A Hybrid Orthogonal Polynomial Approach for Optimal Control of Fractional Parabolic PDEs: Combining Legendre, Chebyshev, and Jacobi Polynomials

Al-Hakeem, Muhammed Hassanein; Mahmoudi, Mahmoud; Ahmed Al-Jilawi, Ahmed Sabah

doi:10.30473/coam.2026.76804.1378

Articles in Press

Document Type : Research Article

Authors

¹ Department of Mathematics, College of Science‎, ‎University of Qom‎, ‎Qom‎, ‎Iran

² Department of Mathematics, Faculty of Education, College For Pure Sciences Babylon, University Babylon, Iraq

10.30473/coam.2026.76804.1378

Abstract

This paper presents a novel hybrid orthogonal polynomial method for solving optimal control problems governed by fractional parabolic PDEs. By strategically weighting and combining these polynomial bases, the method adaptively leverages their respective strengths to achieve superior approximation properties. The proposed approach combines the spectral accuracy of Legendre polynomials, the minimax properties of Chebyshev polynomials, and the flexibility of Jacobi polynomials to create a robust numerical framework. The hybrid orthogonal polynomial method is applied to discretize the fractional parabolic PDEs, and an efficient numerical scheme is developed to solve the resulting optimal control problem. Numerical experiments demonstrate the accuracy, efficiency, and applicability of the proposed approach, showing significant improvements over traditional radial basis function methods. The results highlight the potential of the hybrid orthogonal polynomial method for solving complex optimal control problems in science and engineering.

Highlights

Fractional parabolic optimal control problems pose substantial computational challenges due to the intrinsic nonlocal memory effects of fractional derivatives and the tight coupling between state and control variables.
A novel hybrid orthogonal polynomial framework is proposed, systematically integrating Legendre, Chebyshev, and Jacobi polynomials to simultaneously exploit spectral accuracy, minimax optimality, and parametric flexibility.
The hybridization strategy yields a stable and highly accurate spectral scheme, capable of adaptively capturing complex spatial–temporal solution behaviors while maintaining well-conditioned algebraic systems.
The resulting numerical method demonstrates clear superiority over classical radial basis function approaches, delivering lower error norms, faster convergence, and enhanced computational efficiency for fractional optimal control problems in science and engineering.

Keywords

Main Subjects

Optimal Control

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A Hybrid Orthogonal Polynomial Approach for Optimal Control of Fractional Parabolic PDEs: Combining Legendre, Chebyshev, and Jacobi Polynomials

References

References

Articles in Press, Accepted Manuscript Available Online from 09 February 2026

Articles in Press, Accepted Manuscript
Available Online from 09 February 2026