HYBRID FRACTIONAL CALCULUS AND ANN FRAMEWORK FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS

Abstract

Nonlinear differential equations play a fundamental role in modeling complex real-world phenomena across physics, biology, engineering, and finance. However, their analytical solutions are often difficult or impossible to obtain, motivating the development of efficient computational approaches. In recent years, fractional calculus has gained significant attention due to its ability to capture memory and hereditary properties that classical integer-order models fail to represent. Hybrid fractional approaches provide enhanced flexibility in modeling nonlinear dynamics, particularly when combined with advanced computational techniques. At the same time, artificial neural networks (ANNs) have emerged as powerful universal function approximators capable of handling nonlinear mappings with high accuracy. Despite these advances, limited studies have explored the integration of fractional calculus with ANN-based frameworks to solve nonlinear differential equations in a systematic manner.

This paper introduces a novel hybrid framework that integrates fractional-order calculus with ANN architectures to provide an efficient and accurate method for solving nonlinear differential equations. The framework leverages the memory-preserving characteristics of fractional operators while exploiting the learning and generalization power of ANNs. The proposed method constructs a hybrid optimization scheme where ANN parameters are updated using fractional differential operators, leading to enhanced convergence rates and improved stability compared to traditional ANN training approaches. The effectiveness of the framework is evaluated on benchmark nonlinear problems, including chaotic systems and fractional-order physical models, demonstrating superior accuracy and robustness. Furthermore, the hybrid approach provides a scalable methodology that can be extended to higher-dimensional systems, partial differential equations, and real-time dynamic modeling.

The results suggest that integrating fractional calculus into ANN frameworks bridges the gap between classical analytical methods and modern computational intelligence. This synergy not only improves numerical stability and approximation accuracy but also opens new directions in scientific computing, where complex nonlinear behaviors require both memory-aware modeling and adaptive learning mechanisms. Ultimately, the proposed hybrid fractional-ANN framework offers a promising paradigm for advancing the solution of nonlinear differential equations in both theoretical and applied domains.