Highly Effective Iterative Approach for Solving Fractional Riccati Equations via Ujlayan-Dixit Operator

Abstract

The aim of this study is to review the concepts and applications related to fractional calculus. Fractional calculus and its applications constitute an important field in the study of modern mathematical models, and among its most prominent equations is the fractional Riccati equation. This equation represents a fundamental tool for modeling nonlinear phenomena in quantum mechanics and control systems. Despite its significance, traditional approaches based on the Riemann–Liouville or Caputo derivatives often lack closed-form analytical solutions and suffer from considerable computational complexity. In contrast, the Ujlayan–Dixit operator provides a simpler structure for the proportional fractional derivative of order α however, effective iterative methods for solving fractional Riccati equations based on this operator have not yet been sufficiently investigated. In light of this research gap, the present study aims to develop a form of the variational iteration method that is compatible with the Ujlayan–Dixit operator, with the objective of providing a simpler and more efficient numerical approach for solving fractional Riccati equations. The proposed method is applied to a set of numerical examples to demonstrate its effectiveness and its ability to yield accurate approximate solutions.