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Mathematical Biosciences and Engineering

2022, Volume 19, Issue 6: 6234-6251. doi: 10.3934/mbe.2022291
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Research article

Differential equations of arbitrary order under Caputo-Fabrizio derivative: some existence results and study of stability

  • 1.
    Department of Mathematics, University of Tiaret, Tiaret, Algeria
  • 2.
    CITMAga, 15782 Santiago de Compostela, España
  • 3.
    Departamento de Estatística, Análise Matemática e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, España

  • Academic Editor: Yang Kuang
  • Received: 29 October 2021 Revised: 31 March 2022 Accepted: 11 April 2022 Published: 18 April 2022

  • In this work, we consider the problem of the existence and uniqueness of solution, and also the simple existence of solution, for implicit differential equations of arbitrary order involving Caputo-Fabrizio derivative. The main tools for this study are contraction mapping principle and Schaefer's fixed point result. We also study the stability of the equations in the sense of Ulam-Hyers and also from the perspective of Ulam-Hyers-Rassias.

    Citation: Kadda Maazouz, Rosana Rodríguez-López. Differential equations of arbitrary order under Caputo-Fabrizio derivative: some existence results and study of stability[J]. Mathematical Biosciences and Engineering, 2022, 19(6): 6234-6251. doi: 10.3934/mbe.2022291

    Related Papers:

  • In this work, we consider the problem of the existence and uniqueness of solution, and also the simple existence of solution, for implicit differential equations of arbitrary order involving Caputo-Fabrizio derivative. The main tools for this study are contraction mapping principle and Schaefer's fixed point result. We also study the stability of the equations in the sense of Ulam-Hyers and also from the perspective of Ulam-Hyers-Rassias.


