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Jacobi convolution series for Petrov-Galerkin scheme and general fractional calculus of arbitrary order over finite interval

Pavan Pranjivan Mehta, Gianluigi Rozza

TL;DR

The paper extends general fractional calculus on finite intervals to arbitrary orders using the Luchko–Kochubei kernel class, and introduces Jacobi convolution series as basis functions. This enables a Petrov–Galerkin scheme with a diagonal stiffness matrix, where the general fractional derivative of the basis reduces to a shifted Jacobi polynomial, leading to spectral convergence for both polynomial and non-polynomial solutions. The results provide a unifying, efficient numerical framework for general fractional operators defined by arbitrary kernels, with potential extensions to other convolution-type operators. The approach integrates left/right Sonine conditions, explicit basis construction, and rigorous convergence analysis to support practical computations on finite domains.

Abstract

Recently, general fractional calculus was introduced by Kochubei (2011) and Luchko (2021) as a further generalisation of fractional calculus, where the derivative and integral operator admits arbitrary kernel. Such a formalism will have many applications in physics and engineering, since the kernel is no longer restricted. We first extend the work of Al-Refai and Luchko (2023) on finite interval to arbitrary orders. Followed by, developing an efficient Petrov-Galerkin scheme by introducing Jacobi convolution series as basis functions. A notable property of this basis function, the general fractional derivative of Jacobi convolution series is a shifted Jacobi polynomial. Thus, with a suitable test function it results in diagonal stiffness matrix, hence, the efficiency in implementation. Furthermore, our method is constructed for any arbitrary kernel including that of fractional operator, since, its a special case of general fractional operator.

Jacobi convolution series for Petrov-Galerkin scheme and general fractional calculus of arbitrary order over finite interval

TL;DR

The paper extends general fractional calculus on finite intervals to arbitrary orders using the Luchko–Kochubei kernel class, and introduces Jacobi convolution series as basis functions. This enables a Petrov–Galerkin scheme with a diagonal stiffness matrix, where the general fractional derivative of the basis reduces to a shifted Jacobi polynomial, leading to spectral convergence for both polynomial and non-polynomial solutions. The results provide a unifying, efficient numerical framework for general fractional operators defined by arbitrary kernels, with potential extensions to other convolution-type operators. The approach integrates left/right Sonine conditions, explicit basis construction, and rigorous convergence analysis to support practical computations on finite domains.

Abstract

Recently, general fractional calculus was introduced by Kochubei (2011) and Luchko (2021) as a further generalisation of fractional calculus, where the derivative and integral operator admits arbitrary kernel. Such a formalism will have many applications in physics and engineering, since the kernel is no longer restricted. We first extend the work of Al-Refai and Luchko (2023) on finite interval to arbitrary orders. Followed by, developing an efficient Petrov-Galerkin scheme by introducing Jacobi convolution series as basis functions. A notable property of this basis function, the general fractional derivative of Jacobi convolution series is a shifted Jacobi polynomial. Thus, with a suitable test function it results in diagonal stiffness matrix, hence, the efficiency in implementation. Furthermore, our method is constructed for any arbitrary kernel including that of fractional operator, since, its a special case of general fractional operator.

Paper Structure

This paper contains 7 sections, 11 theorems, 90 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. General Fractional Calculus
  3. Basis function
  4. Jacobi convolution series
  5. Petrov-Galerkin scheme for general fractional derivative
  6. Convergence analysis
  7. Summary

Key Result

Lemma 2.1

If ($k_n, \kappa_n$) are a Sonine kernel from $\mathbb{L}_n(a,b]$, $f \in C^n [a, b]$ and $x \in (a, b]$ then,

Figures (3)

  • Figure 1: Sonine Kernel obtained using (\ref{['eq:son_pair']}) for $\alpha = 0.5$ and $a = \{0.5, 0.25, 0.25\}$ results in $b = \{2, -1, -0.83333\}$ are singular functions with singularity at $x=0$
  • Figure 2: The first eight left Jacobi convolution series (with $\alpha =\beta =0$) obtained using for the Sonine pair using (\ref{['eq:son_pair']}) for $\alpha = 0.5$ and $a = \{0.5, 0.25, 0.25\}$ results, $b = \{2, -1, -0.83333\}$ and the shifted Legendre polynomials
  • Figure 3: The rate of convergence with respect to $N$ for our Petrov-Galerkin scheme for the functions (a) $f(x) = x^{15}$ and (b) $f(x) = x^{15.5}$. Our method converges spectrally for both polynomial and non-polynomial function.

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • ...and 11 more