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Fractional Quadrature rule and using its Exactness for the Müntz-Legendre Scaling Functions for Solving Fractional Differential Equations

Ritu Kumari, Mani Mehra, Abhishek Kumar Singh

TL;DR

The paper addresses solving left Caputo fractional differential equations by introducing a fractional quadrature rule that is exact for Müntz polynomials and by leveraging Müntz-Legendre scaling functions to build an accurate collocation method. It develops an RL fractional integration operational matrix via a transformation from piecewise fractional powers, enabling an efficient representation $$_0I_t^{\alpha}\Phi(t)\approx P(t,\alpha)\Phi(t)$$ and solves the resulting algebraic system for coefficients. Key contributions include proofs of exactness and error bounds for the new quadrature, construction of the transformation matrix, and comprehensive numerical validation showing superior $L^2$ accuracy over Block-Pulse methods, including nonlinear problems. This framework provides a high-accuracy, scalable approach for fractional models with non-integer orders and fractional power function bases, with potential extensions to optimal control and other applied settings.

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 Müntz 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, Müntz-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.

Fractional Quadrature rule and using its Exactness for the Müntz-Legendre Scaling Functions for Solving Fractional Differential Equations

TL;DR

The paper addresses solving left Caputo fractional differential equations by introducing a fractional quadrature rule that is exact for Müntz polynomials and by leveraging Müntz-Legendre scaling functions to build an accurate collocation method. It develops an RL fractional integration operational matrix via a transformation from piecewise fractional powers, enabling an efficient representation and solves the resulting algebraic system for coefficients. Key contributions include proofs of exactness and error bounds for the new quadrature, construction of the transformation matrix, and comprehensive numerical validation showing superior accuracy over Block-Pulse methods, including nonlinear problems. This framework provides a high-accuracy, scalable approach for fractional models with non-integer orders and fractional power function bases, with potential extensions to optimal control and other applied settings.

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 Müntz 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, Müntz-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 -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.

Paper Structure

This paper contains 14 sections, 7 theorems, 102 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Motivation and Contribution of the Paper
  3. Preliminaries
  4. Müntz-Legendre Wavelet
  5. Function Approximation
  6. Fractional Quadrature Rule
  7. Properties of Fractional Quadrature Rule
  8. Error Analysis
  9. Operational Matrix
  10. Fractional Integration of Piecewise Fractional Power Functions
  11. Transformation Matrix
  12. Müntz-Legendre Scaling Functions Operational Matrix
  13. Description of the Numerical Method and Test Examples
  14. Conclusion

Key Result

Lemma 3.1

Let $w>0$ be a given weight in $L^1(a,b)$. Then for any $n \geq 1$, we can find a unique sequence of orthogonal monic Müntz polynomials $\{p_n^\lambda\}$ using the recurrence relation where

Figures (3)

  • Figure 1: Distribution of roots of orthogonal Müntz polynomials of degree $5$ and $7$ for different values of $\lambda.$
  • Figure 2: $L_2$ norm of distance of the roots of orthogonal Müntz polynomial of degree $5$ from the zero of the weight function $(2-t)^p$.
  • Figure 3: $L_2$ error in the approximation of the function $p_\lambda(t)=e^{-2x}sin(3t^{\lambda})$ with orthogonal Müntz polynomial of degree $N+1$, using the fractional quadrature rule (\ref{['4.5']}) for different values of $\lambda$.

Theorems & Definitions (23)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Example 3.1
  • Theorem 3.3
  • ...and 13 more