Generalizations of higher-order Taylor method: Fractional and q-fractional approaches for initial value problems

Iqbal M. Batiha; Sharaf Aldeen O. Albteiha; Hamzah O. Al-Khawaldeh; Shaher Momani; Iqbal M. Batiha; Sharaf Aldeen O. Albteiha; Hamzah O. Al-Khawaldeh; Shaher Momani

doi:10.3934/math.2026021

AIMS Mathematics

2026, Volume 11, Issue 1: 483-510. doi: 10.3934/math.2026021

Previous Article Next Article

Research article Topical Sections

Generalizations of higher-order Taylor method: Fractional and q-fractional approaches for initial value problems

1.
Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan
2.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, UAE
3.
Department of Mathematics, Irbid National University, Irbid 2600, Jordan
4.
Department of Mathematics, Al al-Bayt University, Mafraq, Jordan
5.
Department of Mathematics, The University of Jordan, Amman, Jordan

Received: 02 August 2025 Revised: 14 November 2025 Accepted: 24 November 2025 Published: 07 January 2026
MSC : 26A33, 34A08

Motivated by the challenges faced by standard methods in solving nonlinear fractional and q-fractional models with strong memory effects, this study develops numerical approaches capable of handling these complex behaviors more effectively. The proposed techniques are tested on four representative fractional and q-fractional initial value problems for several values of the order $ \alpha \in $ $ [0.5, 1] $ and $ q \in (0, 1) $. In particular, the major aim of this work is to propose two generalizations of the higher-order Taylor method: The first one is the fractional Taylor method, and the second one is the q-fractional Taylor method. These methods will then be used to find approximate solutions for several fractional and q-fractional initial value problems. Numerous numerical comparisons will be performed to verify the effectiveness of our proposed generalizations.
- fractional calculus,
- fractional q-calculus,
- fractional Taylor method,
- fractional q-Taylor method,
- numerical solutions
Citation: Iqbal M. Batiha, Sharaf Aldeen O. Albteiha, Hamzah O. Al-Khawaldeh, Shaher Momani. Generalizations of higher-order Taylor method: Fractional and q-fractional approaches for initial value problems[J]. AIMS Mathematics, 2026, 11(1): 483-510. doi: 10.3934/math.2026021

Related Papers:

Abstract

Motivated by the challenges faced by standard methods in solving nonlinear fractional and q-fractional models with strong memory effects, this study develops numerical approaches capable of handling these complex behaviors more effectively. The proposed techniques are tested on four representative fractional and q-fractional initial value problems for several values of the order $ \alpha \in $ $ [0.5, 1] $ and $ q \in (0, 1) $. In particular, the major aim of this work is to propose two generalizations of the higher-order Taylor method: The first one is the fractional Taylor method, and the second one is the q-fractional Taylor method. These methods will then be used to find approximate solutions for several fractional and q-fractional initial value problems. Numerous numerical comparisons will be performed to verify the effectiveness of our proposed generalizations.

