Narrow Results

Framelets and their attractive features in many disciplines have attracted a great interest in the recent years. This paper intends to show the advantages of using bi-framelet systems in the context of numerical fractional differential equations (FDEs). We present a computational method based on the quasi-affine bi-framelets with high vanishing moments constructed using the generalized (mixed) oblique extension principle. We use this system for solving some types of FDEs by solving a series of important examples of FDEs related to many mathematical applications. The quasi-affine bi-framelet-based methods for numerical FDEs show the advantages of using sparse matrices and its accuracy in numerical analysis.

In wavelet analysis, refinable functions are the bases of extension principles for constructing (weak) dual wavelet frames for $L^2({ℝ}_{+})$ and its reducing subspaces. This paper addresses refinable function-based dual wavelet frames construction in Walsh reducing subspaces of $L^2({ℝ}_{+})$. We obtain a Walsh–Fourier transform domain characterization for weak $p$-adic nonhomogeneous dual wavelet frames; and present a mixed oblique extension principle for constructing weak $p$-adic nonhomogeneous dual wavelet frames in Walsh reducing subspaces of $L^2({ℝ}_{+})$.