Optimized numerical solutions for a certain fractional dynamic systems using generalized Laguerre collocation technique Optimized numerical solutions for a certain fractional

• Published in: International journal of dynamics and control Vol. 13; no. 9
• Main Authors: Adel, M., Khader, M. M.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 28.08.2025
• Subjects: Complexity; Control; Control and Systems Theory; Dynamical Systems; Engineering; Vibration; Smoking system; 47H10; 41A30; 34A12; Generalized Laguerre collocation method; Dynamical system of marriage; RK4 method; Caputo fractional derivative; Convergence analysis; 65N20
• ISSN: 2195-268X; 2195-2698
• DOI: 10.1007/s40435-025-01830-3

In this study, we introduce a powerful computational algorithm for simulating two significant models, namely the fractional smoking model (FSM) and the fractional dynamic marriage model (FDMM). Smoking is a prevalent global practice, stands as the third leading cause of human mortality, and is a significant contributor to disease burden. While the dynamic marriage model, representing the intricate dynamics of romantic partnerships, plays a pivotal role in understanding and predicting the complexities of human relationships, shaping the fabric of societies and individual lives across the globe, and couples’ love affairs. The fractional derivative considered here is in the Caputo sense. The proposed approach relies on generalized Laguerre polynomials (GLPs) within a spectral collocation method. We give an approximate formula for the fractional derivatives and use it with the help of properties of the GLPs to convert each model into a system of algebraic equations, considered as a constrained optimization problem, which is subsequently optimized to determine unknown coefficients. Our study includes convergence analysis, supported by numerical simulations, demonstrating the validity of the proposed algorithm. We confirm the accuracy and effectiveness of our approach by evaluating the residual error function (REF). Additionally, we provide a comparative analysis with results obtained using the RK4 method, and other published results using the SCM based on the Gegenbauer wavelet polynomials. Finally, we will provide some necessary recommendations for each model individually through comments on the results we obtain, to clarify the importance of these solutions and this numerical simulation.