Modern Calculus

Abstract

Calculus, since its inception by Newton and Leibniz, has served as a foundational tool for understanding change, motion, and accumulation. Over the centuries, its scope has expanded beyond the classical framework, giving rise to modern extensions that address new mathematical and scientific challenges. This article provides a comprehensive review of three significant branches: fractional calculus, multiplicative calculus, and difference calculus. Fractional calculus generalizes differentiation and integration to non-integer orders, offering powerful models for viscoelastic materials, anomalous diffusion, and memory effects in physical and biological systems. Multiplicative calculus, rooted in proportional change, reinterprets differentiation in terms of ratios, providing natural formulations for processes in finance, population dynamics, and biological growth. Difference calculus, along with its quantum variants, extends the principles of calculus to discrete settings, enabling applications in numerical methods, dynamical systems, and quantum models. By highlighting their theoretical foundations and practical applications, this paper illustrates how these non-Newtonian approaches enrich the classical view of calculus, bridging continuous and discrete analysis and offering versatile tools for modern science and engineering.