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A Spectrally Accurate Shifted Lucas Collocation Framework for Fractional Lanchester Combat Dynamics with Time-Dependent Variable Coefficients | ||
| Journal of Computational Applied Mechanics | ||
| دوره 57، شماره 3، مهر 2026، صفحه 425-442 اصل مقاله (1.43 M) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22059/jcamech.2026.412544.1799 | ||
| نویسندگان | ||
| M. H. Salama1؛ H. A. Zedan1؛ W. M. Abd-Elhameed2؛ Y. H. Youssri* 2، 3، 4 | ||
| 1Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt | ||
| 2Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt | ||
| 3Faculty of Engineering, Egypt University of Informatics, Knowledge City, New Administrative Capital 19519, Egypt | ||
| 4Associate Fellow (AFHEA) of the Higher Education Academy (Advance HE), UK | ||
| چکیده | ||
| This paper is confined to developing a rigorous computational framework using the shifted Lucas polynomials for the numerical treatment of the generalized fractional-order Lanchester combat model characterized by time-dependent variable coefficients. The method uses an exact operational matrix for the Caputo derivative to handle the singular kernel, thereby eliminating the need for numerical quadrature. A global polynomial projection at shifted Chebyshev–Gauss–Lobatto nodes eliminates predictor–corrector errors and preserves high-order accuracy under memory effects. A rigorous analysis employing a generalized Gronwall inequality establishes well-posedness and derives sharp stability bounds via Mittag–Leffler functions. Numerical investigations validate enhanced stability and efficiency, especially for memory effects and heavy-tail decay, and error estimates indicate super-geometric convergence. | ||
| کلیدواژهها | ||
| Fractional dynamical systems؛ Spectral collocation؛ Caputo derivative؛ Operational matrix؛ Coupled fractional ODEs؛ Mittag-Leffler stability؛ Numerical methods | ||
| مراجع | ||
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[8] J. Shen, T. Tang, L. Wang, 2011, Spectral Methods: Algorithms, Analysis and Applications, Springer Science & Business Media,
[9] W. M. Abd-Elhameed, Y. H. Youssri, New connection formulae between Chebyshev and Lucas polynomials: New expressions involving Lucas numbers via hypergeometric functions, Adv. Stud. Contemp. Math., Vol. 28, No. 3, pp. 357-368, 2018.
[10] M. H. Salama, H. A. Zedan, W. M. Abd-Elhameed, Y. H. Youssri, Galerkin method with modified shifted lucas polynomials for solving the 2d poisson equation, J. Comput. Appl. Mech, Vol. 56, pp. 737-775, 2025.
[11] M. H. Salama, H. A. Zedan, W. M. Abd-Elhameed, Y. H. Youssri, Spectral Treatment of the Fractional Bratu Equation via Shifted Lucas Polynomials: A Precise Collocation Approach with Error Quantification, Contemp. Math., Vol. 6, No. 5, pp. 6832-6870, 2025.
[12] W. M. Abd-Elhameed, Y. H. Youssri, A. G. Atta, Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials, J. Math. Model., Vol. 12, No. 2, pp. 277-299, 2024.
[13] Y. H. Youssri, W. M. Abd-Elhameed, H. M. Ahmed, New fractional derivative expression of the shifted third-kind Chebyshev polynomials: Application to a type of nonlinear fractional pantograph differential equations, J. Funct. Spaces, Vol. 2022, pp. 9836512, 2022.
[14] W. M. Abd-Elhameed, Y. H. Youssri, New formulas of the high-order derivatives of fifth-kind Chebyshev polynomials: Spectral solution of the convection-diffusion equation, Numer. Methods Partial Differ. Equ., Vol. 40, No. 2, 2024.
[15] Y. H. Youssri, A. G. Atta, Spectral Collocation Approach via Normalized Shifted Jacobi Polynomials for the Nonlinear Lane--Emden Equation with Fractal-Fractional Derivative, Fractal and Fractional, Vol. 7, No. 2, pp. 133, 2023.
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[18] H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., Vol. 328, No. 2, pp. 1075-1081, 2007.
[19] W. M. Abd-Elhameed, Y. H. Youssri, Generalized Lucas polynomial sequence approach for fractional differential equations, Nonlinear Dyn., Vol. 89, No. 2, pp. 1341-1355, 2017.
[20] W. M. Abd-Elhameed, O. M. Alqubori, A. G. Atta, A Collocation Procedure for Treating the Time-Fractional FitzHugh-Nagumo Differential Equation Using Shifted Lucas Polynomials, Mathematics, Vol. 12, No. 23, pp. 3672, 2024.
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