Fractional calculus is a generalization of classical (integer-order) derivatives, where the order can be arbitrary. Paradoxically, these are nonlocal operators, addressing anomalous diffusion processes. Although fractional derivatives have found applications in real world phenomena, such as turbulence [1], it begs the question for physics and engineers, can a power-law kernel describe for all physical processes? Indeed, the answer is, No; and one needs to seek further generalisation. One such mathematical foundation is laid down by Kochubei [2] and Luchko [3] for arbitrary kernels following Sonine. In our recent work [4], we extended the theory for general fractional operators to finite intervals for arbitrary orders. Furthermore, we introduced a novel basis function, namely, the Jacobi convolution polynomial. The notable property of such a basis function, the general fractional derivative of this basis function, is a shifted Jacobi polynomial. Thus, allowing the construction of an efficient Petrov-Galerkin scheme where the stiffness matrix is diagonal and our error analysis shows convergence rate is spectral. In this talk, we will introduce the operators, followed by our results for constructing efficient spectral methods for these new types of operators, which will potentially have many applications in physics and engineering. Needless to mention, since fractional operators are a special case of general fractional operators, our methods are valid for fractional cases too. References: [1] Pavan Pranjivan Mehta. Fractional and tempered fractional models for reynolds averaged navier–stokes equations. Journal of Turbulence, 24(11-12):507–553, 2023 [2] Anatoly N Kochubei. General fractional calculus, evolution equations, and renewal processes. Integral Equations and Operator Theory, 71(4):583–600, 2011. [3] Yuri Luchko. General fractional integrals and derivatives of arbitrary order. Symmetry, 13(5):755, 2021. [4] Pavan Pranjivan Mehta and Gianluigi Rozza. Jacobi convolution polynomial for petrov-galerkin scheme and general fractional calculus of arbitrary order over finite interval. arXiv:2411.08080, 2024.