This work constructs and analyzes a spectral collocation procedure for the classical and time-fractional nonlinear fifth-order Korteweg–de Vries equations (NFOKDVEs). The approximation is built by using Fibonacci level $ m $ polynomials, a generalized form of the standard Fibonacci polynomial sequence. We derive several auxiliary results for this polynomial family, including an inversion relation, an integer-order differentiation formula, a fractional differentiation formula, and the corresponding operational matrices. These ingredients are then used to convert the nonlinear models under consideration into algebraic systems whose unknown coefficients are numerically determined using Newton's method. The performance of the resulting algorithms is tested through several examples. The reported errors, residuals, and comparisons with available methods show that the proposed polynomial basis produces accurate approximations with few expansion terms.
Citation: Mohamed Adel, Ahmed Gamal Atta, Naher Mohammed A. Alsafri, Mohamed Abbas El-Naggar, Waleed Mohamed Abd-Elhameed. Fibonacci level $ m $ polynomial collocation algorithms for classical and time-fractional fifth-order Korteweg–de Vries equations[J]. AIMS Mathematics, 2026, 11(9): 30509-30544. doi: 10.3934/math.20261209
This work constructs and analyzes a spectral collocation procedure for the classical and time-fractional nonlinear fifth-order Korteweg–de Vries equations (NFOKDVEs). The approximation is built by using Fibonacci level $ m $ polynomials, a generalized form of the standard Fibonacci polynomial sequence. We derive several auxiliary results for this polynomial family, including an inversion relation, an integer-order differentiation formula, a fractional differentiation formula, and the corresponding operational matrices. These ingredients are then used to convert the nonlinear models under consideration into algebraic systems whose unknown coefficients are numerically determined using Newton's method. The performance of the resulting algorithms is tested through several examples. The reported errors, residuals, and comparisons with available methods show that the proposed polynomial basis produces accurate approximations with few expansion terms.
| [1] |
R. C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 51 (1984), 299–307. https://doi.org/10.1115/1.3167616 doi: 10.1115/1.3167616
|
| [2] | K. B. Oldham, J. Spanier, The fractional calculus: theory and applications of differentiation and integration to arbitrary order, New York: Academic Press, 1974 |
| [3] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006 |
| [4] |
Y. Li, Z. H. Rui, B. Hu, Monotone iterative and quasilinearization method for a nonlinear integral impulsive differential equation, AIMS Math., 10 (2025), 21–37. https://doi.org/10.3934/math.2025002 doi: 10.3934/math.2025002
|
| [5] |
B. Hu, Y. K. Liao, Convergence conditions for extreme solutions of an impulsive differential system, AIMS Math., 10 (2025), 10591–10604. https://doi.org/10.3934/math.2025481 doi: 10.3934/math.2025481
|
| [6] |
D. Damircheli, M. Razzaghi, Time-space collocation based on fractional-order generalized Chebyshev wavelets for fractional advection problem, J. Comput. Appl. Math., 488 (2026), 117776. https://doi.org/10.1016/j.cam.2026.117776 doi: 10.1016/j.cam.2026.117776
|
| [7] |
