A Haar wavelet operational matrix method for fractional Riccati differential equations with Atangana’s beta derivative

Najeeb Alam  Khan; Mumtaz  Ali; Asmat  Ara; Samreen  Ahmed; Ali Saleh  Alshomrani; Faris  Alzahrani

doi:10.59400/adecp3476

A Haar wavelet operational matrix method for fractional Riccati differential equations with Atangana’s beta derivative

Najeeb Alam Khan
Department of Mathematics, University of Karachi, Karachi 75270, Pakistan
Mumtaz Ali
Department of Mathematics, University of Karachi, Karachi 75270, Pakistan; Department of Basic Sciences, Balochistan University of Engineering and Technology, Khuzdar 89100, Pakistan
Asmat Ara
Department of Mathematics, University of Karachi, Karachi 75270, Pakistan
Samreen Ahmed
Department of Mathematics, University of Karachi, Karachi 75270, Pakistan
Ali Saleh Alshomrani
Mathematical Modeling and Applied Computation (MMAC) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Faris Alzahrani
Mathematical Modeling and Applied Computation (MMAC) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Article ID: 3476

Keywords: operational matrix method; Haar wavelet; Atangana’s beta derivative; fractional Riccati differential equations

Abstract

In this paper, we present an operational matrix method for integrating fractional Riccati differential equations (FRDEs) based on Haar wavelets. The fractional derivative is considered in the sense of Atangana’s beta derivative, which effectively captures the memory and nonlocal characteristics of complex dynamical systems. The proposed technique employs a truncated Haar wavelet series and an operational matrix of integration to convert the governing FRDEs into a system of algebraic equations. These equations are then formulated as objective functions, and the unknown Haar wavelet coefficients are determined using a random search optimization procedure. This transformation reduces the computational complexity and provides an efficient framework for handling nonlinear fractional-order problems. The convergence and validity of the proposed method are demonstrated using several illustrative examples. The numerical results obtained with the proposed approach are compared with those from the Adams–Bashforth method, and the results show that the present technique provides more accurate approximations. Furthermore, to assess the performance and reliability of the method, several error metrics were computed, including the mean absolute deviation, root mean square error, Theil’s inequality coefficient, Nash–Sutcliffe efficiency (NSE), and variance account for (VAF), for different numbers of collocation points. The results confirm that the Haar wavelet operational matrix method is simple to implement, computationally efficient, and highly accurate for solving fractional Riccati differential equations.

Published

2026-06-15

How to Cite

Khan, N. A., Ali, M., Ara, A., Ahmed, S., Alshomrani, A. S., & Alzahrani, F. (2026). A Haar wavelet operational matrix method for fractional Riccati differential equations with Atangana’s beta derivative. Advances in Differential Equations and Control Processes, 33(2). https://doi.org/10.59400/adecp3476

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Vol. 33 No. 2 (2026): In Progress

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This work is licensed under a Creative Commons Attribution 4.0 International License.

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