Moduli of continuity for functions in Sobolev spaces and Haar wavelet solutions to fractional Basset equation

Abstract

In this paper, the method for solving fractional differential equations have been proposed using the Haar wavelet operational matrix of fractional integration. An operational matrix of fractional integration using the Haar wavelet is designed to solve a linear multi-term fractional differential equation as well as a system of fractional differential equations. The Basset equation for different fractional orders and a system of fractional differential equations both have been solved in order to validate and show the viability of the proposed method. Furthermore, it has also been demonstrated to approximate functions in Sobolev space via the Haar wavelet approach with the help of moduli of continuity. By treating fractional differential equations as a set of algebraic equations, this study significantly advances both the moduli of continuity and numerical solutions of fractional differential equations.