M. Hossain, M. A. Ullah, A new hybrid Laplace Adomian variational iteration method for solving nonlinear time-fractional partial differential equations, Ann. Univ. Ferrara, 72 (2026), 37. https://doi.org/10.1007/s11565-026-00658-9 doi: 10.1007/s11565-026-00658-9
|
| [8] |
P. E. Kloeden, D. T. Son, H. T. Tuan, Lyapunov methods for dissipative Caputo fractional differential equations with two indices, Commun. Nonlinear Sci., 157 (2026), 109737. https://doi.org/10.1016/j.cnsns.2026.109737 doi: 10.1016/j.cnsns.2026.109737
|
| [9] |
M. Boumaaza, A. Boutiara, O. Djidel, R. Guefaifia, S. Boulaaras, Analysis of delay differential equations with dual Caputo-type fractional derivatives using Laplace transform methods, Sci. Rep., 16 (2026), 11181. https://doi.org/10.1038/s41598-026-41584-2 doi: 10.1038/s41598-026-41584-2
|
| [10] |
Z. W. Wang, Y. G. Zhu, Optimal control method for uncertain fractional switched systems and application to glucose-insulin regulatory, Commun. Nonlinear Sci., 159 (2026), 109900. https://doi.org/10.1016/j.cnsns.2026.109900 doi: 10.1016/j.cnsns.2026.109900
|
| [11] |
R. Brociek, A. Wajda, M. Błasik, D. Słota, An application of the homotopy analysis method for the time-or space-fractional heat equation, Fractal Fract., 7 (2023), 224. https://doi.org/10.3390/fractalfract7030224 doi: 10.3390/fractalfract7030224
|
| [12] |
H. Jafari, B. F. Malidareh, V. R. Hosseini, Collocation discrete least squares meshless method for solving nonlinear multi-term time fractional differential equations, Eng. Anal. Bound. Elem., 158 (2024), 107–120. https://doi.org/10.1016/j.enganabound.2023.10.014 doi: 10.1016/j.enganabound.2023.10.014
|
| [13] |
S. M. Sivalingam, P. Kumar, H. Trinh, V. Govindaraj, A novel L1-predictor–corrector method for the numerical solution of the generalized-Caputo type fractional differential equations, Math. Comput. Simulat., 220 (2024), 462–480. https://doi.org/10.1016/j.matcom.2024.01.017 doi: 10.1016/j.matcom.2024.01.017
|
| [14] |
C. P. Li, N. N'Gbo, F. Su, Finite difference methods for nonlinear fractional differential equation with $\psi$-Caputo derivative, Physica D, 460 (2024), 134103. https://doi.org/10.1016/j.physd.2024.134103 doi: 10.1016/j.physd.2024.134103
|
| [15] |
H. Sun, M. Zheng, Z. Jin, Numerical method for fractional variational problems by Müntz-Legendre polynomials, J. Comput. Appl. Math., 483 (2026), 117390. https://doi.org/10.1016/j.cam.2026.117390 doi: 10.1016/j.cam.2026.117390
|
| [16] |
M. M. AlBaidani, Numerical solution of fractional third-order nonlinear Emden–Fowler delay differential equations via Chebyshev polynomials, Axioms, 15 (2026), 64. https://doi.org/10.3390/axioms15010064 doi: 10.3390/axioms15010064
|
| [17] |
W. M. Abd-Elhameed, M. A. Abdelkawy, O. M. Alqubori, A. G. Atta, Pell coefficient polynomials for solving linear hyperbolic first-order partial differential equations via the tau approach, Electron. Res. Arch., 33 (2025), 6012–6035. https://doi.org/10.3934/era.2025267 doi: 10.3934/era.2025267
|
| [18] |
H. S. Osheba, M. A. Ramadan, T. Radwan, An efficient and accurate numerical approach for fractional Bagley–Torvik equations: Hermite polynomials combined with least squares, Fractal Fract., 10 (2026), 37. https://doi.org/10.3390/fractalfract10010037 doi: 10.3390/fractalfract10010037
|
| [19] |
