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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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    1. 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.
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    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. 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.
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    Cited by:

    1. 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).
    2. Turkyilmazoglu, Mustafa & Altanji, Mohamed, 2023. "Fractional models of falling object with linear and quadratic frictional forces considering Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).

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