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A Numerical Approach for Variable Order Fractional Equations

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Numerical Solutions of Realistic Nonlinear Phenomena

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 31))

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Abstract

This work considers numerical solutions of variable fractional order, multi-term differential equations. A second order numerical approach has been used here to approximate fractional order derivative and this approach is similar to the fractional-variable order derivative approach for single-term case (see Cao and Qiu (Appl. Math. Lett. 61, 88–94, 2016)). To construct a multi-term, fractional-variable order differential equation, we first add y(t) term to the single-term equation and then test the convergency of the method. Secondly, we add a second order ordinary derivative term to the equation and this term is approximated by central finite differences, where the variable Riemann–Liouville derivative approximation is based on the shifted GrÜnwald approximation technique and the numerical scheme is still convergent.

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Authors and Affiliations

  1. Faculty of Science, Gazi University, Ankara, Turkey

    Fatma Ayaz & İrem Bektaş Güner

Authors
  1. Fatma Ayaz
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  2. İrem Bektaş Güner
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Correspondence to Fatma Ayaz .

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Editors and Affiliations

  1. Institute of Engineering, Polytechnic of Porto, Porto, Portugal

    J. A. Tenreiro Machado

  2. Department of Mathematics, Balıkesir University, Balıkesir, Turkey

    Necati Özdemir

  3. Department of Mathematics & Computer Science, Çankaya University, Ankara, Turkey

    Dumitru Baleanu

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Ayaz, F., Güner, İ.B. (2020). A Numerical Approach for Variable Order Fractional Equations. In: Machado, J., Özdemir, N., Baleanu, D. (eds) Numerical Solutions of Realistic Nonlinear Phenomena. Nonlinear Systems and Complexity, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-030-37141-8_11

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