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On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative

Year 2023, , 122 - 129, 30.09.2023
https://doi.org/10.32323/ujma.1288015

Abstract

This work aims to study the existing results of mild solutions for a semi-linear Atangana-Baleanu-Caputo fractional differential equation with order $ 0 < \theta < 1 $ in an arbitrary Banach space. We rely on some arguments to present the mild solution to our problem in terms of an $ \theta $-resolvent family. Then we study the existence of this mild solution by using Krasnoselskii's fixed point theorem. Finally, we give an example to prove our results.

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Thanks

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References

  • [1] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Translated from the 1987 Russian original, (1993).
  • [2] R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27.3 (1983), 201-210.
  • [3] N. Chefnaj, T. Abdellah, K. Hilal, S. Melliani, A. Kajouni, Boundary Problems for Fractional Differential Equations Involving the Generalized Caputo-Fabrizio Fractional Derivative in the l-Metric Space, Turkish Journal of Science, 8(1) (2023), 24-36.
  • [4] M. D. Bayrak, A. Demir, On the challenge of identifying space dependent coefficient in space-time fractional diffusion equations by fractional scaling transformations method, Turkish Journal of Science, 7(2) (2022) 132-145.
  • [5] C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, V. Feliu-Batlle., Fractional-Order Systems and Controls: Fundamentals and Applications, Springer Science & Business Media, 2010.
  • [6] M. I. Syam, M. Al-Refai, Fractional differential equations with Atangana–Baleanu fractional derivative: Analysis and applications, Chaos Solitons Fractals: X, 2 (2019), 100013.
  • [7] N. Sweilam, S. Al-Mekhlafi, T. Assiri, A. Atangana, Optimal control for cancer treatment mathematical model using Atangana–Baleanu–Caputo fractional derivative, Adv. Difference Equ., 2020(1) (2020), 1-21.
  • [8] J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515.
  • [9] A. Jajarmi, D. Baleanu, S. S Sajjadi, J. H. Asad, A new feature of the fractional Euler–Lagrange equations for a coupled oscillator using a nonsingular operator approach, Frontiers in Physics, 7 (2019), 196.
  • [10] S. Uc¸ar, E. Uc¸ar, N. Özdemir, Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative, Chaos Solitons Fractals, 118 (2019), 300-306.
  • [11] M. Abdo, K. Shah, H. Wahash, S. Panchal, On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative, Chaos Solitons Fractals, 135 (2020), 109867.
  • [12] M. I. Abbas, M. A. Ragusa, Nonlinear fractional differential inclusions with non-singular Mittag-Leffler kernel, AIMS Math, 7(11) (2022), 20328-20340.
  • [13] F. Jarad, T. Abdeljawad, Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16-20.
  • [14] A. O. Akdemir, A. Karao˘glan, M. A. Ragusa, Fractional integral inequalities via Atangana-Baleanu operators for convex and concave functions, J. Funct. Spaces, 2021 (2021), 1-10.
  • [15] M. A. Dokuyucu, Analysis of a novel finance chaotic model via ABC fractional derivative Numer. Methods Partial Differential Equations, 37(2) (2021), 1583-1590.
  • [16] M. I. Syam, M. Al-Refai, Fractional differential equations with Atangana-Baleanu fractional derivative: Analysis and applications, Chaos Solitons Fractals: X, 2 (2019), 100013.
  • [17] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006.
  • [18] K. Diethelm, N. J. Ford, The Analysis of Fractional Differential equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, 2004. Berlin: Springer, 2010.
  • [19] A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454.
  • [20] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 44, Springer Science & Business Media, 2012.
  • [21] T. A. Burton, A fixed-point theorem of Krasnoselskii, Appl. Math. Lett., 11(1) (1998), 85-88.
  • [22] X. B. Shu, Y. Lai, Y. Chen, The existence of mild solutions for impulsive fractional partial differential equations.Nonlinear Analysis: Theory, Methods & Applications, 74(5) (2011), 2003-2011.
  • [23] E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Eindhoven: Technische Universiteit Eindhoven, 2001.
  • [24] G. M. Bahaa, A. Hamiaz, Optimality conditions for fractional differential inclusions with nonsingular Mittag-Leffler kernel, Adv. Difference Equ., 2018(2018), 1-26.
Year 2023, , 122 - 129, 30.09.2023
https://doi.org/10.32323/ujma.1288015

Abstract

Project Number

.