References

[1]	N. R. Anakira, A. Almalki, D. Katatbeh, G. B. Hani, A. F. Jameel, K. S. Al Kalbani, et al., An algorithm for solving linear and non-linear Volterra integro-differential equations, Int. J. Adv. Soft Comput. Appl., 15 (2023), 69–83.
[2]	G. Farraj, B. Maayah, R. Khalil, W. Beghami, An algorithm for solving fractional differential equations using conformable optimized decomposition method, Int. J. Adv. Soft Comput. Appl., 15 (2023), 187–196.
[3]	M. Berir, Analysis of the effect of white noise on the Halvorsen system of variable-order fractional derivatives using a novel numerical method, Int. J. Adv. Soft Comput. Appl., 16 (2024), 294–306.
[4]	M. A. Hammad, R. Khalil, Conformable fractional heat differential equation, Int. J. Pure Appl. Math., 94 (2014), 215–221. https://doi.org/10.12732/IJPAM.V94I2.8 doi: 10.12732/IJPAM.V94I2.8
[5]	M. W. Alomari, I. M. Batiha, S. Momani, P. Agarwal, M. Odeh, Surpassing Taylor method: Generalized Taylor method for solving initial value problems, Bol. Soc. Parana. Mat., 43 (2025), 1–13. https://doi.org/10.5269/bspm.75572 doi: 10.5269/bspm.75572
[6]	Y. F. Luchko, H. M. Srivastava, The exact solution of certain differential equations of fractional order by using operational calculus, Comput. Math. Appl., 29 (1995), 73–85. https://doi.org/10.1016/0898-1221(95)00031-S doi: 10.1016/0898-1221(95)00031-S
[7]	N. Allouch, I. M. Batiha, I. H. Jebril, S. Hamani, A. Al-Khateeb, S. Momani, A new fractional approach for the higher-order q-Taylor method, Image Anal. Stereol., 43 (2024), 249–257. https://doi.org/10.5566/ias.3286 doi: 10.5566/ias.3286
[8]	A. A. Al-Nana, M. W. Alomari, I. M. Batiha, The modified Taylor method for approximating solutions of initial value problems, Int. Rev. Model. Simul., 17 (2024), 5. https://doi.org/10.15866/iremos.v17i5.25176 doi: 10.15866/iremos.v17i5.25176
[9]	I. M. Batiha, I. H. Jebril, A. Abdelnebi, Z. Dahmani, S. Alkhazaleh, N. Anakira, A new fractional representation of the higher-order Taylor scheme, Comput. Math. Methods, 2024 (2024), 2849717. https://doi.org/10.1155/2024/2849717 doi: 10.1155/2024/2849717
[10]	I. M. Batiha, S. A. O. Albteiha, S. Momani, On q-fractional calculus: Foundations and theoretical insights, In: Recent Developments in Fractional Calculus: Theory, Applications, and Numerical Simulations, Springer, Cham, 235 (2025), 123–145. https://doi.org/10.1007/978-3-031-84955-8_6
[11]	M. W. Alomari, W. G. Alshanti, I. M. Batiha, L. Guran, I. H. Jebril, Differential q-calculus of several variables, Results Nonlinear Anal., 7 (2024), 109–129.
[12]	L. T. Watson, Numerical linear algebra aspects of globally convergent homotopy methods, SIAM Rev., 28 (1986), 529–545. https://doi.org/10.1137/1028157 doi: 10.1137/1028157
[13]	A. M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput., 111 (2000), 33–51. https://doi.org/10.1016/S0096-3003(99)00063-6 doi: 10.1016/S0096-3003(99)00063-6
[14]	H. F. Ahmed, Fractional Euler method: An effective tool for solving fractional differential equations, J. Egypt. Math. Soc., 26 (2018), 38–43.
[15]	P. L. Butzer, U. Westphal, An introduction to fractional calculus, Appl. Fract. Calc. Phys., 2000, 1–85. https://doi.org/10.1142/3779 doi: 10.1142/3779
[16]	K. K. Ali, S. Tarla, M. R. Ali, A. Yusuf, Modulation instability analysis and optical solutions of an extended (2+1)-dimensional perturbed nonlinear Schrödinger equation, Results Phys., 45 (2023), 106255. https://doi.org/10.1016/j.rinp.2023.106255 doi: 10.1016/j.rinp.2023.106255
[17]	F. H. Jackson, On $q$-difference integrals, Am. J. Math., 32 (1910), 305–2314. https://doi.org/10.2307/2370183 doi: 10.2307/2370183
[18]	W. A. Al-Salam, Some fractional $q$-integrals and $q$-derivatives, P. Edinburgh Math. Soc., 15 (1966), 135–140. https://doi.org/10.1017/S0013091500011469 doi: 10.1017/S0013091500011469
[19]	R. P. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Math. Proc. Cambridge, 66 (1969), 365–370. https://doi.org/10.1017/S0305004100045060 doi: 10.1017/S0305004100045060
[20]	R. P. Agarwal, V. Lupulescu, D. O'Regan, G. U. Rahman, Fractional calculus and fractional differential equations in nonreflexive Banach spaces, Commun. Nonlinear Sci., 20 (2015), 59–73. https://doi.org/10.1016/j.cnsns.2013.10.010 doi: 10.1016/j.cnsns.2013.10.010
[21]	P. Stempin, W. Sumelka, Approximation of fractional Caputo derivative of variable order and variable terminals with application to initial/boundary value problems, Fractal Fract., 9 (2025), 269. https://doi.org/10.3390/fractalfract9050269 doi: 10.3390/fractalfract9050269
[22]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, San Diego: Academic Press, 198 (1998).
[23]	T. Blaszczyk, K. Bekus, K. Szajek, W. Sumelka, On numerical approximation of the Riesz–Caputo operator with the fixed/short memory length, J. King Saud Univ. Sci., 33 (2021), 101220. https://doi.org/10.1016/j.jksus.2020.10.017 doi: 10.1016/j.jksus.2020.10.017
[24]	M. Stynes, E. O'Riordan, J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057–1079. https://doi.org/10.1137/16M1082329 doi: 10.1137/16M1082329
[25]	D. Li, C. Wu, Z. Zhang, Linearized Galerkin FEMs for nonlinear time-fractional parabolic problems with non-smooth solutions in time direction, J. Sci. Comput., 80 (2019), 403–419. https://doi.org/10.1007/s10915-019-00943-0 doi: 10.1007/s10915-019-00943-0
[26]	D. Li, W. Sun, C. Wu, A novel numerical approach to time-fractional parabolic equations with nonsmooth solutions, Numer. Math.-Theory Me., 14 (2021), 355–376. https://doi.org/10.4208/NMTMA.OA-2020-0129 doi: 10.4208/NMTMA.OA-2020-0129
[27]	W. Yuan, D. Li, C. Zhang, Linearized transformed $L^{1}$ Galerkin FEMs with unconditional convergence for nonlinear time-fractional Schrödinger equations, Numer. Math.-Theory Me., 16 (2023), 398–423. https://doi.org/10.4208/nmtma.OA-2022-0087 doi: 10.4208/nmtma.OA-2022-0087

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)