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An efficient operation matrix method for solving fractal–fractional differential equations with generalized Caputo-type fractional–fractal derivative

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  • Shloof, A.M.
  • Senu, N.
  • Ahmadian, A.
  • Salahshour, Soheil

Abstract

In this study, we present the new generalized derivative and integral operators which are based on the newly constructed new generalized Caputo fractal–fractional derivatives (NGCFFDs). Based on these operators, a numerical method is developed to solve the fractal–fractional differential equations (FFDEs). We approximate the solution of the FFDEs as basis vectors of shifted Legendre polynomials (SLPs). We also extend the derivative operational matrix of SLPs to the generalized derivative operational matrix in the sense of NGCFFDs. The efficiency of the developed numerical method is tested by taking various test examples. We also compare the results of our proposed method with the methods existed in the literature In this paper, we specified the fractal–fractional differential operator of new generalized Caputo in three categories: (i) different values in ρ and fractal parameters, (ii) different values in fractional parameter while fractal and ρ parameters are fixed, and (iii) different values in fractal parameter controlling fractional and ρ parameters.

Suggested Citation

  • Shloof, A.M. & Senu, N. & Ahmadian, A. & Salahshour, Soheil, 2021. "An efficient operation matrix method for solving fractal–fractional differential equations with generalized Caputo-type fractional–fractal derivative," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 415-435.
    • Handle: RePEc:eee:matcom:v:188:y:2021:i:c:p:415-435
    DOI: 10.1016/j.matcom.2021.04.019
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    References listed on IDEAS

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    1. Constantin Bota & Bogdan Căruntu, 2015. "Approximate Analytical Solutions of the Fractional-Order Brusselator System Using the Polynomial Least Squares Method," Advances in Mathematical Physics, Hindawi, vol. 2015, pages 1-5, March.
    2. Erturk, Vedat Suat & Kumar, Pushpendra, 2020. "Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    3. Atangana, Abdon & Qureshi, Sania, 2019. "Modeling attractors of chaotic dynamical systems with fractal–fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 320-337.
    4. Atangana, Abdon, 2017. "Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 396-406.
    5. Odibat, Zaid & Momani, Shaher, 2008. "Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 167-174.
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    Cited by:

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    2. Rahimkhani, Parisa & Heydari, Mohammad Hossein, 2023. "Fractional shifted Morgan–Voyce neural networks for solving fractal-fractional pantograph differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 175(P2).
    3. S M, Sivalingam & Kumar, Pushpendra & Govindaraj, V., 2023. "A novel numerical scheme for fractional differential equations using extreme learning machine," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 622(C).
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    5. Ma, Zhiying & Hou, Jie & Zhu, Wenhao & Peng, Yaxin & Li, Ying, 2023. "PMNN: Physical model-driven neural network for solving time-fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).

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