Fractional calculus with respect to function $\psi$, also named as $\psi$-fractional calculus, generalizes the Hadamard and the Riemann-Liouville fractional calculi, which causes challenge in numerical treatment. In this paper we study spectral-type methods using mapped Jacobi functions (MJFs) as basis functions and obtain efficient algorithms to solve $\psi$-fractional differential equations. In particular, we setup the Petrov-Galerkin spectral method and spectral collocation method for initial and boundary value problems involving $\psi$-fractional derivatives. We develop basic approximation theory for the MJFs and conduct the error estimates of the derived methods. We also establish a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. Numerical examples confirm the theoretical results and demonstrate the effectiveness of the spectral and collocation methods.

翻译：以函数$\psi$为基础的分数阶微积分（亦称$\psi$-分数阶微积分）推广了Hadamard和Riemann-Liouville分数阶微积分，这给数值求解带来了挑战。本文研究以映射Jacobi函数（MJFs）为基函数的谱类方法，并建立求解$\psi$-分数阶微分方程的高效算法。具体而言，我们针对涉及$\psi$-分数阶导数的初边值问题，构建了Petrov-Galerkin谱方法和谱配置方法。我们发展了MJFs的基本逼近理论，并对所提方法进行了误差估计。同时，建立了递归关系用于计算配置微分矩阵，以实现谱配置算法。数值算例验证了理论结果，并表明了谱方法和配置方法的有效性。