The Numerical Solution of the L-Fractional Abel Differential Equations

Handan Yaslan

doi:10.33187/jmsm.1749492

The Numerical Solution of the L-Fractional Abel Differential Equations

Abstract

In this paper, approximate solutions of the L (Leibniz)-fractional Abel differential equations of the first and second kind are computed by using the shifted Morgan-Voyce (SMV) polynomials. Fractional derivatives are defined in the Leibniz sense. Firstly, the unknown function and its L-fractional derivative are represented by truncated series of SMV polynomials. After applying the collocation procedure, matrix relations are obtained for the unknown function and its derivatives. These relations transform the L-fractional Abel differential equation into a nonlinear system of algebraic equations, which is efficiently solved by using the Newton's method. Finally, five numerical examples are considered to demonstrate validity of the method. The results show excellent agreement with exact solutions, with maximum absolute errors of $10^{-16}$ for $\alpha=1$.Comparisons with existing methods show that the proposed scheme is both accurate and computationally efficient.

Keywords

Shifted Morgan-Voyce polynomials, The L (Leibniz)- fractional derivative, The fractional Abel differential equations

References

[1] G. Alobaidi, R. Mailler, On the Abel equation of the second kind with sinusoidal forcing, Nonlinear Anal. Control, 12 (2007), 33-44. https://doi.org/10.15388/NA.2007.12.1.14720
[2] J. Lebrun, On two coupled Abel-type differential equations arising in a magnetostatic problem, Il Nuovo Cimento, A 103(10) (1990), 1369-1379. https://doi.org/10.1007/BF02820566
[3] J. M. Olm, X. Ros-Oton, Y. B. Shtessel, Stable Inversion of Abel equation, applications to tracking control in DC-DC nonlinear phase boost converters, J. Autom., 47(1) (2011), 221-226. https://doi.org/10.1016/j.automatica.2010.10.035
[4] T. Harko, M. Mak, Relativistic dissipative cosmological models and Abel differential equation, Comput. Math. Appl., 46 (2003), 849-853. https://doi.org/10.1016/S0898-1221(03)90147-7
[5] B. Bulbul, M. Sezer, A numerical approach for solving generalized Abel-type nonlinear differential equations, Appl. Math. Comput., 262 (2015), 169-177. https://doi.org/10.1016/j.amc.2015.04.057
[6] M. Al-Smadi, N. Djeddi, S. Momani, et. al., An attractive numerical algorithm for solving nonlinear Caputo-Fabrizio fractional Abel differential equation in a Hilbert space, Adv. Difference Equ., 2021(271) (2021), Article ID 271. https://doi.org/10.1186/s13662-021-03428-3
[7] M. Fattahi, M. Matinfar, The numerical solution of the second kind of Abel equations by the modified matrix-exponential method, Int. J. Nonlinear Anal. Appl., 14(12) (2023), 139-144.
[8] S. Yuzbasi, G. Yildirim, A Pell-Lucas approximation to solve the Abel equation of the second kind, Ricerche Mat., 74 (2025), 495-518. https://doi.org/10.1007/s11587-022-00723-3
[9] M. Gulsu, Y. Ozturk, M. Sezer, On the solution of the Abel equation of the second kind by the shifted Chebyshev polynomials, Appl. Math. Comput., 217(9) (2011), 4827-4833. https://doi.org/10.1016/j.amc.2010.11.044
[10] J. M. Olm, X. Ros-Oton, T. M. Seara, Periodic solutions with nonconstant sign in Abel equations of the second kind, J. Math. Anal. Appl., 381 (2011), 582-589. https://doi.org/10.1016/j.jmaa.2011.02.084

