Fractional Quadrature rule and using its Exactness for the M\"untz-Legendre Scaling Functions for Solving Fractional Differential Equations

Abstract

Fractional operators (derivatives/integrals) are defined via the integration of the functions. When the function is produced by a spanning set of fractional power functions, traditional quadrature rules often need to be revised, failing to provide exact evaluations for fractional power functions and thus introducing approximation errors. In this paper, we have formulated a fractional quadrature rule that achieves exact integration for functions within this specific set to address this issue. Some properties of the fractional quadrature rule have been proved, and the absolute error bound in the proposed fractional quadrature rule has been derived. The behavior of roots of the orthogonal M\"untz polynomial has also been observed for its application as nodes in the fractional quadrature rule. To illustrate the effectiveness of the newly proposed fractional quadrature rule, we focus on fractional differential equations that incorporate the left Caputo fractional derivative. In this context, M\"untz-Legendre scaling functions are utilized to approximate the Caputo derivative of functions involved in these equations. Additionally, we have derived an operational matrix for Riemann-Liouville integration to approximate the respective functions with the help of the fractional quadrature rule. To demonstrate the practical utility of our method, we provide illustrative examples that compare the $L_2$-error estimates in the solutions of fractional differential equations using our approach against those obtained with the Block-pulse method. These comparisons underscore the superior accuracy of our proposed method.