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A Review of Definitions for Fractional Derivatives and Integral

Author

Listed:
  • Edmundo Capelas de Oliveira
  • José António Tenreiro Machado

Abstract

This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering.

Suggested Citation

  • Edmundo Capelas de Oliveira & José António Tenreiro Machado, 2014. "A Review of Definitions for Fractional Derivatives and Integral," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-6, June.
    • Handle: RePEc:hin:jnlmpe:238459
    DOI: 10.1155/2014/238459
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    Cited by:

    1. Qiushuang Wang & Run Xu, 2022. "On Hilfer Generalized Proportional Nabla Fractional Difference Operators," Mathematics, MDPI, vol. 10(15), pages 1-16, July.
    2. Shahid Saleem & Shahbaz Ahmad & Junseok Kim, 2023. "Total Fractional-Order Variation-Based Constraint Image Deblurring Problem," Mathematics, MDPI, vol. 11(13), pages 1-26, June.
    3. Jacek Gulgowski & Tomasz P. Stefański & Damian Trofimowicz, 2020. "On Applications of Elements Modelled by Fractional Derivatives in Circuit Theory," Energies, MDPI, vol. 13(21), pages 1-17, November.
    4. Mohra Zayed & Mahmoud Abul-Ez & Mohamed Abdalla & Nasser Saad, 2020. "On the Fractional Order Rodrigues Formula for the Shifted Legendre-Type Matrix Polynomials," Mathematics, MDPI, vol. 8(1), pages 1-23, January.
    5. Machado, J. A. Tenreiro & Lopes, António M., 2016. "The N-link pendulum: Embedding nonlinear dynamics into the multidimensional scaling method," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 130-138.
    6. Duarte Valério & Manuel D. Ortigueira & António M. Lopes, 2022. "How Many Fractional Derivatives Are There?," Mathematics, MDPI, vol. 10(5), pages 1-18, February.
    7. Aneesh S. Deogan & Roeland Dilz & Diego Caratelli, 2024. "On the Application of Fractional Derivative Operator Theory to the Electromagnetic Modeling of Frequency Dispersive Media," Mathematics, MDPI, vol. 12(7), pages 1-17, March.
    8. Torres-Hernandez, A. & Brambila-Paz, F. & Montufar-Chaveznava, R., 2022. "Acceleration of the order of convergence of a family of fractional fixed-point methods and its implementation in the solution of a nonlinear algebraic system related to hybrid solar receivers," Applied Mathematics and Computation, Elsevier, vol. 429(C).
    9. Ayşegül Daşcıoğlu & Serpil Salınan, 2019. "Comparison of the Orthogonal Polynomial Solutions for Fractional Integral Equations," Mathematics, MDPI, vol. 7(1), pages 1-10, January.
    10. Sergio Adriani David & Carlos Alberto Valentim, 2015. "Fractional Euler-Lagrange Equations Applied to Oscillatory Systems," Mathematics, MDPI, vol. 3(2), pages 1-15, April.
    11. Manuel D. Ortigueira, 2022. "A New Look at the Initial Condition Problem," Mathematics, MDPI, vol. 10(10), pages 1-17, May.
    12. Meshach Kumar & Utkal Mehta & Giansalvo Cirrincione, 2023. "A Novel Approach to Modeling Incommensurate Fractional Order Systems Using Fractional Neural Networks," Mathematics, MDPI, vol. 12(1), pages 1-14, December.
    13. Hamid, Muhammad & Usman, Muhammad & Haq, Rizwan Ul & Tian, Zhenfu, 2021. "A spectral approach to analyze the nonlinear oscillatory fractional-order differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    14. Mudassir Shams & Bruno Carpentieri, 2023. "On Highly Efficient Fractional Numerical Method for Solving Nonlinear Engineering Models," Mathematics, MDPI, vol. 11(24), pages 1-30, December.
    15. Maria de Jesus Estudillo-Ayala & Hugo Aguirre-Ramos & Juan Gabriel Avina-Cervantes & Jorge Mario Cruz-Duarte & Ivan Cruz-Aceves & Jose Ruiz-Pinales, 2020. "Algorithmic Analysis of Vesselness and Blobness for Detecting Retinopathies Based on Fractional Gaussian Filters," Mathematics, MDPI, vol. 8(5), pages 1-19, May.
    16. Daniele Mortari, 2023. "Representation of Fractional Operators Using the Theory of Functional Connections," Mathematics, MDPI, vol. 11(23), pages 1-16, November.
    17. Panda, Sumati Kumari & Vijayakumar, Velusamy, 2023. "Results on finite time stability of various fractional order systems," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    18. Oscar Martínez-Fuentes & Fidel Meléndez-Vázquez & Guillermo Fernández-Anaya & José Francisco Gómez-Aguilar, 2021. "Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities," Mathematics, MDPI, vol. 9(17), pages 1-29, August.

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