Fractional beam dynamics with nonlinear foundation : Theory and singular perturbation analysis

Jebril, Iqbal; Dahmani, Zoubir

doi:10.47974/JIM-2430

Open Access Research Article

Fractional beam dynamics with nonlinear foundation : Theory and singular perturbation analysis

Iqbal Jebriliqbal501@hotmail.comDepartment of MathematicsAl-Zaytoonah University of JordanAmman, 11733, Jordan0000-0003-4348-6197View full profile → , ^*Zoubir DahmaniCorresponding authorzzdahmani@yahoo.frLaboratory of Discrete and Applied Mathematics and Operations ResearchLAMDA-ROUniversity of Blida 1Blida, 09000, Algeria0000-0003-4659-0723View full profile →

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pp. 1–10Online FirstMay 2026DOI: 10.47974/JIM-2430XML

Received:: 01 Jul 2025
Published Online:: 26 May 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2430
Pages:: 1–10

Abstract

This paper investigates the dynamics of a nonlinear elastic beam governed by a space-fractional differential equation of Caputo type, where the fractional order lies in the range z ∈ (3,4). A small parameter ε is incorporated to capture singular perturbation effects, and a composite approximation is constructed through matched asymptotic expansions. The leading-order solution is derived using the Tanh method for the reduced integer-order problem. The fractional perturbation introduces algebraically decaying corrections, which are analyzed and matched with the outer-region behavior. Numerical plots illustrate how varying ε and the fractional order z affects the beam profile, especially the far-field tails. The analysis shows that higher fractional orders amplify long-range perturbative effects in the solution profile.

Keywords

Beam equation Caputo derivative Singular perturbation Tanh method Matched asymptotic expansions

Subject Classifications

35C0735Q5334E10

References

[1] A. A. Alikhanov and C. Huang, “A class of time-fractional diffusion equations with generalized fractional derivatives,” Journal of Computational and Applied Mathematics, vol. 414, Art. no. 114424 (Nov. 2022).
[2] D. Baleanu and K. Diethelm, Fractional calculus: models and numerical approaches, Mathematics, vol. 8, no. 3, Art. no. 369 (2020).
[3] I. M. Batiha, S. A. O. Albteiha, and S. Momani, “On Q-fractional calculus: Foundations and theoretical insights,” in Recent Developments in Fractional Calculus: Theory, Applications, and Numerical Simulations, L. Kaur and P. Kumar, Eds., Studies in Systems, Decision and Control, vol. 235. Cham, Switzerland: Springer (2025), doi: 10.1007/978-3-031-84955-8_6.
[4] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Geophys. J. Int., vol. 13, no. 5, pp. 529-539 (1967).
[5] A. Carpinteri, P. Cornetti, and A. Sapora, Nonlocal elasticity: an appraoch based on fractional calculus, Meccanica, vol. 49 (2014).
[6] K. Diethelm, The Analysis of Fractional Differential Equations, Springer (2010).
[7] D. Dönmez Demir, N. Bildik, and B. G. Sinir, Application of fractional calculus in the dynamics of beams, Boundary Value Problems, vol. 2012, Art. no. 135 (2012).
[8] L. Euler, A method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense, Lausanne and Geneva (1744).
[9] T. Feng and M. Ni, “Asymptotic solution for a system of singularly perturbed delay differential equations,” Journal of Applied Analysis & Computation, vol. 16, no. 1, pp. 458-478 (2026). doi:10.11948/20250097.
[10] R. E. George and B. O. Oyelami, “Analytical solution of the fractional-order Euler–Bernoulli beam equation,” Electronic Journal of Mathematics, vol. 9, pp. 102-112 (2025).
[11] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific (2000).
[12] R.F. Kadam and K.M.M Al-Abrahemee, Neuro-fuzzy system for solving fuzzy singular perturbation problems, Journal of Interdisciplinary Mathematics, 25(5), 1509-1524 (2022).
[13] R. Kareem and K.M.M Al-Abraheme, Modification artificial neural networks for solving singular perturbation problems, Journal of Interdisciplinary Mathematics, 25(5), 1535-1549, 2022.
[14] A. N. Khan, U. Younis, M. Q. Mehmood, and M. Zubair, Atmospheric propagation of space-fractional Gaussian-beam waves in a FSO communication system, Opt. Express, vol. 30, no. 2, pp. 1570-1586, Jan. 2022.
[15] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier (2006).
[16] J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Springer (1996).
[17] M. A. Langthjem and H.M. Jensen, On matched asymptotic expansion solutions of a buckle wave propagation problem, ZAMM“ J. Appl. Math. Mech. (2024).
[18] C. Li and F. Zeng, “Finite difference methods for fractional differential equations,” International Journal of Bifurcation and Chaos, vol. 22, no. 04, p. 1230014 (2012).
[19] J. A. T. Machado, Some applications of fractional calculus in engineering, Math. Probl. Eng., Art. ID 639801 (2010).
[20] I. Podlubny, Fractional Differential Equations, Academic Press (1999).
[21] P. D. Spanos and G. Malara, Nonlinear vibrations of beams and plates with fractional derivative elements subject to combined harmonic and random excitations, Probabilist. Eng. Mech., vol. 59, Art. no. 103043 (Jan. 2020).
[22] P. Stempin and W. Sumelka, “Space-fractional Euler-Bernoulli beam model--theory and identification for silver nanobeam bending,” Int. J. Mech. Sci., vol. 186, Art. no. 105902 (Nov. 2020).
[23] P. Stempin and W. Sumelka, “Dynamics of space-fractional Euler-Bernoulli and Timoshenko beams,” Materials, vol. 14, no. 8, Art. no. 1817 (Apr. 2021).
[24] W. Sumelka, “Fractional calculus for continuum mechanics—anisotropic non-locality,” Bull. Polish Acad. Sci. Tech. Sci., vol. 64, no. 2, pp. 1-10 (2016).
[25] S. Timoshenko, History of Strength of Materials, McGraw-Hill, New York (1953).
[26] C. Truesdell, The Rational Mechanics of Flexible or Elastic Bodies 1638-1788, Venditioni Exponunt Orell Fussli Turici (1960).
[27] L. Wang, G. Wang, X. Dong, X. Gao, and S. Zhuang, Creation of a stable vector vortex beam with dual fractional orbital angular momentum, Sci. Rep., vol. 15, Art. no. 576 (2025).

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