Offline–Online Decoupled Architecture for Projected Microsecond-Latency and near Machine-Precision Caputo Fractional Derivative Evaluation, with an Edge-Computing Feasibility Study
- 1Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman P.O. Box 346, United Arab Emirates
- 2Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman P.O. Box 346, United Arab Emirates
Abstract
Accurate evaluation of the Caputo fractional derivative (CFD) of arbitrary non-integer order is essential for modeling memory-dependent phenomena. However, the classical Shifted Gegenbauer Pseudospectral (SGPS) method attains exponential convergence only on a zero-measure set of the fractional order parameter; for all other orders, the transformed integrand exhibits reduced endpoint regularity, limiting quadrature convergence to algebraic rates. We introduce a ϕ-generalized algebraic transformation that enforces Cr endpoint regularity for any prescribed integer r>0, and establish a convergence hierarchy for the interpolation-induced truncation error (ITE) across analytic, Gevrey, C∞, and finitely smooth data. For data analytic in a disc around the evaluation point, the transformed integrand has a convergent fractional power series with geometrically decaying coefficients, which explains the fast pre-asymptotic quadrature convergence observed numerically. Precomputable weighted Fractional Shifted Gegenbauer Integration Matrices (FSGIMs), assembled by a batched formulation, reduce each online evaluation to one O(Mn) matrix–vector product whose cost does not depend on the history length, apart from the retained sample buffer. Double-precision MATLAB benchmarks give excellent accuracies for problems with known solutions and improve on several finite-difference, diffusive-representation, and adaptive-quadrature baselines by orders of magnitude. Microsecond-scale latency on a Cortex-M4-class microcontroller is projected from operation counts and desktop timings; it has not been measured on hardware. A two-cycle NASA battery case study illustrates the feasibility of CFD-based State-of-Health features.
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BibTeX
@article{elgindy2026offlineonline,
title = {{Offline–Online Decoupled Architecture for Projected Microsecond-Latency and near Machine-Precision Caputo Fractional Derivative Evaluation, with an Edge-Computing Feasibility Study}},
author = {Kareem T. Elgindy},
journal = {Fractal and Fractional},
year = {2026},
month = oct,
doi = {10.3390/fractalfract10110752},
url = {https://www.mdpi.com/2504-3110/10/11/752},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Offline–Online Decoupled Architecture for Projected Microsecond-Latency and near Machine-Precision Caputo Fractional Derivative Evaluation, with an Edge-Computing Feasibility Study
AU - Kareem T. Elgindy
PY - 2026
DA - 2026/10/28
JO - Fractal and Fractional
DO - 10.3390/fractalfract10110752
UR - https://www.mdpi.com/2504-3110/10/11/752
AB - Accurate evaluation of the Caputo fractional derivative (CFD) of arbitrary non-integer order is essential for modeling memory-dependent phenomena. However, the classical Shifted Gegenbauer Pseudospectral (SGPS) method attains exponential convergence only on a zero-measure set of the fractional order parameter; for all other orders, the transformed integrand exhibits reduced endpoint regularity, limiting quadrature convergence to algebraic rates. We introduce a ϕ-generalized algebraic transformation that enforces Cr endpoint regularity for any prescribed integer r>0, and establish a convergence hierarchy for the interpolation-induced truncation error (ITE) across analytic, Gevrey, C∞, and finitely smooth data. For data analytic in a disc around the evaluation point, the transformed integrand has a convergent fractional power series with geometrically decaying coefficients, which explains the fast pre-asymptotic quadrature convergence observed numerically. Precomputable weighted Fractional Shifted Gegenbauer Integration Matrices (FSGIMs), assembled by a batched formulation, reduce each online evaluation to one O(Mn) matrix–vector product whose cost does not depend on the history length, apart from the retained sample buffer. Double-precision MATLAB benchmarks give excellent accuracies for problems with known solutions and improve on several finite-difference, diffusive-representation, and adaptive-quadrature baselines by orders of magnitude. Microsecond-scale latency on a Cortex-M4-class microcontroller is projected from operation counts and desktop timings; it has not been measured on hardware. A two-cycle NASA battery case study illustrates the feasibility of CFD-based State-of-Health features.
SN - 2504-3110
ER -