Error estimation of the finite difference and Galerkin spectral for the optimal control problem of time-fractional fourth-order partial differential equations

Guiying Jiang; Ziqiang Wang; Junying Cao; Guiying Jiang; Ziqiang Wang; Junying Cao

doi:10.3934/math.20261217

AIMS Mathematics

2026, Volume 11, Issue 9: 30714-30758. doi: 10.3934/math.20261217

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Research article

Error estimation of the finite difference and Galerkin spectral for the optimal control problem of time-fractional fourth-order partial differential equations

School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China

Received: 27 April 2026 Revised: 01 September 2026 Accepted: 15 September 2026 Published: 20 September 2026
MSC : 65L12, 65M06, 65M12

This paper proposes a numerical method with uniform temporal accuracy for solving an optimal control problem (OCP) governed by time-fractional fourth-order partial differential equations. The approach combines a high-order finite difference scheme in time and a Legendre-Galerkin spectral method in space to provide an efficient and accurate framework for the discretization and solution of the given OCP. The first-order optimal condition theorem is derived by using calculus variations. A fully discrete scheme for OCP is established by combining the first-order optimality conditions, spatial spectral discretization, and L2 numerical scheme of time-fractional derivatives, which achieves a uniform temporal convergence order $ (3-\theta) $ for time-fractional derivatives order of $ 0 < \theta < 1 $ by using block-by-block techniques to achieve high-precision time convergence order only for the first two time layers. The error estimation for the OCP is rigorously completed by using first-order optimal condition and three proposed auxiliary equations with uniform accuracy $ (3-\theta) $-order in time and spatial spectral accuracy. The stability analysis for the full discrete is also provided rigorously. The numerical results confirm the error estimation theory of the presented fully discrete scheme.
- optimal condition,
- time-fractional fourth-order partial differential equations,
- error estimation,
- spectral method
Citation: Guiying Jiang, Ziqiang Wang, Junying Cao. Error estimation of the finite difference and Galerkin spectral for the optimal control problem of time-fractional fourth-order partial differential equations[J]. AIMS Mathematics, 2026, 11(9): 30714-30758. doi: 10.3934/math.20261217

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Abstract

This paper proposes a numerical method with uniform temporal accuracy for solving an optimal control problem (OCP) governed by time-fractional fourth-order partial differential equations. The approach combines a high-order finite difference scheme in time and a Legendre-Galerkin spectral method in space to provide an efficient and accurate framework for the discretization and solution of the given OCP. The first-order optimal condition theorem is derived by using calculus variations. A fully discrete scheme for OCP is established by combining the first-order optimality conditions, spatial spectral discretization, and L2 numerical scheme of time-fractional derivatives, which achieves a uniform temporal convergence order $ (3-\theta) $ for time-fractional derivatives order of $ 0 < \theta < 1 $ by using block-by-block techniques to achieve high-precision time convergence order only for the first two time layers. The error estimation for the OCP is rigorously completed by using first-order optimal condition and three proposed auxiliary equations with uniform accuracy $ (3-\theta) $-order in time and spatial spectral accuracy. The stability analysis for the full discrete is also provided rigorously. The numerical results confirm the error estimation theory of the presented fully discrete scheme.

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