| Optimal control for a variable-order diffusion-wave equation with a reaction term; A numerical study |
| Paper ID : 1030-ISCHU |
| Authors |
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Salma Asaad Mohammed Shatta *1, Nasser Hassan sweilam2, Fouad Megahed3 1T.A at Mathematics department, faculty of Science, Helwan university, Cairo, Egypt. 2Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt. 3Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt. |
| Abstract |
| Recently, Baleanu et al., in [1] introduced a new type of derivative known as a hybrid fractional operator, which can be expressed as a linear combination of the Caputo fractional derivative and the Riemann-Liouville fractional integral. On the other hand, the theory of fractional optimal control PDEs has been widely applied in various areas such as science, engineering, and economics. In [2] and [3], Agrawal suggested a generalized formulation and approach for solving the fractional scheme of optimal control problems using the Lagrange multiplier technique and the fractional variation principle. Several studies have analyzed many the optimal control for integer-order PDEs problems, but there have been few studies on the fractional order optimal control systems of PDEs and the variational calculus, [4-9]. In this talk, a novel variable-order diffusion wave with a reaction term given in [9] by applying the new hybrid fractional operator derivative. This operator has two cases, there are the (PC) stands for Proportional-Caputo and the (CPC) stands for Constant-Proportional-Caputo. Moreover, a control variable to minimize the objective cost will be introduced. To approximate the obtained fractional optimality system, two numerical approaches will be constructed. These methods are: the nonstandard leap frog method (NLFM) and the nonstandard weighted average finite difference method (NWAFDM). The stability of the proposed methods will be proved. Numerical simulations show that the NWAFDM can be applied to solve such the optimal control for VOD-Wave with reaction term (OCVOD-wave) and the cost functional solution is smallest than the other method. References [1] Baleanu D., Fernandez A., Akgül A. Math, 8(3):360.2020. [2] Agrawal OP., J Math Anal Appl, 272(1):368-379. 2002. [3] Agrawal OP., Nonlinear Dyn. 38:323-337. 2004. [4] Araz I., Suba M., Int J Differ Equ, (2018):7417590. 2018. [5] Bahaa G M., Filomat, 30 (8):2177-2189. 2016. [6] Mophou G., Comput Math Appl, 61(1):68-78. 2011. [7] Mophou G., Comput Math Appl, 62(3):1413-1426. 2011. [8] Mophou G., Joseph C., Dyn Contin Disc Impul Syst Ser A, 23:341-364. 2016. [9] Yaseen M., Abbas M., Nazir T., Baleanu D., Adv Differ Equ, 2017:274. 2017. |
| Keywords |
| Lagrange method for the optimal control of partial differential equations; Optimality system; Diffusion-wave equation with a reaction term; Variable-order fractional derivatives; Constant-proportional-Caputo hybrid fractional operator; Nonstandard leap-frog method; Nonstandard weighted average finite difference method. |
| Status: Abstract Accepted (Poster Presentation) |