[11] K. Parand, M. Nikarya, Application of Bessel functions for solving differential and integro-differential equations of the fractional order, Appl. Math. Model., 38 (2014), 4137-4147. https://doi.org/10.1016/j.apm.2014.02.001
[12] Y. Xu, Z. He, The short memory principle for solving Abel differential equation of fractional order, Comput. Math. Appl., 62(12) (2011), 4796-4805. https://doi.org/10.1016/j.camwa.2011.10.071
[13] K. Parand, M. Nikarya, New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind, Nonlinear Eng., 8 (2019), 438-448. https://www.degruyterbrill.com/document/doi/10.1515/nleng-2018-0095/html
[14] K. Parand, M. Delkhosh, M. Nikarya, Novel orthogonal functions for solving differential equations of arbitrary order, Tbil. Math. J. 10(1) (2017), 31-55. https://doi.org/10.1515/tmj-2017-0004
[15] K. A. Lazopoulos, A. A. Lazopoulos, Fractional vector calculus and fractional continuum mechanics, Progr. Fract. Differ. Appl., 2(2) (2016), 85-104. https://doi.org/10.18576/pfda/020202
[16] A. K. Lazopoulos, D. Karaoulanis, On L-fractional derivatives and L-fractional homogeneous equations, Commun. Appl. Anal., 21(2) (2017), 249-268. https://www.acadsol.eu/en/articles/21/2/6.pdf
[17] M. Jornet, J. J. Nieto, Power-series solution of the L-fractional logistic equation, Appl. Math. Lett., 154 (2024), Article ID 109085. https://doi.org/10.1016/j.aml.2024.109085
[18] M. Jornet, Theory on linear L-fractional differential equations and a new Mittag-Leffler-type function, Fractal Fract., 8(7) (2024), Article ID 411. https://doi.org/10.3390/fractalfract8070411
[19] K. A. Lazopoulos, D. Karaoulanis, A. K. Lazopoulos, On fractional modeling of viscoelastic mechanical systems, Mech. Res. Commun., 78(A)(2016), 1-5. https://doi.org/10.1016/j.mechrescom.2016.10.002
[20] D. E. Mouzakis, A. K. Lazopoulos, Fractional modelling and the Leibniz (L-fractional) derivative as Viscoelastic respondents in polymer biomaterials, Progr. Fract. Differ. Appl., 5(1) (2019), 43-48. http://dx.doi.org/10.18576/pfda/050105
[21] M. Jornet, J. J. Nieto, Probabilistic Interpretation of the L and Prabhakar integrals of fractional Order, Progr. Fract. Differ. Appl., 11(2) (2025), 351-355. http://dx.doi.org/10.18576/pfda/110209
[22] M. Jornet, Theory on new fractional operators using normalization and probability tools, Fractal Fract., 8(11) (2024), Article ID 665. https://doi.org/10.3390/fractalfract8110665
[23] S. Yazici, B. Cekim, J. J. Nieto, Proportional L-fractional forms of Malthusian and Verhulst equations, Qual. Theory Dyn. Syst., 24 (2025), Article ID 208. https://doi.org/10.1007/s12346-025-01364-1
[24] H. M. Srivastava, W. Adel, M. Izadi, et. al., Solving some physics problems involving fractional-order differential equations with the Morgan-Voyce polynomials, Fractal Fract., 7(4) (2023), Article ID 301. https://doi.org/10.3390/fractalfract7040301
[25] P. Rahimkhani, M. H. Heydari, Fractional shifted Morgan-Voyce neural networks for solving fractal-fractional pantograph differential equations, Chaos Soliton. Fract., 175 (2023), Article ID 114070. https://doi.org/10.1016/j.chaos.2023.114070
[26] M. Izadi, S. Yuzbasi, W. Adel, Accurate and efficient matrix techniques for solving the fractional Lotka-Volterra population model, Physica A, 600 (2022), Article ID 127558. https://doi.org/10.1016/j.physa.2022.127558
[27] M. Tarakci, M. Ozel, M. Sezer, Solution of nonlinear ordinary differential equations with quadratic and cubic terms by Morgan-Voyce matrix-collocation method, Turk. J. Math., 44(3) (2020), 906-918. https://doi.org/10.3906/mat-1908-102
[28] D. Zwillinger, Handbook of Differential Equations, Academic Press, Boston, 1997.
[29] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[30] M. Izadi, D. Zeidan, A convergent hybrid numerical scheme for a class of nonlinear diffusion equations, Comput. Appl. Math., 41 (2022), Article ID 318. https://doi.org/10.1007/s40314-022-02033-8
[31] C. Guler, A new numerical algorithm for the Abel equation of the second kind, Int. J. Comput. Math., 84(1) (2007), 109-119. https://doi.org/10.1080/00207160601176889

Details

Primary Language

English

Subjects

Partial Differential Equations, Applied Mathematics (Other)

Journal Section

Research Article

Authors

Handan Yaslan ^*
0000-0002-3243-3703
Türkiye

Early Pub Date

March 6, 2026

Publication Date

March 6, 2026

Submission Date

July 23, 2025

Acceptance Date

February 22, 2026

Published in Issue

Year 2026 Volume: 9 Number: 1

DOI

https://doi.org/10.33187/jmsm.1749492

IZ

https://izlik.org/JA72BC53CC

APA

Yaslan, H. (2026). The Numerical Solution of the L-Fractional Abel Differential Equations. Journal of Mathematical Sciences and Modelling, 9(1), 47-54. https://doi.org/10.33187/jmsm.1749492

AMA

1.Yaslan H. The Numerical Solution of the L-Fractional Abel Differential Equations. Journal of Mathematical Sciences and Modelling. 2026;9(1):47-54. doi:10.33187/jmsm.1749492

Chicago

Yaslan, Handan. 2026. “The Numerical Solution of the L-Fractional Abel Differential Equations”. Journal of Mathematical Sciences and Modelling 9 (1): 47-54. https://doi.org/10.33187/jmsm.1749492.

EndNote

Yaslan H (March 1, 2026) The Numerical Solution of the L-Fractional Abel Differential Equations. Journal of Mathematical Sciences and Modelling 9 1 47–54.

IEEE

[1]H. Yaslan, “The Numerical Solution of the L-Fractional Abel Differential Equations”, Journal of Mathematical Sciences and Modelling, vol. 9, no. 1, pp. 47–54, Mar. 2026, doi: 10.33187/jmsm.1749492.

ISNAD

Yaslan, Handan. “The Numerical Solution of the L-Fractional Abel Differential Equations”. Journal of Mathematical Sciences and Modelling 9/1 (March 1, 2026): 47-54. https://doi.org/10.33187/jmsm.1749492.

JAMA

1.Yaslan H. The Numerical Solution of the L-Fractional Abel Differential Equations. Journal of Mathematical Sciences and Modelling. 2026;9:47–54.

MLA

Yaslan, Handan. “The Numerical Solution of the L-Fractional Abel Differential Equations”. Journal of Mathematical Sciences and Modelling, vol. 9, no. 1, Mar. 2026, pp. 47-54, doi:10.33187/jmsm.1749492.

Vancouver

1.Handan Yaslan. The Numerical Solution of the L-Fractional Abel Differential Equations. Journal of Mathematical Sciences and Modelling. 2026 Mar. 1;9(1):47-54. doi:10.33187/jmsm.1749492