In this paper, the time fractional diffusion equations optimal control problem is solved by $3-\alpha$ order with uniform accuracy scheme in time and finite element method (FEM) in space. For the state and adjoint state equation, the piecewise linear polynomials are used to make the space variables discrete, and obtain the semidiscrete scheme of the state and adjoint state. The priori error estimates for the semidiscrete scheme for state and adjoint state equation are established. Furthermore, the $3-\alpha$ order uniform accuracy scheme is used to make the time variable discrete in the semidiscrete scheme and construct the full discrete scheme for the control problems based on the first optimal condition and ‘first optimize, then discretize’ approach. The fully discrete scheme’s stability and truncation error are analyzed. Finally, two numerical examples are denoted to show that the theoretical analysis are correct.

Based on the spatial compact finite difference (SCFD) method, an improved high-order temporal accuracy scheme for high-dimensional time-fractional diffusion equations (TFDEs) is presented in this work. Combining the temporal piecewise quadratic interpolation and the high-dimensional SCFD method, the proposed numerical method is described. In order to establish the stability and convergence analysis, we introduce a norm $\left|\right|·{\left|\right|}_{{\tilde{H}}^1}$, which is rigorously proved equivalent to the standard $H^1$-norm. Considering that the coefficients of high-order numerical schemes are not entirely positive, we introduce an appropriate parameter to transform the numerical scheme into an equivalent form with positive coefficients. Based on the equivalent form, we prove that the temporal and spatial convergence orders are $(3-\gamma )$ and 4 by applying the convergence of geometric progression. The proposed scheme ensures that the theoretical convergence accuracy at each time step is of order $(3-\gamma )$ without requiring any additional processing techniques. Ultimately, the convergence of the proposed high-order accurate scheme is verified through numerical experiments involving (non-)linear high-dimensional TFDEs.

In this paper, Multiquadric quasi-interpolation method is used to approximate fractional integral equations and fractional differential equations. Firstly, we construct two operators for approximating the Hadamard integral-differential equation based on quasi interpolators, and verify their properties and order of convergence. Secondly, we obtain that the approximation order of the numerical integral scheme is 3, and the approximation order of the numerical scheme is 3-μ for μ(0 < μ < 1) order fractional Hadamard derivative. Finally, the results of numerical experiments show that the numerical results are in agreement with the theoretical analysis.