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

TL;DR

This paper introduces a fractional quadrature rule that exactly integrates fractional power functions, improving the numerical solution of fractional differential equations using M"untz-Legendre functions and demonstrating superior accuracy over existing methods.

Contribution

The paper formulates a new fractional quadrature rule with exactness for fractional power functions, along with properties, error bounds, and application to solving fractional differential equations.

Findings

The fractional quadrature rule achieves exact integration for fractional power functions.

The method provides higher accuracy in solving fractional differential equations.

Comparative results show improved $L_2$-error estimates over existing approaches.

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.