Research article

Comparison principles of fractional differential equations with non-local derivative and their applications

  • Received: 17 September 2020 Accepted: 10 November 2020 Published: 20 November 2020
  • MSC : 34A08, 35B50, 26A33

  • In this paper, we derive and prove a maximum principle for a linear fractional differential equation with non-local fractional derivative. The proof is based on an estimate of the non-local derivative of a function at its extreme points. A priori norm estimate and a uniqueness result are obtained for a linear fractional boundary value problem, as well as a uniqueness result for a nonlinear fractional boundary value problem. Several comparison principles are also obtained for linear and nonlinear equations.

    Citation: Mohammed Al-Refai, Dumitru Baleanu. Comparison principles of fractional differential equations with non-local derivative and their applications[J]. AIMS Mathematics, 2020, 6(2): 1443-1451. doi: 10.3934/math.2021088

    Related Papers:

  • In this paper, we derive and prove a maximum principle for a linear fractional differential equation with non-local fractional derivative. The proof is based on an estimate of the non-local derivative of a function at its extreme points. A priori norm estimate and a uniqueness result are obtained for a linear fractional boundary value problem, as well as a uniqueness result for a nonlinear fractional boundary value problem. Several comparison principles are also obtained for linear and nonlinear equations.
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    [1] A. M. A. El-Sayed, M. Gaber, On the finite Caputo and finite Riesz derivatives, Electron. J. Theor. Phys., 3 (2006), 81-95.
    [2] A. B. Malinowska, D. F. Torres, Fractional calculus of variations for a combined Caputo derivative, Fract. Calc. Appl. Anal., 14 (2011), 523-537.
    [3] T. Odzijewics, A. B. Malinowska, D. F. Torres, Fractional variable calculus with classical and combined Caputo derivatives, Nonlinear Anal.: Theory, Method. Appl., 75 (2012), 1507-1515.
    [4] T. Odzijewics, A. B. Malinowska, D. F. Torres, Fractional calculus of variations in terms of a generalized fractional integral with applications to physics, Abstr. Appl. Anal., 2012 (2012), 1-24.
    [5] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach, Yverdon, 1993.
    [6] F. Chen, D. Baleanu, G. C. Wu, Existence results of fractional differential equations with RieszCaputo derivative, Eur. Phys. J. Spec. Top., 226 (2017), 3411-3425.
    [7] H. Ye, F. Liu, V. Anh, I. Turner, Maximum principle and numerical method for the multi-term time space Riesz-Caputo fractional differential equations, Appl. Math. Comput., 227 (2014), 531-540.
    [8] M. Al-Refai, Basic results on nonlinear eigenvalue problems of fractional order, Electron. J. Differ. Equ., 2012 (2012), 1-12.
    [9] M. Al-Refai, Y. Luchko, Maximum principles for the fractional diffusion equations with the Riemann-Liouville fractional derivative and their applications, Fract. Calc. Appl. Anal., 17 (2014), 483-498.
    [10] M. Al-Refai, Y. Luchko, Analysis of fractional diffusion equations of distributed order: maximum principles and its applications, Analysis, 36 (2016), 123-133.
    [11] M. Al-Refai, K. Pal, A maximum principle for a fractional boundary value problem with convection term and applications, Math. Model. Anal., 24 (2019), 62-71.
    [12] M. Al-Refai, On the fractional derivative at extreme points, Elect. J. Qual. Theory Differ. Equ., 2012 (2012), 1-5.
    [13] Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), 218-223.

    © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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