Document Type : Applied Article
Authors
1 Department of Mathematics, College of Science, University of Qom, Qom, Iran. Department of Information and Communication Technology, Baghdad Institute of Technology, Middle Technical University, Baghdad, Iraq.
2 Department of Mathematics, College of Science, University of Qom, Qom, Iran.
3 Department of Mathematics, Behbahan Khatam Alanbia University of Technology, Khouzestan, Iran.
10.30473/coam.2025.73804.1289
Abstract
This study addresses the numerical solution of an optimal control problem governed by a fractional convection–reaction–diffusion partial differential equation. The approach utilizes Radial Basis Function–Partition of Unity (RBF-PU) methods combined with the Grünwald-Letnikov approximation for fractional derivatives, which provides a fundamental extension of classical derivatives in fractional calculus. To enhance sparsity in the control strategy, an $L_2$ norm is integrated into the objective function, along with quadratic penalties to reduce deviations from the desired state. This hybrid formulation facilitates the effective management of spatially sparse controllers, relevant in many practical applications. The RBF-PU technique offers a flexible and efficient framework by partitioning the domain into overlapping subregions, applying local RBF approximations, and synthesizing the global solution with compactly supported weight functions. Numerical experiments demonstrate the accuracy and effectiveness of this method.
Highlights
- Developed a computational framework for solving an optimal control problem governed by a fractional convection–reaction–diffusion PDE, capturing memory-dependent and nonlocal phenomena.
- Employed Radial Basis Function–Partition of Unity (RBF-PU) methodology to achieve high accuracy, mesh-free flexibility, and efficient spatial discretization in complex two-dimensional domains.
- Utilized the Grünwald-Letnikov approximation for effective discretization of fractional derivatives, with temporal derivatives handled via a direct time-stepping scheme using the Caputo definition to preserve causality and initial conditions.
- Incorporated an $L_2$ norm into the objective function to promote spatial sparsity in the control variables, combined with quadratic penalties to minimize deviations from target states.
- The RBF-PU technique partitions the domain into overlapping subregions, applies local RBF approximations, and synthesizes the global solution through compactly supported weight functions, enabling sparse and efficient computations suitable for real-world applications.
- Numerical experiments validate the accuracy, robustness, and efficiency of the proposed method in solving fractional PDEs, demonstrating its potential for complex systems with nonlocal dynamics.
Keywords
Main Subjects
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