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Computing Hadamard type operators of variable fractional order

Author

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  • Almeida, Ricardo
  • Torres, Delfim F.M.

Abstract

We consider Hadamard fractional derivatives and integrals of variable fractional order. A new type of fractional operator, which we call the Hadamard–Marchaud fractional derivative, is also considered. The objective is to represent these operators as series of terms involving integer-order derivatives only, and then approximate the fractional operators by a finite sum. An upper bound formula for the error is provided. We exemplify our method by applying the proposed numerical procedure to the solution of a fractional differential equation and a fractional variational problem with dependence on the Hadamard–Marchaud fractional derivative.

Suggested Citation

  • Almeida, Ricardo & Torres, Delfim F.M., 2015. "Computing Hadamard type operators of variable fractional order," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 74-88.
    • Handle: RePEc:eee:apmaco:v:257:y:2015:i:c:p:74-88
    DOI: 10.1016/j.amc.2014.12.071
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    Cited by:

    1. Duarte Valério & Manuel D. Ortigueira, 2023. "Variable-Order Fractional Scale Calculus," Mathematics, MDPI, vol. 11(21), pages 1-13, November.
    2. Cai, Ruiyang & Ge, Fudong & Chen, YangQuan & Kou, Chunhai, 2019. "Regional observability for Hadamard-Caputo time fractional distributed parameter systems," Applied Mathematics and Computation, Elsevier, vol. 360(C), pages 190-202.
    3. John R. Graef & Kadda Maazouz & Moussa Daif Allah Zaak, 2023. "A Generalized Lyapunov Inequality for a Pantograph Boundary Value Problem Involving a Variable Order Hadamard Fractional Derivative," Mathematics, MDPI, vol. 11(13), pages 1-16, July.
    4. Ahmed Refice & Mohammed Said Souid & Ivanka Stamova, 2021. "On the Boundary Value Problems of Hadamard Fractional Differential Equations of Variable Order via Kuratowski MNC Technique," Mathematics, MDPI, vol. 9(10), pages 1-16, May.
    5. Pei, Ke & Wang, Guotao & Sun, Yanyan, 2017. "Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain," Applied Mathematics and Computation, Elsevier, vol. 312(C), pages 158-168.

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