International Journal of Pure & Applied Mathematical Research

EXTENDED HYBRID ANALYTICAL METHODS FOR COUPLED NONLINEAR FRACTIONAL DYNAMICAL SYSTEMS: ADAPTIVE RESIDUAL CORRECTION, SPECTRAL DECOMPOSITION, AND MULTI-ORDER STABILITY

D. Sugumar, S. Balakrishnan & A. Ganesh

D. Sugumar¹,²,* S. Balakrishnan¹,† A. Ganesh³,‡

¹ PG & Research Department of Mathematics, Islamiah College (Autonomous), Vaniyambadi – 635752, Tamil Nadu (Affiliated to Thiruvalluvar University, Vellore – 632115)

² PG & Research Department of Mathematics, Government Arts College for Men, Krishnagiri – 635 001

³ Department of Mathematics, Government Arts and Science College, Hosur – 635 110

DOI : https://doi.org/10.5281/zenodo.20195205 Page No : 01-12

Published Online : 2026-05-15

Download Full Article : PDF Check for Updates

ABSTRACT

This paper presents a significant extension of the Hybrid Analytical Framework (HAF) and the Unified Analytical Framework previously established by the authors for nonlinear fractional differential equations (NFDEs). The present work introduces the Extended Hybrid Analytical Method (EHAM), which incorporates three principal innovations: (i) an Adaptive Residual Spectral Correction (ARSC) mechanism that dynamically adjusts correction parameters using spectral energy norms; (ii) a rigorous theory of Coupled Multi-Order Fractional Systems that extends previous single-order existence and uniqueness results to systems involving simultaneous Caputo derivatives of distinct rational orders; and (iii) a new Fractional Spectral Decomposition Theorem (FSDT) that decomposes nonlinear forcing terms along eigenfunctions of fractional Sturm–Liouville operators. Five new theorems are established and rigorously proved, including the Multi-Order Contraction Theorem, the Adaptive Convergence Rate Theorem, the Spectral Stability Theorem, the Fractional Coupling Regularity Theorem, and the Generalized Error Propagation Theorem. The EHAM is applied to five complex engineering systems: a coupled fractional thermo-viscoelastic oscillator, a multi-order fractional epidemiological model, a fractional nonlinear Schrödinger-type equation, a coupled fractional electrochemical circuit with nonlinear resistance, and a fractional anomalous diffusion–reaction system. Numerical validation confirms exponential decay of the residual error and asymptotic Mittag-Leffler stability across all fractional orders α ∈ (0, 1]. Comparative analysis demonstrates that EHAM achieves a superior convergence rate and tighter error bounds relative to classical HAF, the Adomian Decomposition Method (ADM), and the Homotopy Perturbation Method (HPM).

Keywords: Fractional differential equations; Caputo derivative; Hybrid analytical method; Adaptive residual correction; Spectral decomposition; Coupled systems; Mittag-Leffler stability; Multi-order fractional systems; Convergence analysis; Engineering dynamics

References

[1] Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego.

[2] Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.

[3] Diethelm, K. (2010). The Analysis of Fractional Differential Equations. Springer, Berlin.

[4] Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London.

[5] Oldham, K. B., & Spanier, J. (1974). The Fractional Calculus. Academic Press, New York.

[6] Baleanu, D., Diethelm, K., Scalas, E., & Trujillo, J. J. (2012). Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore.

[7] He, J. H. (1999). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178(3–4), 257–262.

[8] Adomian, G. (1994). Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic, Dordrecht.

[9] Li, C., & Zeng, F. (2015). Numerical Methods for Fractional Calculus. CRC Press, Boca Raton.

[10] Atangana, A., & Baleanu, D. (2016). New fractional derivatives with nonlocal and non-singular kernel. Thermal Science, 20(2), 763–769.

[11] Luchko, Y. (2012). Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fractional Calculus and Applied Analysis, 12(4), 409–422.

[12] Zhou, Y. (2014). Basic Theory of Fractional Differential Equations. World Scientific, Singapore.

[13] Gorenflo, R., Kilbas, A. A., Mainardi, F., & Rogosin, S. V. (2014). Mittag-Leffler Functions: Related Topics and Applications. Springer, Berlin.

[14] Sayevand, K., & Pichaghchi, K. (2016). Analysis of nonlinear fractional KdV equation based on the homotopy perturbation method. Communications in Nonlinear Science and Numerical Simulation, 35, 115–125.

[15] Duan, J. S., Rach, R., Baleanu, D., & Wazwaz, A. M. (2012). A review of the Adomian decomposition method and its applications to fractional differential equations. Communications in Fractional Calculus, 3(2), 73–99.

[16] Abbas, S., Benchohra, M., & N'Guérékata, G. M. (2012). Topics in Fractional Differential Equations. Springer, New York.

[17] Zayernouri, M., & Karniadakis, G. E. (2013). Fractional Sturm–Liouville eigen-problems: Theory and numerical approximation. Journal of Computational Physics, 252, 495–517.

[18] Liu, F., Zhuang, P., & Liu, Q. (Eds.) (2015). Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term. SIAM Journal on Numerical Analysis, 47(3), 1760–1781.

[19] Uchaikin, V. V. (2013). Fractional Derivatives for Physicists and Engineers. Springer, Berlin.

[20] Yang, X. J., Baleanu, D., & Srivastava, H. M. (2015). Local Fractional Integral Transforms and Their Applications. Academic Press, Oxford.

[21] Tarasov, V. E. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Berlin.

[22] Klimek, M., & Agrawal, O. P. (2013). Fractional Sturm–Liouville problem. Computers & Mathematics with Applications, 66(5), 795–812.

[23] Ye, H., Gao, J., & Ding, Y. (2007). A generalized Gronwall inequality and its application to a fractional differential equation. Journal of Mathematical Analysis and Applications, 328(2), 1075–1081.

[24] Sugumar, D., Balakrishnan, S., & Ganesh, A. (2024). A novel analytical approach for solving nonlinear fractional differential equations with applications in engineering systems. [Paper I of this series].

[25] Sugumar, D., Balakrishnan, S., & Ganesh, A. (2025). Hybrid analytical techniques for nonlinear fractional differential equations in complex engineering dynamics. [Paper II of this series].

[26] Sugumar, D., Balakrishnan, S., & Ganesh, A. (2025). A unified analytical framework for nonlinear fractional dynamical models in engineering applications. [Paper III of this series].

Cite this Article as : D. Sugumar, S. Balakrishnan & A. Ganesh (2026), “Extended Hybrid Analytical Methods for Coupled Nonlinear Fractional Dynamical Systems: Adaptive Residual Correction, Spectral Decomposition, and Multi-Order Stability”, International Journal of Pure & Applied Mathematical Research, Vol. 10, Issue 1, 2026, pp 1-12, DOI: https://doi.org/10.5281/zenodo.20195205

Article View: 22
PDF Download: 3