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Efficient numerical approach for solving fractional partial differential equations with non-singular kernel derivatives

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  • Koca, Ilknur

Abstract

Adams-Bashforth was recognized as powerful numerical method to solve linear and non-linear ordinary differential equations. Nevertheless the method was applicable only for ordinary differential equations mostly with integer order. Atangana and Batogna have extended this method for partial differential equation with the Atangana-Baleanu fractional derivative. In this paper, to accommodate partial differential equation with Caputo-Fabrizio derivative, we suggest the corresponding method with this derivative. We applied the method to solve numerically a very interesting non-linear partial differential equation accounting for the motion of a viscous fluid. Some simulations are presented to test the efficiency of the numerical method.

Suggested Citation

  • Koca, Ilknur, 2018. "Efficient numerical approach for solving fractional partial differential equations with non-singular kernel derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 278-286.
    • Handle: RePEc:eee:chsofr:v:116:y:2018:i:c:p:278-286
    DOI: 10.1016/j.chaos.2018.09.038
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    References listed on IDEAS

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    1. Atangana, Abdon & Koca, Ilknur, 2016. "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 447-454.
    2. 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.
    3. Atangana, Abdon & Gómez-Aguilar, J.F., 2018. "Fractional derivatives with no-index law property: Application to chaos and statistics," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 516-535.
    4. 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.
    5. Norberg, Ragnar, 1995. "Differential equations for moments of present values in life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 17(2), pages 171-180, October.
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    1. 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).
    2. Maike A. F. dos Santos, 2019. "Mittag–Leffler Memory Kernel in Lévy Flights," Mathematics, MDPI, vol. 7(9), pages 1-13, August.

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