M. H. Heydari, T. Baghban, M. Bayram, M. A. Zaky, Discrete Chebyshev polynomials for two-dimensional fractional optimal control problems involving Caputo-Hadamard derivative, Int. J. Syst. Sci., 57 (2026), 4262–4277. https://doi.org/10.1080/00207721.2025.2600349 doi: 10.1080/00207721.2025.2600349
|
| [20] |
M. Pourbabaee, Artificial neural networks method based on Gegenbauer polynomials for solving some types of integral equations, Comp. Appl. Math., 45 (2026), 84. https://doi.org/10.1007/s40314-025-03564-6 doi: 10.1007/s40314-025-03564-6
|
| [21] |
T. T. Zhao, T. J. Wang, J. Tan, Composite spectral collocation method of Legendre-Legendre rational polynomials for the Fokker–Planck equation on unbounded domain, Math. Comput. Simulat., 246 (2026), 188–208. https://doi.org/10.1016/j.matcom.2026.01.035 doi: 10.1016/j.matcom.2026.01.035
|
| [22] |
W. A. Khan, S. A. Wani, M. Ayman-Mursaleen, K. Kotecha, P. Jadhav, Extended forms of Legendre-Laguerre-based hybrid polynomials and their characteristics via fractional operator approach, J. Inequal. Appl., 2026 (2026), 24. https://doi.org/10.1186/s13660-026-03437-4 doi: 10.1186/s13660-026-03437-4
|
| [23] |
U. L. Manikanta, K. J. Gowtham, B. J. Gireesha, P. Venkatesh, A novel Hosoya polynomial collocation approach for evaluating the thermal performance of porous annular fins with internal heat generation employing trihybrid nanofluids, Int. Commun. Heat Mass, 172 (2026), 110242. https://doi.org/10.1016/j.icheatmasstransfer.2025.110242 doi: 10.1016/j.icheatmasstransfer.2025.110242
|
| [24] |
N. Samadyar, P. Rahimkhani, Numerical treatment of tumor-immune interaction model of fractional order related to lung cancer via Chelyshkov polynomials, Math. Methods Appl. Sci., 49 (2026), 4056–4073. https://doi.org/10.1002/mma.70333 doi: 10.1002/mma.70333
|
| [25] |
A. AbuGneam, S. Nemati, A. Babaei, Application of generalized fractional-order Bernoulli wavelets in solving distributed-order fractional differential equations, J. Supercomput., 81 (2025), 1348. https://doi.org/10.1007/s11227-025-07760-3 doi: 10.1007/s11227-025-07760-3
|
| [26] |
H. Hassani, Z. Avazzadeh, A. Turan-Dincel, R. Katani, Application of fractional shifted vieta-fibonacci polynomials in nonlinear reaction diffusion equation with variable order time-space fractional derivative, Phys. Scr., 100 (2025), 025231. https://doi.org/10.1088/1402-4896/adaa38 doi: 10.1088/1402-4896/adaa38
|
| [27] |
P. Sawangtong, M. Taghipour, A. Najafi, Enhanced numerical solution for time fractional Kuramoto–Sivashinsky dynamics via shifted companion Morgan–Voyce polynomials, Comp. Appl. Math., 44 (2025), 214. https://doi.org/10.1007/s40314-025-03160-8 doi: 10.1007/s40314-025-03160-8
|
| [28] | J. Shen, T. Tang, L. L. Wang, Spectral methods: algorithms, analysis and applications, Berlin: Springer, 2011. https://doi.org/10.1007/978-3-540-71041-7 |
| [29] | J. P. Boyd, Chebyshev and Fourier spectral methods, 2Eds., New York: Dover Publications, 2001. |
| [30] |
M. Sahu, P. Roul, S. Khandagale, A B-spline collocation method for a time fractional Black–Scholes model under jump-diffusion, J. Comput. Appl. Math., 487 (2026), 117717. https://doi.org/10.1016/j.cam.2026.117717 doi: 10.1016/j.cam.2026.117717
|
| [31] |
N. Sarkar, M. Sen, A hybrid Daubechies wavelet collocation approach for a fractional-order SIR epidemic model with delay effects, Sci. Rep., 16 (2026), 2716. https://doi.org/10.1038/s41598-025-32483-z doi: 10.1038/s41598-025-32483-z
|
| [32] |
