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 polynomials as basis functions. A notable property of this basis function, the general fractional derivative of Jacobi convolution polynomial 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.

翻译：近年来，Kochubei (2011) 与 Luchko (2021) 引入了广义分数阶微积分作为分数阶微积分的进一步推广，其导数与积分算子允许采用任意核函数。由于核函数不再受限，该形式体系将在物理与工程领域具有广泛的应用。我们首先将 Al-Refai 与 Luchko (2023) 在有限区间上的工作推广至任意阶。随后，通过引入雅可比卷积多项式作为基函数，发展了一种高效的 Petrov-Galerkin 格式。该基函数的一个显著性质是：雅可比卷积多项式的广义分数阶导数是一个平移后的雅可比多项式。因此，配合适当的测试函数可得到对角刚度矩阵，从而保证了算法实现的高效性。此外，由于分数阶算子是广义分数阶算子的特例，我们的方法适用于包括分数阶算子核在内的任意核函数。