IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v126y2019icp7-11.html
   My bibliography  Save this article

Fundamental results on weighted Caputo–Fabrizio fractional derivative

Author

Listed:
  • Al-Refai, Mohammed
  • Jarrah, Abdulla M.

Abstract

In this paper, we define the weighted Caputo–Fabrizio fractional derivative of Caputo sense, and study related linear and nonlinear fractional differential equations. The solution of the linear fractional differential equation is obtained in a closed form, and has been used to define the weighted Caputo–Fabrizio fractional integral. We study main properties of the weighted Caputo–Fabrizio fractional derivative and integral. We also, apply the Banach fixed point theorem to establish the existence of a unique solution to the nonlinear fractional differential equation. Two examples are presented to illustrate the efficiency of the obtained results.

Suggested Citation

  • Al-Refai, Mohammed & Jarrah, Abdulla M., 2019. "Fundamental results on weighted Caputo–Fabrizio fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 7-11.
    • Handle: RePEc:eee:chsofr:v:126:y:2019:i:c:p:7-11
    DOI: 10.1016/j.chaos.2019.05.035
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077919302000
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2019.05.035?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Atangana, Abdon, 2016. "On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 948-956.
    2. Alkahtani, B.S.T. & Atangana, A., 2016. "Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 539-546.
    3. Gómez-Aguilar, J.F. & López-López, M.G. & Alvarado-Martínez, V.M. & Reyes-Reyes, J. & Adam-Medina, M., 2016. "Modeling diffusive transport with a fractional derivative without singular kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 467-481.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Acay, Bahar & Bas, Erdal & Abdeljawad, Thabet, 2020. "Fractional economic models based on market equilibrium in the frame of different type kernels," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    2. Panda, Sumati Kumari & Abdeljawad, Thabet & Ravichandran, C., 2020. "A complex valued approach to the solutions of Riemann-Liouville integral, Atangana-Baleanu integral operator and non-linear Telegraph equation via fixed point method," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    3. Dehestani, H. & Ordokhani, Y. & Razzaghi, M., 2020. "Application of fractional Gegenbauer functions in variable-order fractional delay-type equations with non-singular kernel derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    4. Oscar Martínez-Fuentes & Fidel Meléndez-Vázquez & Guillermo Fernández-Anaya & José Francisco Gómez-Aguilar, 2021. "Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities," Mathematics, MDPI, vol. 9(17), pages 1-29, August.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Saad, Khaled M. & Gómez-Aguilar, J.F., 2018. "Analysis of reaction–diffusion system via a new fractional derivative with non-singular kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 703-716.
    2. Sun, HongGuang & Hao, Xiaoxiao & Zhang, Yong & Baleanu, Dumitru, 2017. "Relaxation and diffusion models with non-singular kernels," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 590-596.
    3. Abdulhameed, M. & Vieru, D. & Roslan, R., 2017. "Modeling electro-magneto-hydrodynamic thermo-fluidic transport of biofluids with new trend of fractional derivative without singular kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 484(C), pages 233-252.
    4. Yu, Xiangnan & Zhang, Yong & Sun, HongGuang & Zheng, Chunmiao, 2018. "Time fractional derivative model with Mittag-Leffler function kernel for describing anomalous diffusion: Analytical solution in bounded-domain and model comparison," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 306-312.
    5. Owolabi, Kolade M. & Atangana, Abdon, 2017. "Numerical approximation of nonlinear fractional parabolic differential equations with Caputo–Fabrizio derivative in Riemann–Liouville sense," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 171-179.
    6. Costa, F.S. & Oliveira, D.S. & Rodrigues, F.G. & de Oliveira, E.C., 2019. "The fractional space–time radial diffusion equation in terms of the Fox’s H-function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 403-418.
    7. Liu, Zhengguang & Cheng, Aijie & Li, Xiaoli, 2017. "A second order Crank–Nicolson scheme for fractional Cattaneo equation based on new fractional derivative," Applied Mathematics and Computation, Elsevier, vol. 311(C), pages 361-374.
    8. Cai, Rui-Yang & Zhou, Hua-Cheng & Kou, Chun-Hai, 2021. "Boundary control strategy for three kinds of fractional heat equations with control-matched disturbances," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    9. Gómez-Aguilar, J.F., 2017. "Irving–Mullineux oscillator via fractional derivatives with Mittag-Leffler kernel," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 179-186.
    10. Xu, Xuefang & Li, Bo & Qiao, Zijian & Shi, Peiming & Shao, Huaishuang & Li, Ruixiong, 2023. "Caputo-Fabrizio fractional order derivative stochastic resonance enhanced by ADOF and its application in fault diagnosis of wind turbine drivetrain," Renewable Energy, Elsevier, vol. 219(P1).
    11. Fouladi, Somayeh & Dahaghin, Mohammad Shafi, 2022. "Numerical investigation of the variable-order fractional Sobolev equation with non-singular Mittag–Leffler kernel by finite difference and local discontinuous Galerkin methods," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    12. Shuai Yang & Qing Wei & Lu An, 2022. "Fractional Advection Diffusion Models for Radionuclide Migration in Multiple Barriers System of Deep Geological Repository," Mathematics, MDPI, vol. 10(14), pages 1-7, July.
    13. Owolabi, Kolade M. & Atangana, Abdon, 2017. "Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 111-119.
    14. Amiri, Pari & Afshari, Hojjat, 2022. "Common fixed point results for multi-valued mappings in complex-valued double controlled metric spaces and their applications to the existence of solution of fractional integral inclusion systems," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    15. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    16. Abdalla, Bahaaeldin & Abdeljawad, Thabet, 2019. "On the oscillation of Caputo fractional differential equations with Mittag–Leffler nonsingular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 173-177.
    17. Yadav, Swati & Pandey, Rajesh K., 2020. "Numerical approximation of fractional burgers equation with Atangana–Baleanu derivative in Caputo sense," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    18. Shah, Kamal & Alqudah, Manar A. & Jarad, Fahd & Abdeljawad, Thabet, 2020. "Semi-analytical study of Pine Wilt Disease model with convex rate under Caputo–Febrizio fractional order derivative," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).
    19. Coronel-Escamilla, A. & Gómez-Aguilar, J.F. & López-López, M.G. & Alvarado-Martínez, V.M. & Guerrero-Ramírez, G.V., 2016. "Triple pendulum model involving fractional derivatives with different kernels," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 248-261.
    20. Yusuf, Abdullahi & Inc, Mustafa & Isa Aliyu, Aliyu & Baleanu, Dumitru, 2018. "Efficiency of the new fractional derivative with nonsingular Mittag-Leffler kernel to some nonlinear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 220-226.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:126:y:2019:i:c:p:7-11. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.