Construction of Mechanically Preserved Optical Travelling Wave Solution for Fractional Generalized Reaction Duffing Model

Document Type : Research Paper

Authors
Ameer Said 1
Abu Bakkar 1
Hassan Khan 1 , 2
Carlo Cattani 3
Fairouz Tchier 4
1 Department of Mathematics, Abdul Wali khan University, Mardan, Pakistan
2 Department of Mathematics, Near East University, Mersin 10, Turkey
3 Engineering School, DEIM, University of Tuscia, 01100 Viterbo, Italy
4 Mathematics Department, College of Science, King Saud University, P.O Box 22452, Riyadh 11495, Saudi Arabia
10.22055/jacm.2025.48049.4941
Abstract
This research paper presents an efficient analytical method, namely the Sardar Sub-Equation Method (SSM), to construct new families of traveling solitary wave solutions of Fractional Partial Differential Equations (FPDEs), i.e., the Fractional Phi-Four Equation (FPFE) and the Fractional Generalized Duffing Equation (FGDE). By applying fractional wave transformations, these FPDEs are transformed into nonlinear ordinary differential equations. The SSM solutions of the reduced nonlinear ordinary differential equations are further investigated to attain traveling (mechanical) wave solutions to the selected problems in terms of hyperbolic and trigonometric functions. The propagating behavior of some traveling wave solutions is graphically illustrated through 3D, 2D, and contour graphs, which reveal Cuspon solitons as well as dark and bright traveling wave solutions. The obtained soliton solutions play a vital role in the practical discussion of mechanical procedures. We believe that the current results can be utilized to capture many dynamics in mechanical engineering. Moreover, the traveling wave solutions attained by the Sardar Sub-Equation Method are highly efficient and accurate. The results demonstrate that the employed method has a high degree of reliability, with a straightforward methodology, and can thus be used to solve other types of nonlinear equations.
Keywords
Sardar Sub-equation
Solitary waves
Waves transformation
Non-linear fractional partial differential equations
Ordinary differential equations
Conformable fractional derivative
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Salam, Md.A., Sharif Uddin, Md., Dey, P., Generalized Bernoulli Sub-ODE method and its applications, Journals of Pure and Applied Mathematics, 10, 2015, 1-6.
[2] Siddique, I., et al., Diverse precise traveling wave solutions possessing beta derivative of the fractional differential equations arising in mathematical physics, Journal of Function Spaces, 2022, 2022, 5613708.
[3] Rehman, H.U., et al., New soliton solutions for the space-time fractional modified third order Korteweg–de Vries equation, Journal of Ocean Engineering and Science, 2022, DOI: 10.1016/j.joes.2022.05.032.
[4] Muhammad, T., et al., Traveling wave solutions to the Boussinesq equation via Sardar sub-equation technique, AIMS Mathematics, 7, 2022, 11134-11149.
[5] Rezazadeh, H., Inc, M., Baleanu, D., New solitary wave solutions for variants of (3+1)-dimensional Wazwaz-Benjamin-Bona-Mahony equations, Frontiers in Physics, 8, 2020, 332.
[6] Ullah, N., et al., Dynamics of nonlinear optics with different analytical approaches, Fractal and Fractional, 7, 2023, 138.
[7] Katsikadelis, J.T., Nonlinear dynamic analysis of viscoelastic membranes described with fractional differential models, Journal of Theoretical and Applied Mechanics, 50(3), 2012, 743-753
[8] Mabrouk, S.M., Wazwaz, A., Rashed, A.S., Monitoring dynamical behavior and optical solutions of space-time fractional order double-chain deoxyribonucleic acid model considering the Atangana’s conformable derivative, Journal of Applied and Computational Mechanics, 10, 2024, 383-391.
[9] Zhou, Y., Wang, M., Wang, Y., Periodic wave solutions to a coupled KdV equations with variable coefficients, Physics Letters A, 308, 2003, 31-36.
[10] Alquran, M., Alqawaqneh, A., New bidirectional wave solutions with different physical structures to the complex coupled Higgs model via recent ansatze methods: applications in plasma physics and nonlinear optics, Optical and Quantum Electronics, 54, 2022, 301.
