Multipoint fraccional iterative methods with (2a+1)th-order of convergence for solving nonlinear problems

[EN] In the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence ¿ + 1 and compare it with the existing fractional Newton method with order 2¿. Moreover, we also introduce a multipoint fractional Traub-type method with order 2¿ + 1 and compare its performance with that of its first step. Some numerical tests and analysis of the dependence on the initial estimations are made for each case, including a comparison with classical Newton (¿ = 1 of the first step of the class) and classical Traub¿s scheme (¿ = 1 of fractional proposed multipoint method). In this comparison, some cases are found where classical Newton and Traub¿s methods do not converge and the proposed methods do, among other advantages.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE) and FONDOCYT 029-2018 Republica Dominicana.Candelario, G.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2020). Multipoint fraccional iterative methods with (2a+1)th-order of convergence for solving nonlinear problems. Mathematics. 8(3):1-15. https://doi.org/10.3390/math803045211583Mathai, A. M., & Haubold, H. J. (2017). Fractional and Multivariable Calculus. Springer Optimization and Its Applications. doi:10.1007/978-3-319-59993-9Altaf Khan, M., Ullah, S., & Farhan, M. (2019). The dynamics of Zika virus with Caputo fractional derivative. AIMS Mathematics, 4(1), 134-146. doi:10.3934/math.2019.1.134Akgül, A., Cordero, A., & Torregrosa, J. R. (2019). A fractional Newton method with 2αth-order of convergence and its stability. Applied Mathematics Letters, 98, 344-351. doi:10.1016/j.aml.2019.06.028Cordero, A., Girona, I., & Torregrosa, J. R. (2019). A Variant of Chebyshev’s Method with 3αth-Order of Convergence by Using Fractional Derivatives. Symmetry, 11(8), 1017. doi:10.3390/sym11081017Odibat, Z. M., & Shawagfeh, N. T. (2007). Generalized Taylor’s formula. Applied Mathematics and Computation, 186(1), 286-293. doi:10.1016/j.amc.2006.07.102Trujillo, J. J., Rivero, M., & Bonilla, B. (1999). On a Riemann–Liouville Generalized Taylor’s Formula. Journal of Mathematical Analysis and Applications, 231(1), 255-265. doi:10.1006/jmaa.1998.6224Jumarie, G. (2006). Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications, 51(9-10), 1367-1376. doi:10.1016/j.camwa.2006.02.001Lanczos, C. (1964). A Precision Approximation of the Gamma Function. Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, 1(1), 86-96. doi:10.1137/0701008Magreñán, Á. A. (2014). A new tool to study real dynamics: The convergence plane. Applied Mathematics and Computation, 248, 215-224. doi:10.1016/j.amc.2014.09.06

Semilocal convergence of a family of iterative methods in Banach spaces

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