Research Projects

Fractional order differential equations (FDEs) offer powerful modeling capabilities for real-world systems exhibiting memory and hereditary properties, such as viscoelastic materials, control systems, and biological processes. These systems, often nonlinear in nature, are analytically intractable and require advanced numerical and computational methods for their study. This project proposes a hybrid framework for analyzing and solving fractional order nonlinear dynamical systems using a combination of wavelet methods, Artificial Neural Networks (ANNs), and Physics-Informed Neural Networks (PINNs). Wavelet transforms will be used to handle multiscale aspects of the problem and reduce computational complexity by sparsely representing the dynamics in the time-frequency domain. Artificial Neural Networks will be employed to approximate the solutions of FDEs by learning the underlying patterns in the data generated from the systems. Furthermore, PINNs will be developed to incorporate the physical laws governing the system directly into the loss function of the neural network, enabling high-precision solutions without requiring large labeled datasets. This approach preserves physical consistency while significantly improving accuracy and generalization. The research will involve: The formulation of benchmark fractional nonlinear systems; Development of wavelet-based ANN and PINN models; Validation of results through numerical comparisons with existing methods; Sensitivity and stability analysis of the proposed models. This project bridges analytical mathematics, computational modeling, and machine learning, contributing to the development of robust and efficient numerical techniques for fractional systems. The findings are expected to provide a generalized framework that can be extended to a wide range of nonlinear systems across science and engineering disciplines.