Abstract

This paper presents a rigorous and systematic investigation of fractional-order derivatives and their role in the formulation and analysis of fractional differential equations. The main definitions of fractional calculus, including the Riemann–Liouville, Caputo, and Grünwald–Letnikov approaches, are examined within a unified theoretical framework. Their mathematical properties, structural differences, and areas of applicability are comparatively analyzed. Particular attention is devoted to the Caputo derivative due to its compatibility with classical initial and boundary conditions, which makes it especially suitable for physical and engineering applications. The fundamental properties of fractional derivatives, such as linearity, non-local behavior, memory effects, and their relationship with classical integer-order differentiation, are discussed in detail. Analytical solution methods for fractional differential equations are developed using the Laplace transform and the Mittag–Leffler function, which generalizes the exponential function in fractional dynamics. Step-by-step solution procedures for complex initial–boundary value problems are provided to ensure mathematical consistency and clarity. The obtained results demonstrate that fractional-order models provide enhanced accuracy and flexibility in describing anomalous diffusion, hereditary phenomena, and memory-dependent processes. The study confirms the effectiveness of fractional calculus as a powerful mathematical tool for modeling complex dynamical systems in applied sciences and engineering.

References

[1]. Podlubny I., Fractional Differential Equations, Academic Press - San Diego, 1999, 340 p.
[2]. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier - Amsterdam, 2006, 523 p.
[3]. Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives,Gordon and Breach Science Publishers, Amsterdam, 1993, 976 p.
[4]. Diethelm, K., The Analysis of Fractional Differential Equations, Springer, Berlin, 2010, 247 p.
[5]. Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993, 384 p.
[6]. Oldham, K.B. & Spanier, J. (1974). The fractional calculus: theory and applications of differentiation and integration to arbitrary order. Mathematics in Science and Engineering, Vol. 111. Academic Press, New York.
[7]. Samko, S.G., Kilbas, A.A. & Marichev, O.I. (1993). Fractional integrals and derivatives: theory and applications. Gordon and Breach, New York.
[8]. Podlubny, I. (1999). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering, Vol. 198. Academic Press, San Diego.
[9]. Kilbas, A.A., Srivastava, H.M. & Trujillo, J.J. (2006). Theory and applications of fractional differential equations. North-Holland Mathematics Studies, Vol. 204. Elsevier, Amsterdam.
[10]. Diethelm, K. (2010). The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Lecture Notes in Mathematics, Vol. 2004. Springer, Berlin.
[11]. Mainardi, F. (2010). Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. Imperial College Press, London.
[12]. Atanacković, T.M., Pilipović, S., Stanković, B. & Zorica, D. (2014). Fractional calculus with applications in mechanics: vibrations and diffusion processes. ISTE-Wiley, London.
[13]. Li, C. & Cai, M. (2020). Theory and numerical approximations of fractional integrals and derivatives. Other Titles in Applied Mathematics, Vol. 163. SIAM, Philadelphia.
[14]. Agarwal, P., Cattani, C. & Momani, S. (eds.) (2024). Fractional differential equations: theoretical aspects and applications. Elsevier/Academic Press, Amsterdam.
[15]. Teso, F. & Gómez-Castro, D. (2025). "Fractional derivatives: an extension of classical analysis to non-integer orders". Gaceta de la Real Sociedad Española, translated from Spanish.