An Improved Spectral-Element Method for the High-Dimensional Spatial Fourth-Order Differential Equations with Temporal Fractional Derivative

An efficient numerical algorithm is formulated in this work to resolve the high-dimensional spatial fourth-order differential equations with temporal fractional derivatives. The Caputo-type temporal fractional derivative is discretized utilizing a uniform-accuracy L2 scheme. For spatial discretization, the Legendre spectral element method is adopted to derive the fully discrete scheme. The fully discrete scheme achieves efficient implementation by combining order-reduction that converts the original fourth-order problem into a second-order system, and eigenvalue decomposition that diagonalizes the stiffness matrix. The stability of the presented numerical scheme is given by using the properties of the inverse Laplace operator. Rigorous error analysis is carried out to establish the convergence of the presented numerical scheme with a (3−θ) uniform temporal convergence order. Four numerical examples have verified the theoretical temporal and spatial convergence order. Numerical experiments demonstrate that the proposed scheme exhibits remarkable precision and computational performance. Furthermore, the simulation data show strong consistency with analytical solutions.