Numerical Simulation and Analysis of Fractional Differential Equations through Wavelets

Abstract

The main objective of this work is to develop efficient and accurate computational methods for solving a variety of fractional models by using various wavelet-based numerical techniques. The research explores different wavelet families, including Taylor, modified Taylor and Bell wavelets, and constructs operational matrices of fractional integration to convert complex fractional systems into simpler algebraic equations using collocation and Galerkin methods. The study includes several important applications such as modeling and simulation of fractional-order electrical circuits like RL, RC, LC, and RLC circuits under the generalized Caputo fractional derivative. It also analyzes the dynamical behavior of non-delay and delay fractional optimal control problems by using fractional-order Taylor wavelets together with the method of Lagrange multipliers. Furthermore, it develops a numerical framework for solving higher-order fractional Emden-Fowler equations through fractional Bell wavelets in combination with the Caputo-Fabrizio operator, which helps to remove the singularity problem present in traditional formulations. The work also presents a complete discretization approach for solving fractional Volterra and Fredholm integro-differential equations of the second kind by using modified Taylor wavelets under the Riemann Liouville fractal fractional derivative. In each case, the proposed method demonstrates high accuracy, stability, and fast convergence, supported by numerical results, tables, and graphical representations that compare favourably with existing numerical approaches. In addition to the numerical modeling, the study explores the role of artificial intelligence and robotics in mathematics education, showing how these technologies enhance computational understanding and make learning more interactive. Overall, this research integrates fractional calculus, wavelet analysis, and artificial intelligence to create a unified and effective framework for modeling complex dynamical systems and improving modern co