Abstract
The application of fractional calculus for solving nonlinear equations through iterative techniques is an emerging area of research. In recent times, several Newton-type methods have been proposed which particularly utilize the concept of fractional order derivatives. However, convergence of such methods essentially require the existence of at least first order derivative. Accordingly, in the case where derivative is not feasible to obtain, the derivative-free methods are of much significance. In this paper, a fractional Traub-Steffensen-type method is developed, the formulation of which is based on the idea of conformable fractional derivatives of order \(\alpha \in (0,1]\). In addition, the scheme is extended to its multi-dimensional case to solve the systems of equations. The proposed derivative-free scheme is further investigated for its dynamical aspects and convergence characteristics by varying the value of \(\alpha \) in given range. In this concern, the convergence planes are presented in a well-defined region which in general provide the fundamental information about the stability of given method. Furthermore, the numerical performance is analyzed by locating the solutions of variety of nonlinear equations, including some applied problems.
Similar content being viewed by others
Availability of data and materials
All data generated or analyzed during this study is included in this article.
References
Traub, J.F.: Iterative methods for the solution of equations. Chelsea Publishing Company, New York (1982)
Hueso, J.L., Martínez, E., Teruel, C.: Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear systems. J. Comput. Appl. Math. 275, 412–420 (2015)
Ahmad, F., Soleymani, F., Haghani, F.K., Serra-Capizzano, S.: Higher order derivative-free iterative methods with and without memory for systems of nonlinear equations. Appl. Math. Comput. 314, 199–211 (2017)
Narang, M., Bhatia, S., Kanwar, V.: New efficient derivative free family of seventh-order methods for solving systems of nonlinear equations. Numer. Algor. 76, 283–307 (2017)
Behl, R., Salimi, M., Ferrara, M., Sharifi, S., Alharbi, S.K.: Some real-life applications of a newly constructed derivative free iterative scheme. Symmetry 11, 239 (2019)
Kumar, D., Sharma, J.R., Singh, H.: Higher order Traub-Steffensen type methods and their convergence analysis in Banach spaces. Int. J. Nonlinear Sci. Numer. Simul. (2022). https://doi.org/10.1515/ijnsns-2021-0202
Singh, H., Sharma, J.R.: Simple yet highly efficient numerical techniques for systems of nonlinear equations. Comput. Appl. Math. 42, 22 (2023)
Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley, New York (1993)
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)
Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)
Atangana, A., Baleanu, D., Alsaedi, A.: New properties of conformable derivative. Open Math. 13, 889–898 (2015)
Toprakseven, Ş: Numerical solutions of conformable fractional differential equations by Taylor and finite difference methods. J. Nat. Appl. Sci. 23, 850–863 (2019)
Candelario, G., Cordero, A., Torregrosa, J.R., Vassileva, M.P.: An optimal and low computational cost fractional Newton-type method for solving nonlinear equations. Appl. Math. Lett. 124, 107650 (2022)
Candelario, G., Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Generalized conformable fractional Newton-type method for solving nonlinear systems. Numer. Algor. (2023). https://doi.org/10.1007/s11075-022-01463-z
Fischer, P., Stegeman, J.D.: Fractional Hadamard powers of positive semidefinite matrices. Linear Algebra Appl. 371, 53–74 (2003)
Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press, Cambridge (1990)
Magreñán, Á.A.: A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014)
Geum, Y.H., Kim, Y.I., Magreñán, Á.A.: A study of dynamics via Möbius conjugacy map on a family of sixth-order modified Newton-like multiple-zero finders with bivariate polynomial weight functions. J. Comput. Appl. Math. 344, 608–623 (2018)
Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)
Author information
Authors and Affiliations
Contributions
Methodology and conceptualization: H.S., J.R.S. Formal analysis: H.S. Writing—Original draft preparation: H.S. Review and editing: J.R.S.
Corresponding author
Ethics declarations
Ethical approval
Not applicable
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Singh, H., Sharma, J.R. A fractional Traub-Steffensen-type method for solving nonlinear equations. Numer Algor 95, 1103–1126 (2024). https://doi.org/10.1007/s11075-023-01601-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-023-01601-1