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Predictor–corrector for non-linear differential and integral equation with fractal–fractional operators

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Abstract

Fractal–fractional differential and integral operators have been recognized recently as superior operators as they are able to depict physical problem with both memory effect and self-similar properties. Therefore, differential and integral equations constructed from these new operators are of great importance. In this paper, we extend the method of predictor–corrector to obtain numerical solution of non-linear differential and integral equations. Some examples are presented to illustrate the efficiency of the new method for solving these new equations.

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References

  1. Hammouch Z, Mekkaoui T (2014) Chaos synchronization of a fractional nonautonomous system. Nonauton Dyn Syst 1:61–71

    MathSciNet  MATH  Google Scholar 

  2. Atangana A, Baleanu D (2016) New fractional derivative with non-local and non-singular kernel. Therm Sci 20:757–763

    Article  Google Scholar 

  3. Qureshi S, Yusuf A (2019) Mathematical modeling for the impacts of deforestation on wildlife species using Caputo differential operator. Chaos Solitons Fractals 126:32–40

    Article  MathSciNet  Google Scholar 

  4. Caputo M, Fabrizio M (2016) Applications of new time and spatial fractional derivatives with exponential kernels. Progr Fract Differ Appl 2:1–11

    Article  Google Scholar 

  5. Gomez-Aguilar JF, Atangana A (2017) New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications. Eur Phys J Plus 132:13

    Article  Google Scholar 

  6. Owolabi KM, Pindza E (2019) Modeling and simulation of non-linear dynamical system in the frame of non-local and non-singular derivatives. Chaos Solitons Fractals 127:146–157

    Article  MathSciNet  Google Scholar 

  7. Atangana A, Igret Araz S (2019) Analysis of a new partial integro-differential equation with mixed fractional operators. Chaos Solitons Fractals 127:257–271

    Article  MathSciNet  Google Scholar 

  8. Toufik M, Atangana A (2017) New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models. Eur Phys J Plus 132(10):444

    Article  Google Scholar 

  9. Owolabi KM, Atangana A (2019) Computational study of multi-species fractional reaction-diffusion system with ABC operator. Chaos Solitons Fractals 128:280–289

    Article  MathSciNet  Google Scholar 

  10. Igret Araz S (2020) Numerical analysis of a new Volterra integro-differential equation involving fractal–fractional operators. Chaos Solitons Fractals 130:109396

    Article  MathSciNet  Google Scholar 

  11. Atangana A, Khan MA (2019) Validity of fractal derivative to capturing chaotic attractors. Chaos Solitons Fractals 126:50–59

    Article  MathSciNet  Google Scholar 

  12. Owolabi KM (2019) Mathematical modelling and analysis of love dynamics: a fractional approach. Phys A 525:849–865

    Article  MathSciNet  Google Scholar 

  13. Atangana A, Igret Araz S (2019) New numerical method for ordinary differential equations: Newton polynomial. J Comput Appl Math

  14. Owolabi KM (2019) Behavioural study of symbiosis dynamics via the Caputo and Atangana–Baleanu fractional derivatives. Chaos Solitons Fractals 122:89–101

    Article  MathSciNet  Google Scholar 

  15. Ul Haq R, Ahmed Soomro FA, Mekkaoui T, Al-Mdallal Q (2018) MHD natural convection flow enclosure in a corrugated cavity filled with a porous medium. Int J Heat Mass Transf

  16. Abro KA, Mirbhar MN, Gómez-Aguilar JF (2019) Functional application of Fourier sine transform in radiating gas flow with non-singular and non-local kernel. J Braz Soc Mech Sci Eng 41:400

    Article  Google Scholar 

  17. Gómez-Aguilar JF, Abro KA, Kolebaje O, Yıldırım A (2019) Chaos in a calcium oscillation model via Atangana–Baleanu operator with strong memory. Eur Phys J Plus 134(4):140

    Article  Google Scholar 

  18. Abro KA, Abro IA, Yıldırım A (2019) A comparative analysis of sulfate \(\left( SO_{4}^{-2}\right)\) ion concentration via modern fractional derivatives: an industrial application to cooling system of power plant. Phys A 2019:123306

    Google Scholar 

  19. Atangana A (2017) Fractal–fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fractals 102:396–406

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, FSTE Moulay Ismail University, Meknes, Morocco

    Toufik Mekkaoui

  2. Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, South Africa

    Abdon Atangana

  3. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

    Abdon Atangana

  4. Department of Mathematics Education, Siirt University, Siirt, Turkey

    Seda İğret Araz

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  1. Toufik Mekkaoui
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  2. Abdon Atangana
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  3. Seda İğret Araz
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Correspondence to Seda İğret Araz.

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Mekkaoui, T., Atangana, A. & Araz, S.İ. Predictor–corrector for non-linear differential and integral equation with fractal–fractional operators. Engineering with Computers 37, 2359–2368 (2021). https://doi.org/10.1007/s00366-020-00948-6

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  • DOI: https://doi.org/10.1007/s00366-020-00948-6

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