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Journal of Information and Optimization Sciences cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-0103·ISSN (Print): 0252-2667

WoS  JIF 2026 : 0.4 (Q4)

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Open Access Research Article

Generalized operational matrices to analyze inhomogeneous multi-term fractional differential equations : An efficient direct method

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pp. 951–976Vol. 47Issue 3March 2026DOI: 10.47974/JIOS-1602XML
Received:
09 Oct 2023
Published Online:
15 Jul 2025
Article type:
Research Article
Language:
EN
Article no.:
JIOS-1602
Pages:
951–976

Abstract

Fractional calculus has gained popularity as a modeling and control tool for dynamical systems. For systems with commensurate and non-commensurate generalized fractional orders, some control theory techniques have been proposed. In this study, a novel iterative method for dealing with fractional order systems is provided. An orthogonal function set, known as hybrid function (HF) set is linearly integrated with the sample and hold function (SHF) set and triangular function (TF) set, is a novel technique for evaluating fractional order systems, used to represent fractional order plants. To illustrate how the suggested approach has been validated, a few instances are given. A few numerical examples are provided, along with relevant graphs and tables, to validate the proposed approach.

Keywords

Subject Classifications

15A3034A3034A4541A6547A08

References

[1] T. Mahmood, M. ur Rahman, M. Arfan, S. I. Kayani, and M. Sun, Mathematical study of Algae as a bio-fertilizer using fractal–fractional dynamic model, Mathematics and Computers in Simulation, vol. 203, pp. 207-222 (2023).
[2] B. Abdullaeva, M. J. C. Opulencia, V. Borisov, K. F. Uktamov, W. K. Abdelbasset, A. K. J. Al-Nussair, and A. H. Jabbar, Optimal variable estimation of a Li-ion battery model by fractional calculus and bio-inspired algorithms, Journal of Energy Storage, vol. 54, 105323 (2022).
[3] H. P. Ren, X. Wang, J. T. Fan, and O. Kaynak, Fractional order sliding mode control of a pneumatic position servo system, Journal of the Franklin Institute, vol. 356, no. 12, pp. 6160-6174 (2019).
[4] L. Liu, L. Zhang, G. Pan, and S. Zhang, Robust yaw control of autonomous underwater vehicle based on fractional-order PID controller, Ocean Engineering, vol. 257, 111493 (2022). 
[5] K. Vanchinathan, and N. Selvaganesan, Adaptive fractional order PID controller tuning for brushless DC motor using artificial bee colony algorithm, Results in Control and Optimization, vol. 4, 100032 (2021). 
[6] B. Hekimoğlu, Optimal tuning of fractional order PID controller for DC motor speed control via chaotic atom search optimization algorithm, IEEE Access, vol. 7, pp. 38100-38114 (2019).
[7] S. Roychoudhury, U. Rahman and Md. S. H. Rabbi, Estimating Response of Fractional Order Systems and Control: An Orthogonal HF-Based Approach, International Conference on Signal Processing, Information, Communication and Systems 2024 (SPICSCON-2024), IEEE, Khulna, Bangladesh (2024).
[8] H. Almusawa, and A. Jhangeer, A study of the soliton solutions with an intrinsic fractional discrete nonlinear electrical transmission line, Fractal and Fractional, vol. 6, no. 6, pp. 334 (2022).
[9] H. Almusawa, A. Jhangeer, and M. Munawar, M., Analytical analyses for a fractional low-pass electrical transmission line model with dynamic transition. Symmetry, vol. 14, no. 7, pp. 1377 (2022).
[10] J. P. Hollkamp, and F. Semperlotti, Application of fractional order operators to the simulation of ducts with acoustic black hole terminations, Journal of Sound and Vibration, vol. 465, pp. 115035 (2020).
[11] K. S. Mawonou, A. Eddahech, D. Dumur, D. Beauvois, D., and E. Godoy, Improved state of charge estimation for Li-ion batteries using fractional order extended Kalman filter, Journal of Power Sources, vol. 435, pp. 226710 (2019).
[12] J. I. Hidalgo-Reyes, J. F. Gómez-Aguilar, R. F. Escobar-Jiménez, V. M. Alvarado-Martínez, and M. G. López-López, Classical and fractional-order modeling of equivalent electrical circuits for supercapacitors and batteries, energy management strategies for hybrid systems and methods for the state of charge estimation: A state of the art review, Microelectronics Journal, vol. 85, pp. 109-128 (2019).
[13] K. S. Mawonou, A. Eddahech, D. Dumur, D. Beauvois, and E. Godoy, Improved state of charge estimation for Li-ion batteries using fractional order extended Kalman filter, Journal of Power Sources, vol. 435, pp. 226710 (2019).
[14] X. Wu, and Y. Huang, Adaptive fractional-order non-singular terminal sliding mode control based on fuzzy wavelet neural networks for omnidirectional mobile robot manipulator, ISA transactions, vol. 121, pp. 258-267 (2022).
[15] M. I. Haro-Olmo, I. Tejado, B. M. Vinagre, and V. Feliu-Batlle, Fractional-Order Models of Damping Phenomena in a Flexible Sensing Antenna Used for Haptic Robot Navigation, Fractal and Fractional, vol. 7, no. 8, pp. 621 (2023).
[16] S. Roychoudhury, A. Deb, and G. Sarkar, Analysis and synthesis of time-varying systems via orthogonal hybrid functions (HF) in state space environment, International Journal of Dynamics and Control, vol. 3, pp. 389-402 (2015).
[17] S. Roychoudhury, and A. Deb, Identification of multi-delay systems using orthogonal hybrid functions in state space environment, International Journal of Modelling, Identification and Control, vol. 30, no. 2, pp. 93-104 (2018).
[18] T. F. Nonnenmacher, and R. Metzler, On the Riemann-Liouville fractional calculus and some recent applications, Fractals, vol. 3, no. 3, pp. 557-566 (1995).
[19] T. Abdeljawad, On Riemann and Caputo fractional differences, Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602-1611 (2011).
[20] M. D. Ortigueira, L. Rodríguez-Germá, and J. J. Trujillo, Complex Grünwald–Letnikov, Liouville, Riemann–Liouville, and Caputo derivatives for analytic functions, Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4174-4182 (2011).
[21] S. Roychoudhury, A. Deb, and G. Sarkar, Analysis and synthesis of homogeneous/non-homogeneous control systems via orthogonal hybrid functions (HF) under state space environment, Journal of Information and Optimization Sciences, vol. 35, no. 5-6, pp. 431-482 (2014).
[22] J. F. Neville, and A. C. Joseph, Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations, Journal of Computational and Applied mathematics, vol. 229, pp. 382-391 (2009).
[23] S. C. Shiralashetti and A. B. Deshi, An efficient Haar Wavelet collocation method for the numerical solution of multi-term fractional differential equations, Nonlinear Dyn, vol. 83, pp. 293-303 (2016).

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