A Meshless Method for Optimal Control of Parabolic PDEs Using Rational Radial Basis Functions

Saud Al-tameemi, Afrah Kadhim; Mahmoudi, Mahmoud; Darehmiraki, Majid

doi:10.30473/coam.2025.73126.1277

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	A Meshless Method for Optimal Control of Parabolic PDEs Using Rational Radial Basis Functions
Control and Optimization in Applied Mathematics
مقاله 9، دوره 10، شماره 1 - شماره پیاپی 19، شهریور 2025، صفحه 139-161 اصل مقاله (2.25 M)
نوع مقاله: Research Article
شناسه دیجیتال (DOI): 10.30473/coam.2025.73126.1277
نویسندگان
Afrah Kadhim Saud Al-tameemi¹؛ Mahmoud Mahmoudi^* ¹؛ Majid Darehmiraki²
¹Department of Applied Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Qom‎, ‎Qom‎, ‎Iran.
²Department of Mathematics‎, ‎Behbahan Khatam Alanbia University of Technology‎, ‎Khouzestan‎, ‎Iran.
چکیده
This study introduces an innovative approach for addressing optimal control problems related to parabolic partial differential equations (PDEs) through the application of rational radial basis functions (RBFs). Parabolic PDEs, which are instrumental in modeling time-dependent processes such as heat transfer and diffusion, pose significant computational challenges in optimal control due to the requirement for precise approximations of both state and adjoint equations. The proposed approach exploits the adaptability and spectral accuracy of rational RBFs within a meshless framework, effectively addressing the limitations of traditional discretization methods. By enhancing the accuracy and efficiency of control strategies, this method significantly contributes to advancing the theory and application of optimal control in dynamic systems. The tunable shape parameters of rational RBFs allow for accurate representation of solution characteristics, including steep gradients and localized behaviors. Additionally, their meshless framework adeptly accommodates complex geometries and boundary conditions, ensuring computational efficiency through the generation of sparse and well-conditioned system matrices. This paper also introduces a novel hybrid rational RBF, termed the Gaussian rational hybrid RBF. The efficacy of the proposed approach is validated through a series of benchmark tests and practical applications, highlighting its ability to achieve high accuracy with reduced computational effort. The findings illustrate the potential of rational RBFs as a robust and versatile tool for solving optimal control problems governed by parabolic PDEs, paving the way for further exploration of advanced rational RBF-based techniques in the field of computational optimal control.
تازه های تحقیق
Introduced a novel hybrid rational Radial Basis Function (Gaussian rational hybrid RBF) specifically designed to address the challenges of optimal control problems governed by parabolic PDEs, leveraging the spectral accuracy and adaptability of rational RBFs. Developed a fully meshless computational framework that exploits the spectral accuracy of rational RBFs, effectively capturing steep gradients, localized behaviors, and complex solution features inherent in dynamic parabolic systems. Demonstrated the method’s high accuracy and computational efficiency through extensive benchmark tests and practical applications, highlighting its ability to produce solutions with small ( L₂ ) and ( L_infty ) errors while reducing computational effort compared to traditional discretization techniques. Showcased the flexibility of the hybrid RBF approach in handling intricate geometries and boundary conditions within a sparse, well-conditioned system matrix, emphasizing its robustness for complex optimal control scenarios. Highlighted promising directions for future research by exploring advanced RBF-based techniques, including adaptive methods and extensions to nonlinear and multi-physics PDEs, to further enhance the scalability, precision, and applicability of meshless spectral approaches in computational optimal control.
کلیدواژه‌ها
Meshless methods‎؛ ‎Parabolic PDEs‎؛ ‎Rational RBFs‎؛ ‎Optimization‎؛ ‎Gaussian rational hybrid RBF

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A Meshless Method for Optimal Control of Parabolic PDEs Using Rational Radial Basis Functions