[11] Darwish, A., et al., Optical solitons of Biswas–Arshed equation in birefringent fibers using improved modified extended tanh-function method, Optik, 227, 2021, 165385.
[12] Wazwaz, A., The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Applied Mathematics and Computation, 184, 2007, 1002-1014.
[13] Alquran, M., Optical bidirectional wave-solutions to new two-mode extension of the coupled KdV–Schrodinger equations, Optical and Quantum Electronics, 53, 2021, 588.
[14] Akram, G., Sarfraz, M., Multiple optical soliton solutions for CGL equation with Kerr law nonlinearity via extended modified auxiliary equation mapping method, Optik, 242, 2021, 167258.
[15] Khater, M., Lu, D., Attia, R.A.M., Dispersive long wave of nonlinear fractional Wu-Zhang system via a modified auxiliary equation method, AIP Advances, 9, 2019, 025003.
[16] Jing, H., et al., An Analysis of Nonlinear Beam Vibrations with the Extended‎ Rayleigh-Ritz Method, Journal of Applied and Computational Mechanics, 8, 2022, 1299-1306.
[17] Atta, A.G., Abd-Elhameed, W.M., Youssri, Y.H., Approximate collocation solution for the time-fractional Newell-Whitehead-Segel equation, Journal of Applied and Computational Mechanics, 2024, DOI: 10.22055/jacm.2024.47269.4686.
[18] Mandelik, D., et al., Gap solitons in waveguide arrays, Physical Review Letters, 92, 2004, 093904.
[19] Wang, X., Chen, Z., Kevrekidis, P.G., Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices, Physical Review Letters, 96, 2006, 083904.
[20] Ahmad, J., Younas, T., Diverse optical wave structures to the time-fractional phi-four equation in nuclear physics through two powerful methods, Optical and Quantum Electronics, 56, 2024, 606.
[21] Aldandani, M., Altherwi, A.A., Abushaega, M.M., Propagation patterns of dromion and other solitons in nonlinear Phi-Four (φ4) equation, AIMS Mathematics, 9, 2024, 19786-19811.
[22] Uddin, M.H., et al., Close form solutions of the fractional generalized reaction duffing model and the density dependent fractional diffusion reaction equation, Applied and Computational Mathematics, 6, 2017, 177-184.
[23] Güner, Ö., Ahmet, B., Exact solutions of some fractional differential equations arising in mathematical biology, International Journal of Biomathematics, 8, 2015, 1550003.
[24] Baloch, S.A., et al., Multiple Soliton Solutions of Generalized Reaction Duffing Model Arising in Various Mechanical Systems, International Journal of Theoretical Physics, 63, 2024, 234.
[25] Deng, X., Zhao, M., Li, X., Travelling wave solutions for a nonlinear variant of the PHI-four equation, Mathematical and Computer Modelling, 49, 2009, 617-622.
[26] Tariq, K.U., Inc, M., Hashemi, M.S., On the soliton structures to the space-time fractional generalized reaction Duffing model and its applications, Optical and Quantum Electronics, 56, 2024, 708.
[27] Atangana, A., Baleanu, D., Alsaedi, A., Analysis of time-fractional Hunter-Saxton equation: a model of neumatic liquid crystal, Open Physics, 14, 2016, 145-149.
[28] Esen, H., et al., On solitary wave solutions for the perturbed Chen–Lee–Liu equation via an analytical approach, Optik, 245, 2021, 167641.
[29] Pandir, Y., Ulusoy, H., Solutions of nonlinear partial differential equations using generalized hyperbolic functions, Turkish Journal of Mathematics and Computer Science, 1, 2016, 38-46.
[30] Hussain, R., et al., Novel exact and solitary solutions of conformable Klein–Gordon equation via Sardar-subequation method, Journal of Ocean Engineering and Science, 2022, DOI: 10.1016/j.joes.2022.04.036.
Journal of Applied and Computational Mechanics
Volume 11, Issue 4
Autumn 2025
Pages 1139-1148