J. Yang, V. Srinivasan, A domain-overlapping algorithm with nonlinear mappings for Chebyshev collocation-based solutions of eigenvalue problems, Appl. Numer. Math., 226 (2026), 87–105. https://doi.org/10.1016/j.apnum.2026.03.012 doi: 10.1016/j.apnum.2026.03.012
|
| [33] |
A. A. El-Sayed, S. Boulaaras, F. A. Al-Kharousi, Dickson polynomial-based solutions for fractional order physics problems, J. Inequal. Appl., 2025 (2025), 118. https://doi.org/10.1186/s13660-025-03367-7 doi: 10.1186/s13660-025-03367-7
|
| [34] |
W. M. Abd-Elhameed, M. A. Abdelkawy, O. M. Alqubori, A. G. Atta, An accurate tau-based spectral algorithm for the time fractional bioheat transfer model, Bound. Value Probl., 2025 (2025), 124. https://doi.org/10.1186/s13661-025-02118-3 doi: 10.1186/s13661-025-02118-3
|
| [35] |
H. F. Ahmed, W. A. Hashem, A fully spectral tau method for a class of linear and nonlinear variable-order time-fractional partial differential equations in multi-dimensions, Math. Comput. Simulat., 214 (2023), 388–408. https://doi.org/10.1016/j.matcom.2023.07.023 doi: 10.1016/j.matcom.2023.07.023
|
| [36] |
C. X. Niu, H. P. Ma, D. Liang, Legendre-tau Chebyshev collocation spectral element method for Maxwell's equations with material interfaces of two dimensional transverse magnetic mode, Comput. Math. Appl., 147 (2023), 222–238. https://doi.org/10.1016/j.camwa.2023.07.024 doi: 10.1016/j.camwa.2023.07.024
|
| [37] |
Z. Q. Wang, X. Wen, G. Q. Yao, An efficient spectral-Galerkin method for elliptic equations in 2D complex geometries, J. Sci. Comput., 95 (2023), 89. https://doi.org/10.1007/s10915-023-02207-4 doi: 10.1007/s10915-023-02207-4
|
| [38] |
S. U. Arifeen, S. Haq, I. Ali, Galerkin approximation of Burgers–Huxley equation with fractional order arising in reaction mechanism and diffusion transport, Phys. Scr., 99 (2024), 125215. https://doi.org/10.1088/1402-4896/ad8b75 doi: 10.1088/1402-4896/ad8b75
|
| [39] |
W. M. Abd-Elhameed, M. A. Abdelkawy, N. M. A. Alsafri, A. G. Atta, Galerkin-based solution for the time-fractional diffusion-wave equation, Electron. Res. Arch., 33 (2025), 5179–5206. https://doi.org/10.3934/era.2025232 doi: 10.3934/era.2025232
|
| [40] |
U. Bayrakci, S. T. Demiray, H. Yildirim, Obtaining new soliton solutions of the fractional generalized perturbed KdV equation, Phys. Scr., 99 (2024), 125202. https://doi.org/10.1088/1402-4896/ad8846 doi: 10.1088/1402-4896/ad8846
|
| [41] |
H. M. Ahmed, New generalized Jacobi–Galerkin operational matrices of derivatives: an algorithm for solving the time-fractional coupled KdV equations, Bound. Value Probl., 2024 (2024), 144. https://doi.org/10.1186/s13661-024-01948-x doi: 10.1186/s13661-024-01948-x
|
| [42] |
K. Sadri, K. Hosseini, E. Hincal, D. Baleanu, S. Salahshour, A pseudo-operational collocation method for variable-order time-space fractional KdV–Burgers–Kuramoto equation, Math. Method. Appl. Sci., 46 (2023), 8759–8778. https://doi.org/10.1002/mma.9015 doi: 10.1002/mma.9015
|
| [43] |
M. Y. Almusawa, H. Almusawa, Numerical analysis of the fractional nonlinear waves of fifth-order KdV and Kawahara equations under Caputo operator, AIMS Math., 9 (2024), 31898–31925. https://doi.org/10.3934/math.20241533 doi: 10.3934/math.20241533
|
| [44] |
C. J. Li, H. X. Zhang, X. H. Yang, A new $\alpha$-robust nonlinear numerical algorithm for the time fractional nonlinear KdV equation, Communications in Analysis and Mechanics, 16 (2024), 147–168. https://doi.org/10.3934/cam.2024007 doi: 10.3934/cam.2024007
