Mohammad Mutaz, Trounev Alexander, Cattani Carlo
Department of Mathematics & Statistics, Zayed University, Abu Dhabi 144543, UAE.
Department of Computer Technology and Systems, Kuban State Agrarian University, Krasnodar 350044, Russia.
Entropy (Basel). 2020 Jul 28;22(8):824. doi: 10.3390/e22080824.
This paper is devoted to shedding some light on the advantages of using tight frame systems for solving some types of fractional Volterra integral equations (FVIEs) involved by the Caputo fractional order derivative. A tight frame or simply framelet, is a generalization of an orthonormal basis. A lot of applications are modeled by non-negative functions; taking this into account in this paper, we consider framelet systems generated using some refinable non-negative functions, namely, B-splines. The FVIEs we considered were reduced to a set of linear system of equations and were solved numerically based on a collocation discretization technique. We present many important examples of FVIEs for which accurate and efficient numerical solutions have been accomplished and the numerical results converge very rapidly to the exact ones.
本文致力于揭示使用紧框架系统求解一些由卡普托分数阶导数所涉及的分数阶沃尔泰拉积分方程（FVIEs）的优势。紧框架或简称为框架小波，是正交基的一种推广。许多应用由非负函数建模；考虑到这一点，在本文中，我们考虑使用一些可细化的非负函数（即B样条）生成的框架小波系统。我们所考虑的FVIEs被简化为一组线性方程组，并基于配置离散化技术进行数值求解。我们给出了许多FVIEs的重要示例，对于这些示例，已经获得了精确且高效的数值解，并且数值结果非常迅速地收敛到精确解。
