An efficient spectral method is proposed for a new Steklov eigenvalue problem in inverse scattering. Firstly, we establish the weak form and the associated discrete scheme by introducing an appropriate Sobolev space and a corresponding approximation space. Then, according to the Fredholm Alternative, the corresponding operator forms of weak formulation and discrete formulation are derived. After that, the error estimates of approximated eigenvalues and eigenfunctions are proved by using the spectral approximation results of completely continuous operators and the approximation properties of orthogonal projection operators. We also construct an appropriate set of basis functions in the approximation space and derive the matrix form of the discrete scheme based on the tensor product. In addition, we extend the algorithm to the circular domain. Finally, we present plenty of numerical experiments and compare them with some existing numerical methods, which validate that our algorithm is effective and high accuracy.

The 1D and 2D spatial compact finite difference schemes (CFDSs) for time-fractional diffusion equations (TFDEs) were presented in this article with uniform temporal convergence order. Based on the idea of the modified block-by-block method, the CFDSs with uniform temporal convergence order for TFDEs were given by combining the fourth-order CFDSs in space and the high order scheme in time. The stability analysis and convergence order of CFDSs with uniform convergence order in time for TFDEs strictly proved that the provided uniform accuracy time scheme is $(3-\alpha )$ temporal order and spatial fourth-order, respectively. Ultimately, the astringency of 1D and 2D spatial CFDSs was verified by some numerical examples.