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On Fractional Newton‐Type Method for Nonlinear Problems

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  • Mine Aylin Bayrak
  • Ali Demir
  • Ebru Ozbilge

Abstract

The current manuscript is concerned with the development of the Newton–Raphson method, playing a significant role in mathematics and various other disciplines such as optimization, by using fractional derivatives and fractional Taylor series expansion. The development and modification of the Newton–Raphson method allow us to establish two new methods, which are called first‐ and second‐order fractional Newton–Raphson (FNR) methods. We provide convergence analysis of first‐ and second‐order fractional methods and give a general condition for the convergence of higher‐order FNR. Finally, some illustrative examples are considered to confirm the accuracy and effectiveness of both methods.

Suggested Citation

  • Mine Aylin Bayrak & Ali Demir & Ebru Ozbilge, 2022. "On Fractional Newton‐Type Method for Nonlinear Problems," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).
    • Handle: RePEc:wly:jjmath:v:2022:y:2022:i:1:n:7070253
    DOI: 10.1155/2022/7070253
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    References listed on IDEAS

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    1. NESTEROV, Yurii & POLYAK, B.T., 2006. "Cubic regularization of Newton method and its global performance," LIDAM Reprints CORE 1927, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Giro Candelario & Alicia Cordero & Juan R. Torregrosa, 2020. "Multipoint Fractional Iterative Methods with (2 α + 1)th-Order of Convergence for Solving Nonlinear Problems," Mathematics, MDPI, vol. 8(3), pages 1-15, March.
    3. Singh, Jagdev, 2020. "Analysis of fractional blood alcohol model with composite fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
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