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arXiv · 2411.08080

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

Pavan Pranjivan Mehta·Gianluigi Rozza
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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.

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Pavan Pranjivan Mehta, Gianluigi Rozza. 2024-11-12. Jacobi convolution series for Petrov-Galerkin scheme and general fractional calculus of arbitrary order over finite interval. https://arxiv.org/abs/2411.08080

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