operators

TL;DR

This paper develops a theoretical and numerical framework for approximating fractional-order operators in one dimension using the SPH method, enabling new applications in fractional continuum mechanics.

Contribution

It introduces a novel SPH-based formalism for fractional operators, applicable to both constant and variable orders, with validation through numerical examples.

Findings

The method accurately approximates fractional operators in 1D.

Validation shows good agreement with analytical solutions.

The approach facilitates solving fractional continuum mechanics models.

Abstract

This work presents a theoretical formalism and the corresponding numerical techniques to obtain the approximation of fractional-order operators over a 1D domain via the smoothed particle hydrodynamics (SPH) method. The method is presented for both constant- and variable-order operators, in either integral or differential forms. Several numerical examples are presented in order to validate the theory against analytical results and to evaluate the performance of the methodology. This formalism paves the way for the solution of fractional-order continuum mechanics models via the SPH method.