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

TL;DR

This paper introduces a novel Petrov-Galerkin scheme using Jacobi convolution series for general fractional calculus of arbitrary order, enabling efficient computation with diagonal matrices and broad kernel applicability.

Contribution

It develops a new basis function for general fractional calculus, leading to an efficient, flexible Petrov-Galerkin method applicable to any kernel, including fractional derivatives.

Findings

Diagonal stiffness matrix achieved with Jacobi convolution basis

Method applicable to arbitrary kernels, including fractional ones

Enhanced computational efficiency in fractional calculus problems

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.