|
| [45] |
J. Alahmadi, B. Aldossary, E. A. B. Abdel-Salam, Numerical treatment of the coupled fractional mKdV equations based on the Adomian decomposition technique, Appl. Math. Inform. Sci., 18 (2024), 93–99. https://doi.org/10.18576/amis/180110 doi: 10.18576/amis/180110
|
| [46] |
K. Singh, I. Kourakis, Generalized analytical solutions of a Korteweg–de Vries (KdV) equation with arbitrary real coefficients: Association with the plasma-fluid framework and physical interpretation, Wave Motion, 132 (2025), 103443. https://doi.org/10.1016/j.wavemoti.2024.103443 doi: 10.1016/j.wavemoti.2024.103443
|
| [47] |
S. U. Arifeen, S. Haq, Petrov–Galerkin approximation of time-fractional coupled Korteweg–de Vries equation for propagation of long wave in shallow water, Math. Comput. Simulat., 207 (2023), 226–242. https://doi.org/10.1016/j.matcom.2022.12.028 doi: 10.1016/j.matcom.2022.12.028
|
| [48] |
S. W. Yao, A. Zafar, A. Urooj, B. Tariq, M. Shakeel, M. Inc, Novel solutions to the coupled KdV equations and the coupled system of variant Boussinesq equations, Results Phys., 45 (2023), 106249. https://doi.org/10.1016/j.rinp.2023.106249 doi: 10.1016/j.rinp.2023.106249
|
| [49] |
M. L. Rupa, K. Aruna, Soliton solutions of time fractional seventh order Korteweg–de Vries equations arises in applied sciences, Int. J. Appl. Comput. Math., 11 (2025), 119. https://doi.org/10.1007/s40819-025-01939-1 doi: 10.1007/s40819-025-01939-1
|
| [50] |
M. A. Iqbal, M. Hussain, M. Safeer, M. Taj, A. Tounsi, Travelling waves solutions for modified Korteweg-de Vries equation by modified Exp-function method, Comput. Concrete, 35 (2025), 155–161. https://doi.org/10.12989/cac.2025.35.2.155 doi: 10.12989/cac.2025.35.2.155
|
| [51] |
A. Alshareef, H. O. Bakodah, Non-central m-point formula in method of lines for solving the Korteweg-de Vries (KdV) equation, J. Umm Al-Qura Univ. Appl. Sci., 11 (2025), 142–152. https://doi.org/10.1007/s43994-024-00154-w doi: 10.1007/s43994-024-00154-w
|
| [52] |
Z. A. Noor, I. Talib, T. Abdeljawad, M. A. Alqudah, Numerical study of Caputo fractional-order differential equations by developing new operational matrices of Vieta–Lucas polynomials, Fractal Fract., 6 (2022), 79. https://doi.org/10.3390/fractalfract6020079 doi: 10.3390/fractalfract6020079
|
| [53] |
S. C. Shiralashetti, V. R. Pala, S. I. Hanaji, Bernoulli wavelet integration operational matrix method for the numerical solution of nanofluid flow in an asymmetric porous channel with expanding or contracting wall, Int. J. Appl. Comput. Math., 11 (2025), 155. https://doi.org/10.1007/s40819-025-01978-8 doi: 10.1007/s40819-025-01978-8
|
| [54] |
H. M. Ahmed, M. Izadi, C. Cattani, A spectral approach to variable-order fractional differential equations: improved operational matrices for fractional Jacobi functions, Mathematics, 13 (2025), 2544. https://doi.org/10.3390/math13162544 doi: 10.3390/math13162544
|
| [55] |
M. O. Enadi, M. A. AL-Jawary, M. Turkyilmazoglu, The operational matrices methods for solving the Poisson equation with nonlocal boundary conditions, J. Appl. Math. Comput., 71 (2025), 2385–2432. https://doi.org/10.1007/s12190-024-02313-y doi: 10.1007/s12190-024-02313-y
|
| [56] |
A. Turan-Dincel, S. N. Tural-Polat, Operational matrix method approach for fractional partial differential-equations, Phys. Scr., 99 (2024), 125254. https://doi.org/10.1088/1402-4896/ad8f7a doi: 10.1088/1402-4896/ad8f7a
