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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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    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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