Fractional Romanovski–Jacobi tau method for time-fractional partial differential equations with nonsmooth solutions

Abstract

In this paper, we introduce a new finite class of non-polynomial basis functions for the spectral tau approximation of time-fractional partial differential equations on a semi-infinite interval. Different from many other spectral tau approaches, the singular basis functions are based on a finite class of Romanovski–Jacobi polynomials through a fractional coordinate transform. As such, the singularity of the new basis can be tailored to suit that of the singular solutions to a class of time-fractional partial differential equations, leading to spectrally accurate approximations. We introduce the fractional Romanovski-Jacobi-Gauss-type quadrature formulae along with basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We discuss the relationship between such kind of finite orthogonal functions and other classes of infinite orthogonal functions such as fractional Laguerre functions and fractional modified generalized Laguerre functions. Moreover, we derive a new formula expressing explicitly the fractional integral of the fractional Romanovski-Jacobi functions in terms of the same functions themselves. This family of orthogonal systems offers great flexibility to match a wide range of time-fractional differential equations with Robin boundary conditions and with nonsmooth solutions.