|
| [57] |
R. V. Shree, B. P. Mallikarjun, B. Souayeh, S. Bhattacharya, Application of Fibonacci wavelet frame operational matrix for the analysis of Arrhenius-controlled heat transfer flow in a microchannel, Case Stud. Therm. Eng., 63 (2024), 105326. https://doi.org/10.1016/j.csite.2024.105326 doi: 10.1016/j.csite.2024.105326
|
| [58] |
M. A. Abdelkawy, W. M. Abd-Elhameed, S. S. Alzahrani, A. G. Atta, A. Biswas, Numerical solutions via shifted Pell polynomials for third-order Rosenau–Hyman and Gilson–Pickering equations, Mathematics, 14 (2026), 582. https://doi.org/10.3390/math14030582 doi: 10.3390/math14030582
|
| [59] |
O. M. Alqubori, S. A. Alsulami, A. G. Atta, A. K. Amin, W. M. Abd-Elhameed, Numerical scheme for solving the time-fractional Huxley equation using shifted Dickson polynomials, AIMS Math., 11 (2026), 13744–13766. https://doi.org/10.3934/math.2026566 doi: 10.3934/math.2026566
|
| [60] |
W. M. Abd-Elhameed, A. H. Al-Mehmadi, N. M. A. Alsafri, O. M. Alqubori, M. Adel, A. G. Atta, Spectral collocation approach for solving the time-fractional Kuramoto–Sivashinsky equation using the Fibonacci coefficient polynomials, AIMS Math., 10 (2025), 18070–18093. https://doi.org/10.3934/math.2025805 doi: 10.3934/math.2025805
|
| [61] |
S. Kumar, V. Gupta, D. Zeidan, An efficient collocation technique based on operational matrix of fractional-order Lagrange polynomials for solving the space-time fractional-order partial differential equations, Appl. Numer. Math., 204 (2024), 249–264. https://doi.org/10.1016/j.apnum.2024.06.014 doi: 10.1016/j.apnum.2024.06.014
|
| [62] |
M. Kashif, M. Singh, T. Som, E. M. Craciun, Numerical study of variable order model arising in chemical processes using operational matrix and collocation method, J. Comput. Sci., 80 (2024), 102339. https://doi.org/10.1016/j.jocs.2024.102339 doi: 10.1016/j.jocs.2024.102339
|
| [63] | I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, New York: Academic Press, 1998. |
| [64] |
S. Saleem, M. Z. Hussain, Numerical solution of nonlinear fifth-order KdV-type partial differential equations via Haar wavelet, Int. J. Appl. Comput. Math., 6 (2020), 164. https://doi.org/10.1007/s40819-020-00907-1 doi: 10.1007/s40819-020-00907-1
|
| [65] |
H. O. Bakodah, Modified Adomian decomposition method for the generalized fifth order KdV equations, Am. J. Comput. Math., 3 (2013), 53–58. https://doi.org/10.4236/ajcm.2013.31008 doi: 10.4236/ajcm.2013.31008
|
| [66] |
W. M. Abd-Elhameed, H. M. Alshehri, M. H. Alharbi, O. M. Alqubori, A. K. Amin, A. G. Atta, Shifted Pell–Lucas polynomial-based algorithms for fractional and classical Rosenau–Hyman models, Fractal Fract., 10 (2026), 359. https://doi.org/10.3390/fractalfract10060359 doi: 10.3390/fractalfract10060359
|
| [67] |
W. M. Abd-Elhameed, O. M. Alqubori, A. G. Atta, A collocation approach for the nonlinear fifth-order KdV equations using certain shifted Horadam polynomials, Mathematics, 13 (2025), 300. https://doi.org/10.3390/math13020300 doi: 10.3390/math13020300
|
Figures(12) / Tables(12)
Mohamed Adel, Ahmed Gamal Atta, Naher Mohammed A. Alsafri, Mohamed Abbas El-Naggar, Waleed Mohamed Abd-Elhameed. Fibonacci level $ m $ polynomial collocation algorithms for classical and time-fractional fifth-order Korteweg–de Vries equations[J]. AIMS Mathematics, 2026, 11(9): 30509-30544. doi: 10.3934/math.20261209
DownLoad: