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Optimal control problem of advection-diffusion-reaction equation of kind fractal-fractional applying shifted Jacobi polynomials

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  • Shojaeizadeh, T.
  • Mahmoudi, M.
  • Darehmiraki, M.

Abstract

Optimal control is always intended for optimization of different systems. In this study, new introduced differential operators called fractal-fractional derivatives have been used to investigate the behavior of one of the attractions of applied mathematics in physics and engineering. A novel version of the optimal control problem (OCP) generated using a dynamic system of type space-time fractal-fractional advection-diffusion-reaction equation is provided. An exact method based on the operational matrix derived from shifted Jacobi polynomials (SJPs) and collocation schemes is proposed. To define these new class of problems, the fractal-fractional derivative is implemented in to the Atangana-Riemann-Liouville sense whit Mittag-Leffler kernel applied for the time variable, and the fractional derivatives carried out for the space variable in the Caputo sense and the Atangana-Baleano-Caputo concept. In other words, the proposed method turns the principal problems solution into solving a system of algebraic equations. For this purpose, we substitute the approximate state and control values of the variables obtained by SJPs with unknown coefficients into the objective function, the dynamic system, and the initial and Dirichlet boundary conditions. Eventually, this problem becomes a quadratic optimization problem with linear constraint solved by the Lagrange multipliers method. In addition, the convergence analysis of SJPs expansion has been proved. To assess the effectiveness and precision of the process, some scenarios are illustrated.

Suggested Citation

  • Shojaeizadeh, T. & Mahmoudi, M. & Darehmiraki, M., 2021. "Optimal control problem of advection-diffusion-reaction equation of kind fractal-fractional applying shifted Jacobi polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    • Handle: RePEc:eee:chsofr:v:143:y:2021:i:c:s0960077920309590
    DOI: 10.1016/j.chaos.2020.110568
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    References listed on IDEAS

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    1. ARAZ, Seda İĞRET, 2020. "Numerical analysis of a new volterra integro-differential equation involving fractal-fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    2. 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.
    3. 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.
    4. Karaagac, Berat, 2019. "Two step Adams Bashforth method for time fractional Tricomi equation with non-local and non-singular Kernel," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 234-241.
    5. Bahaa, G.M., 2019. "Optimal control problem for variable-order fractional differential systems with time delay involving Atangana–Baleanu derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 129-142.
    6. Goswami, Amit & Singh, Jagdev & Kumar, Devendra & Sushila,, 2019. "An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 563-575.
    7. Chen, W., 2006. "Time–space fabric underlying anomalous diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 923-929.
    8. Bhatter, Sanjay & Mathur, Amit & Kumar, Devendra & Singh, Jagdev, 2020. "A new analysis of fractional Drinfeld–Sokolov–Wilson model with exponential memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
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    1. Saifullah, Sayed & Ali, Amir & Franc Doungmo Goufo, Emile, 2021. "Investigation of complex behaviour of fractal fractional chaotic attractor with mittag-leffler Kernel," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
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    4. Ravi Kanth, A.S.V. & Devi, Sangeeta, 2022. "A computational approach for numerical simulations of the fractal–fractional autoimmune disease model," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).

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