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

TL;DR

This work develops a one-dimensional SPH framework to approximate fractional-order operators, enabling constant- and variable-order definitions (Riemann-Liouville and Caputo) in both integral and differential forms. It formulates SPH approximations for RL integrals/derivatives and Caputo derivatives, and extends to Type-I variable-order operators; a modified SPH integration using auxiliary particles improves accuracy and mitigates kernel singularities. Numerical validations against analytical solutions for sine, cosine, exponential, and shifted polynomial functions demonstrate high accuracy, with the auxiliary-particle approach delivering substantial error reductions in RL integrals. The methodology provides a general SPH-based tool for solving fractional-order continuum mechanics problems in 1D, paving the way for nonlocal and multiscale simulations.

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.

Figures (9)

• Figure 1: Schematic illustration of the discretized physical domain (containing real particles) and of the virtual domain (containing virtual particles). $\Omega$ is the support domain with radius $\kappa h$. Particle $i$ represents a particle with a fully supported domain, while particle $k$ represents a boundary-affected particle with a truncated support domain.
• Figure 2: Plot of the cubic kernel function ($W$), its gradient ($\nabla W$), and its integral ($\Tilde{W}$).
• Figure 3: Left-handed CO Caputo derivative: Comparison between the SPH approximation and the analytical solution for (a1) $\sin (\pi x)$, (b1) $\cos (\pi x)$, (c1) $\exp(x)$, and (d1) $(x-1)^3$. The right column ((a2), (b2), (c2), and (d2)) shows the corresponding absolute error, including the $L_2$-error and the $R^2$-score metrics.
• Figure 4: Left-handed CO RL derivative: Comparison between the SPH approximation and the analytical solution for (a1) $\sin (\pi x)$, (b1) $\cos (\pi x)$, (c1) $\exp(x)$, and (d1) $(x-1)^3$. The right column ((a2), (b2), (c2), and (d2)) shows the corresponding absolute error, including the $L_2$-error and the $R^2$-score metrics.
• Figure 5: Left-handed CO RL integral: Comparison between the SPH approximation and the analytical solution for (a1) $\sin (\pi x)$, (b1) $\cos (\pi x)$, (c1) $\exp(x)$, and (d1) $(x-1)^3$. The right column ((a2), (b2), (c2), and (d2)) shows the corresponding absolute error, including the $L_2$-error and the $R^2$-score metrics.