Numerical solutions to nonlinear fractional differential equations: New results and comparative study in biological and engineering sciences

Abstract

Mathematical models that extend beyond classical differential equations are necessary for understanding complicated dynamical systems, such as circadian rhythm-regulated brain activity or intricate feedback loops in control engineering. Nonlinear fractional differential equations (NFDEs), especially those that are constructed using the Caputo derivative (CD), have become more common in recent years because of their ability to represent memory-dependent phenomena in a variety of biological and engineering fields. Motivated by practical applications like the circadian variation of brain metabolites and the behavior of relaxation-oscillation systems, the present work performs a thorough comparative examination of these advanced models. Three different approaches to solving the NFDEs are used: the Picard Method (PM), the Adomian Decomposition Method (ADM), and the Proposed Numerical Method (PNM). Each approach is rigorously analysed in terms of convergence, accompanied by detailed error estimates that confirm the existence and uniqueness of the solutions.