A comparative numerical study of fractional derivatives and their role in capturing memory effects in viscoelastic materials

Abstract

Fractional calculus is a crucial mathematical tool for understanding complex systems across various fields, such as physics, engineering, and biology. This work delves into the intricate task of computing fractional order derivatives, which typically necessitates numerical methods due to their inherent complexity. We present a comprehensive comparative study of three significant numerical techniques: the Riemann-Liouville, Grunwald-Letnikov, and Caputo derivatives. The study begins with a thorough examination of fractional order derivatives, highlighting their fundamental importance in modern scientific research. We investigate basic transcendental functions—sine, cosine, and exponential—due to their vital role in modeling real-world systems. Our research reveals the complex relationship between sensitivity and precision by consistently computing fractional derivatives across various orders. Principal Component Analysis (PCA) is employed for statistical validation to enhance the credibility of our findings, demonstrating that fractional derivatives can reliably represent underlying data patterns. Consequently, this study significantly advances our understanding of fractional order derivatives and underscores the importance of carefully handling fractional orders in diverse applications. These findings support the reliability of data analytics and deepen our comprehension of the behavior of complex systems. An important application of this work lies in the modeling and control of viscoelastic materials, where fractional calculus provides a powerful framework for accurately capturing memory-dependent behaviors and complex stress-strain relationships.