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Iterative Algorithms and Left-Right Caputo Fractional Derivatives

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Intelligent Numerical Methods: Applications to Fractional Calculus

Part of the book series: Studies in Computational Intelligence ((SCI,volume 624))

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Abstract

We present a local as well as a semilocal convergence analysis for some iterative algorithms in order to approximate a locally unique solution.

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References

  1. S. Amat, S. Busquier, Third-order iterative methods under Kantorovich conditions. J. Math. Anal. Appl. 336, 243–261 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Amat, S. Busquier, S. Plaza, Chaotic dynamics of a third-order Newton-like method. J. Math. Anal. Appl. 366(1), 164–174 (2010)

    Article  MathSciNet  Google Scholar 

  3. G. Anastassiou, Fractional Differentiation Inequalities (Springer, New York, 2009)

    Book  MATH  Google Scholar 

  4. G. Anastassiou, Intelligent Mathematics: Computational Analysis (Springer, Heidelberg, 2011)

    Book  Google Scholar 

  5. G. Anastassiou, I. Argyros, A convergence analysis for some iterative algorithms with applications to fractional calculus, submitted (2015)

    Google Scholar 

  6. I.K. Argyros, Newton-like methods in partially ordered linear spaces. J. Approx. Theory Appl. 9(1), 1–10 (1993)

    MATH  Google Scholar 

  7. I.K. Argyros, Results on controlling the residuals of perturbed Newton-like methods on Banach spaces with a convergence structure. Southwest J. Pure Appl. Math. 1, 32–38 (1995)

    Google Scholar 

  8. I.K. Argyros, Convergence and Applications of Newton-Like Iterations (Springer, New York, 2008)

    Google Scholar 

  9. K. Diethelm, The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics, vol. 2004, 1st edn. (Springer, New York, 2010)

    Google Scholar 

  10. J.A. Ezquerro, J.M. Gutierrez, M.A. Hernandez, N. Romero, M.J. Rubio, The Newton method: from Newton to Kantorovich (Spanish). Gac. R. Soc. Mat. Esp. 13, 53–76 (2010)

    MathSciNet  MATH  Google Scholar 

  11. J.A. Ezquerro, M.A. Hernandez, Newton-like methods of high order and domains of semilocal and global convergence. Appl. Math. Comput. 214(1), 142–154 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. L.V. Kantorovich, G.P. Akilov, Functional Analysis in Normed Spaces (Pergamon Press, New York, 1964)

    MATH  Google Scholar 

  13. A.A. Magrenan, Different anomalies in a Jarratt family of iterative root finding methods. Appl. Math. Comput. 233, 29–38 (2014)

    Article  MathSciNet  Google Scholar 

  14. A.A. Magrenan, A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014)

    Google Scholar 

  15. F.A. Potra, V. Ptak, Nondiscrete Induction and Iterative Processes (Pitman, London, 1984)

    MATH  Google Scholar 

  16. P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. J. Complex. 26, 3–42 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to George A. Anastassiou .

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Anastassiou, G.A., Argyros, I.K. (2016). Iterative Algorithms and Left-Right Caputo Fractional Derivatives. In: Intelligent Numerical Methods: Applications to Fractional Calculus. Studies in Computational Intelligence, vol 624. Springer, Cham. https://doi.org/10.1007/978-3-319-26721-0_14

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  • DOI: https://doi.org/10.1007/978-3-319-26721-0_14

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