References

  • [1] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Translated from the 1987 Russian original, (1993).
  • [2] R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27.3 (1983), 201-210.
  • [3] N. Chefnaj, T. Abdellah, K. Hilal, S. Melliani, A. Kajouni, Boundary Problems for Fractional Differential Equations Involving the Generalized Caputo-Fabrizio Fractional Derivative in the l-Metric Space, Turkish Journal of Science, 8(1) (2023), 24-36.
  • [4] M. D. Bayrak, A. Demir, On the challenge of identifying space dependent coefficient in space-time fractional diffusion equations by fractional scaling transformations method, Turkish Journal of Science, 7(2) (2022) 132-145.
  • [5] C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, V. Feliu-Batlle., Fractional-Order Systems and Controls: Fundamentals and Applications, Springer Science & Business Media, 2010.
  • [6] M. I. Syam, M. Al-Refai, Fractional differential equations with Atangana–Baleanu fractional derivative: Analysis and applications, Chaos Solitons Fractals: X, 2 (2019), 100013.
  • [7] N. Sweilam, S. Al-Mekhlafi, T. Assiri, A. Atangana, Optimal control for cancer treatment mathematical model using Atangana–Baleanu–Caputo fractional derivative, Adv. Difference Equ., 2020(1) (2020), 1-21.
  • [8] J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515.
  • [9] A. Jajarmi, D. Baleanu, S. S Sajjadi, J. H. Asad, A new feature of the fractional Euler–Lagrange equations for a coupled oscillator using a nonsingular operator approach, Frontiers in Physics, 7 (2019), 196.
  • [10] S. Uc¸ar, E. Uc¸ar, N. Özdemir, Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative, Chaos Solitons Fractals, 118 (2019), 300-306.
  • [11] M. Abdo, K. Shah, H. Wahash, S. Panchal, On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative, Chaos Solitons Fractals, 135 (2020), 109867.
  • [12] M. I. Abbas, M. A. Ragusa, Nonlinear fractional differential inclusions with non-singular Mittag-Leffler kernel, AIMS Math, 7(11) (2022), 20328-20340.
  • [13] F. Jarad, T. Abdeljawad, Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16-20.
  • [14] A. O. Akdemir, A. Karao˘glan, M. A. Ragusa, Fractional integral inequalities via Atangana-Baleanu operators for convex and concave functions, J. Funct. Spaces, 2021 (2021), 1-10.
  • [15] M. A. Dokuyucu, Analysis of a novel finance chaotic model via ABC fractional derivative Numer. Methods Partial Differential Equations, 37(2) (2021), 1583-1590.
  • [16] M. I. Syam, M. Al-Refai, Fractional differential equations with Atangana-Baleanu fractional derivative: Analysis and applications, Chaos Solitons Fractals: X, 2 (2019), 100013.
  • [17] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006.
  • [18] K. Diethelm, N. J. Ford, The Analysis of Fractional Differential equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, 2004. Berlin: Springer, 2010.
  • [19] A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454.
  • [20] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 44, Springer Science & Business Media, 2012.
  • [21] T. A. Burton, A fixed-point theorem of Krasnoselskii, Appl. Math. Lett., 11(1) (1998), 85-88.
  • [22] X. B. Shu, Y. Lai, Y. Chen, The existence of mild solutions for impulsive fractional partial differential equations.Nonlinear Analysis: Theory, Methods & Applications, 74(5) (2011), 2003-2011.
  • [23] E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Eindhoven: Technische Universiteit Eindhoven, 2001.
  • [24] G. M. Bahaa, A. Hamiaz, Optimality conditions for fractional differential inclusions with nonsingular Mittag-Leffler kernel, Adv. Difference Equ., 2018(2018), 1-26.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Samira Zerbib 0000-0002-2782-0055

Khalid Hilal 0000-0002-0806-2623

Ahmed Kajounı 0000-0001-8484-6107

Project Number .
Publication Date September 30, 2023
Submission Date April 26, 2023
Acceptance Date September 23, 2023
Published in Issue Year 2023

Cite

APA Zerbib, S., Hilal, K., & Kajounı, A. (2023). On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative. Universal Journal of Mathematics and Applications, 6(3), 122-129. https://doi.org/10.32323/ujma.1288015
AMA Zerbib S, Hilal K, Kajounı A. On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative. Univ. J. Math. Appl. September 2023;6(3):122-129. doi:10.32323/ujma.1288015
Chicago Zerbib, Samira, Khalid Hilal, and Ahmed Kajounı. “On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative”. Universal Journal of Mathematics and Applications 6, no. 3 (September 2023): 122-29. https://doi.org/10.32323/ujma.1288015.
EndNote Zerbib S, Hilal K, Kajounı A (September 1, 2023) On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative. Universal Journal of Mathematics and Applications 6 3 122–129.
IEEE S. Zerbib, K. Hilal, and A. Kajounı, “On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative”, Univ. J. Math. Appl., vol. 6, no. 3, pp. 122–129, 2023, doi: 10.32323/ujma.1288015.
ISNAD Zerbib, Samira et al. “On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative”. Universal Journal of Mathematics and Applications 6/3 (September 2023), 122-129. https://doi.org/10.32323/ujma.1288015.
JAMA Zerbib S, Hilal K, Kajounı A. On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative. Univ. J. Math. Appl. 2023;6:122–129.
MLA Zerbib, Samira et al. “On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative”. Universal Journal of Mathematics and Applications, vol. 6, no. 3, 2023, pp. 122-9, doi:10.32323/ujma.1288015.
Vancouver Zerbib S, Hilal K, Kajounı A. On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative. Univ. J. Math. Appl. 2023;6(3):122-9.

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