In this article, we consider the identification of an unknown steady source in a class of fractional diffusion equations. A modified Tikhonov regularization method based on Hermite expansion is presented to deal with the ill-posedness of the problem. By using the properties of Hermitian functions, we construct a modified penalty term for the Tikhonov functional. It can be proved that the method can adaptively achieve the order optimal results when we choose the regularization parameter by the discrepancy principle. Some examples are also provided to verify the effectiveness of the method.

In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

In this paper, we further extend the Filon-type method to the Bessel function expansion for calculating Fourier integral. By means of complex analysis, this expansion is effective for all the oscillation frequencies. Namely, the errors of the expansion not only decrease as the order of the derivative increases, but also decrease rapidly as the frequency increases. Some numerical experiments are also presented to verify the effectiveness of the method.

This paper proposes a hierarchical Fourier extension framework for the accurate reconstruction of piecewise smooth functions with mixed-order singularities. To address key challenges in spectral approximation–namely boundary-induced artifacts, instability in edge detection, and loss of accuracy near discontinuities–the method integrates three main components: (1) boundary-focused Fourier extensions that isolate endpoint effects while preserving internal structures; (2) a multi-stage edge detection strategy combining spectral mollifiers and coordinate transformations to identify discontinuities in function values and their derivatives; (3) adaptive domain partitioning followed by localized Fourier extensions to retain spectral accuracy on smooth segments. Numerical results demonstrate near machine-precision accuracy (∼10⁻¹⁴–10⁻¹⁵) with significantly improved stability and performance over traditional global methods.