A pseudo−operational collocation method for optimal control problems of fractal−fractional nonlinear Ginzburg−Landau equation

Iranian Journal of Numerical Analysis and Optimization

مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 20 اردیبهشت 1403

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2024.86362.1375

نویسندگان

Tahereh Shojaeizadeh^{* 1}؛ Effat Golpar Raboky^{* 2}؛ P. Rahimkhani³

¹Department of Mathematics, Qom Branch, Islamic Azad University, Qom, Iran

²Department of Mathematics, University of Qom, Qom, Iran

³Faculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran

چکیده

The presented work introduces a new class of nonlinear optimal control problems in two dimensions whose constraints are nonlinear Ginzburg$-$Landau equations with fractal$-$fractional (FF) derivatives. To acquire their approximate solutions, a computational strategy is expressed using the FF derivative in the Atangana$-$Riemann$-$Liouville (A-R-L) concept with the Mittage-Leffler kernel. The mentioned scheme utilizes the shifted Jacobi polynomials (SJPs) and their operational matrices of fractional and FF derivatives. A method based on the derivative operational matrices of SJR and collocation scheme Is suggested and employed to reduce the problem into solving a system of algebraic equations. We approximate state and control functions of the variables derived from  SJPs with unknown coefficients into the objective function, the dynamic system, and the initial and Dirichlet boundary conditions. The effectiveness and efficiency of the suggested approach are investigated through the different types of test problems.

کلیدواژه‌ها

Fractal− fractional (FF) derivative؛ Shifted Jacobi polynomials (SJPs)؛ Operational matrices؛ Nonlinear Ginzburg− Landau equation؛ Optimal control problem