An Efficient Method Based on Framelets for Solving Fractional Volterra Integral Equations

Mutaz Mohammad; Alexander Trounev; Carlo Cattani

doi:10.3390/e22080824

An Efficient Method Based on Framelets for Solving Fractional Volterra Integral Equations

Entropy (Basel). 2020 Jul 28;22(8):824. doi: 10.3390/e22080824.

Authors

Mutaz Mohammad¹, Alexander Trounev², Carlo Cattani³

Affiliations

¹ Department of Mathematics & Statistics, Zayed University, Abu Dhabi 144543, UAE.
² Department of Computer Technology and Systems, Kuban State Agrarian University, Krasnodar 350044, Russia.
³ Engineering School (DEIM), Tuscia University, 01100 Viterbo, Italy.

Abstract

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.

Keywords: fractional calculus; framelets; generalization of Unequal Error Protection (UEP); harmonic numerical analysis; numerical solution; volterra integral equations; wavelets.