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A MATLAB Implementation of Spline Collocation Methods for Fractional Differential Equations

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Computational Science and Its Applications – ICCSA 2021 (ICCSA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12949))

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Abstract

The present paper illustrates a MATLAB program for the solution of fractional differential equations. It is based on a spline collocation method on a graded mesh, introduced by Pedas and Tamme in [J. Comput. Appl. Math. 255, 216–230 (2014)]. This is the first program proposed to implement spline collocation methods for fractional differential equations, and it is one of the few algorithms available in the literature for these functional equations. An explicit formulation of the method is derived, and the computational kernel is a nonlinear system to be solved at each time step. Such system involves some fractional integrals, whose analytical expression is given; their computation requires the knowledge of the coefficients of some polynomials and the evaluation of some special functions. The method is written in a compact matrix form, to improve the efficiency of the MATLAB implementation. The overall algorithm is outlined and then the attention is focused on some routines, which are given. In particular, some MATLAB native routines are used to evaluate special functions and to compute the coefficients of some polynomials. The complete list of the input and output parameters is available. Finally, an example of usage of the MATLAB program on a test problem is provided and some numerical experiments are shown.

Authors are members of the INdAM Research group GNCS and are supported by GNCS-INDAM project. A. Cardone and Dajana Conte are supported by PRIN2017-MIUR project.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Salerno, Fisciano, Italy

    Angelamaria Cardone, Dajana Conte & Beatrice Paternoster

Authors
  1. Angelamaria Cardone
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  2. Dajana Conte
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  3. Beatrice Paternoster
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Corresponding author

Correspondence to Angelamaria Cardone .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Potenza, Italy

    Beniamino Murgante

  3. Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Cagliari, Cagliari, Italy

    Chiara Garau

  5. University of Cagliari, Cagliari, Italy

    Ivan Blečić

  6. Monash University, Clayton, VIC, Australia

    David Taniar

  7. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  8. University of Minho, Braga, Portugal

    Ana Maria A. C. Rocha

  9. Polytechnic University of Bari, Bari, Italy

    Eufemia Tarantino

  10. Polytechnic University of Bari, Bari, Italy

    Carmelo Maria Torre

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Cardone, A., Conte, D., Paternoster, B. (2021). A MATLAB Implementation of Spline Collocation Methods for Fractional Differential Equations. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2021. ICCSA 2021. Lecture Notes in Computer Science(), vol 12949. Springer, Cham. https://doi.org/10.1007/978-3-030-86653-2_29

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  • DOI: https://doi.org/10.1007/978-3-030-86653-2_29

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