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    [1] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
    [2] M. Caputo, M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential, Progr. Fract. Differ. Appl., 2 (2016), 1–11. https://doi.org/10.18576/pfda/020101 doi: 10.18576/pfda/020101
    [3] S. Abbas, M. Benchohra, On the generalized Ulam-Hyers-Rassias stability for Darboux problem for partial fractional implicit differential equations, Appl. Math. E-Notes, 14 (2014), 20–28. Available from: https://www.math.nthu.edu.tw/amen/2014/131113(final).pdf.
    [4] S. M. Aydogan, D. Baleanu, A. Mousalou, S. Rezapoux, On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations, Adv. Differ. Equations, 2017 (2017), 11. https://doi.org/10.1186/s13662-017-1258-3 doi: 10.1186/s13662-017-1258-3
    [5] D. Baleanu, A. Mousalou, S. Rezapoux, On the existence of solutions for some infinite coefficient-symetric Caputo-Fabrizio fractional integro-differential equations, Boundary Value Probl., 2017 (2017), 1–9. https://doi.org/10.1186/s13661-017-0867-9 doi: 10.1186/s13661-017-0867-9
    [6] E. F. D. Goufo, Applications of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Burgers equations, Math. Model. Anal., 21 (2016), 188–198. https://doi.org/10.3846/13926292.2016.1145607 doi: 10.3846/13926292.2016.1145607
    [7] J. Hristov, Derivation of fractional Dodson equation and beyond: Transient mass diffusion with a non-singular memory and exponentially fading-out diffusivity, Progr. Fract. Differ. Appl., 3 (2017), 255–270. https://doi.org/10.18576/pfda/030402 doi: 10.18576/pfda/030402
    [8] J. Losada, J. J. Nieto, Properties of new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 97–92. https://doi.org10.12785/pfda/010202 doi: 10.12785/pfda/010202
    [9] D. Mozyrska, D. F. M. Torres, M. Wyrwas, Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales, Nonlinear Anal. Hybrid Syst., 32 (2019) 168–176. https://doi.org/10.1016/j.nahs.2018.12.001 doi: 10.1016/j.nahs.2018.12.001
    [10] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
    [11] B. S. T. Alkahtani, Model of heat with Caputo-Fabrizio derivative with fractional order, J. Comput. Theor. Nanosci., 13 (2016), 2994–2999. https://doi.org/10.1166/jctn.2016.4948 doi: 10.1166/jctn.2016.4948
    [12] J. Hristov, Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffreys kernel to the Caputo-Fabrizio time-fractional derivative, Therm. Sci., 20 (2016), 757–762. https://doi.org/10.2298/TSCI160112019H doi: 10.2298/TSCI160112019H
    [13] J. Hristov, Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, Front. Fract. Calc., 1 (2017), 270–342. https://doi.org/10.2174/9781681085999118010013 doi: 10.2174/9781681085999118010013
    [14] X. J. Yang, H. M. Srivastava, J. A. T. Machado, A new fractional derivative without singular kernel application to the modelling of the steady heat flow, Therm. Sci., 20 (2016), 753–756. https://doi.org/10.2298/TSCI151224222Y doi: 10.2298/TSCI151224222Y
    [15] D. S. Cimpean, D. Popa, Hyers-Ulam stability of Euler's equation, Appl. Math. Lett., 24 (2011) 1539–1543. https://doi.org/10.1016/j.aml.2011.03.042 doi: 10.1016/j.aml.2011.03.042
    [16] S. M. Jung, Hyers-Ulam stability of linear differential equations of first order Ⅱ, Appl. Math. Lett., 19 (2006), 854–858. https://doi.org/10.1016/j.aml.2005.11.004 doi: 10.1016/j.aml.2005.11.004
    [17] C. Alsina, R. Ger, On some inequalities and stability results related to the exponential functio, J. Inequal. Appl., 2 (1998), 373–380. https://doi.org/10.1155/S102558349800023X doi: 10.1155/S102558349800023X
    [18] E. Capelas de Olivera, J. V. da C. Sousa, Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations, Results Math., 73 (2018), 50–56. https://doi.org/10.1007/s00025-018-0872-z doi: 10.1007/s00025-018-0872-z
    [19] S. M. Jung, On the Hyers-Ulam stability of the functional equation that have the quadratic property, J. Math. Anal. Appl., 222 (1998), 126–137. https://doi.org/10.1006/jmaa.1998.5916 doi: 10.1006/jmaa.1998.5916
    [20] M. Obloza, Hyers stability of the linear differential equation, Rocznik Nauk-Dydakt. Prace Mat., 13 (1993), 295–270.
    [21] J. M. Rassias, Functional Equations, Difference Inequalities and Ulam Stability Notions (F.U.N.), Nova Science Publishers, Inc., New York, 2010.
    [22] T. M. Rassias, J. Brzdek, Functional Equations in Mathematical Analysis, Springer, New York, NY, 2012. https://doi.org/10.1007/978-1-4614-0055-4
    [23] D. L. Kleiman, M. R. Etchechoury, P. Puleston, A simple method for impasse points detection in nonlinear electrical circuits, Math. Probl. Eng., 2018 (2018), 2613890. https://doi.org/10.1155/2018/2613890 doi: 10.1155/2018/2613890
    [24] M. Benchohra, S. Bouriah, Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order, Moroc. J. Pure Appl. Anal., 1 (2015), 22–37. https://doi.org/10.7603/s40956-015-0002-9 doi: 10.7603/s40956-015-0002-9
    [25] M. Benchohra, S. Bouriah, J. R. Graef, Nonlinear implicit differential equations of fractional order at resonance, Electron. J. Differ. Equations, 2016 (2016), 1–10. Available from: https://ejde.math.txstate.edu/Volumes/2016/324/benchohra.pdf.
    [26] S. Abbas, M. Benchohra, J. R. Graef, J. Henderson, Implicit Fractional Differential and Integral Equations, De Gruyter, Berlin, 2018. https://doi.org/10.1515/9783110553819
    [27] M. Benchohra, S. Bouriah, J. J. Nieto, Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative, Demonstr. Math., 52 (2019), 437–450. https://doi.org/10.1515/dema-2019-0032 doi: 10.1515/dema-2019-0032
    [28] M. Alam, D. Shah, Hyers-Ulam stability of coupled implicit fractional integro-differential equations with Riemann–Liouville derivatives, Chaos Solitons Fractals, 150 (2021), 111122. https://doi.org/10.1016/j.chaos.2021.111122 doi: 10.1016/j.chaos.2021.111122
    [29] A. M. Saeed, M. S. Abdo, M. B. Jeelani, Existence and Ulam–Hyers stability of a fractional-order coupled system in the frame of generalized hilfer derivatives, Mathematics, 9 (2021), 2543. https://doi.org/10.3390/math9202543 doi: 10.3390/math9202543
    [30] K. Zhao, S. Ma, Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses, AIMS Math., 7 (2021), 3169–3185. https://doi.org/10.3934/math.2022175 doi: 10.3934/math.2022175
    [31] S. Etemad, B. Tellab, J. Alzabut, S. Rezapour, M. I. Abbas, Approximate solutions and Hyers–Ulam stability for a system of the coupled fractional thermostat control model via the generalized differential transform, Adv. Differ. Equations, 2021 (2021), 428. https://doi.org/10.1186/s13662-021-03563-x doi: 10.1186/s13662-021-03563-x
    [32] E. Bicer, Application of Sumudu transform method for Hyers-Ulam stability of partial differential equation, J. Appl. Math. Inf., 39 (2021), 267–275. https://doi.org/10.14317/jami.2021.267 doi: 10.14317/jami.2021.267
    [33] A. Granas, On the Leray-Schauder alternative, Topol. Methods Nonlinear Anal., 2 (1993), 225–231. https://doi.org/10.12775/TMNA.1993.040 doi: 10.12775/TMNA.1993.040
    [34] T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math., 20 (1919), 292–296. https://doi.org/10.2307/1967124 doi: 10.2307/1967124
    [35] R. Bellman, The stability of solutions of linear differential equations, Duke Math. J., 10 (1943), 643–647. https://doi.org/10.1215/S0012-7094-43-01059-2 doi: 10.1215/S0012-7094-43-01059-2
    [36] E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York, 1956. https://doi.org/10.1063/1.3059875
    [37] K. Liu, M. Fe$ \mathop {{\rm{c}}} \limits^ \smallsmile $kan, D. O'Regan, J. Wang, Hyers-Ulam stability and existence of solutions for differential equations with Caputo-Fabrizio fractional derivative, Mathematics, 7 (2019), 333. https://doi.org/10.3390/math7040333 doi: 10.3390/math7040333
    [38] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific, New York, 2012.
    [39] A. Cabada, K. Maazouz, Results for fractional differential equations with integral boundary conditions involving the Hadamard derivative, NABVP, 292 (2018), 145–155. https://doi.org/10.1007/978-3-030-26987-6_10 doi: 10.1007/978-3-030-26987-6_10
    [40] H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075–1081. https://doi.org/10.1016/j.jmaa.2006.05.061 doi: 10.1016/j.jmaa.2006.05.061
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