Generalized Jacobi functions and their applications to fractional differential equations

Abstract

Abstract. In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus, and can serve as natural basis functions for properly designed spectral methods for FDEs. We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives. We construct efficient GJF-Petrov-Galerkin methods for a class of prototypical fractional initial value problems (FIVPs) and fractional boundary value problems (FBVPs) of general order, and show that with an appropriate choice of the parameters in GJFs, the resulted linear systems can be sparse and well-conditioned. Moreover, we derive error estimates with convergence rate only depending on the smoothness of data, so truly spectral accuracy can be attained if the data are smooth enough. The idea and results presented in this paper will be useful to deal with more general FDEs associated with Riemann-Liouville or Caputo fractional derivatives.

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

“…Unlike the cases with one-sided fractional derivatives considered in [3], it is not clear how the regularity of u for the problem (3.3) depends on the data. Hence, we provide error estimates by assuming u in some non-uniformly weighted Sobolev spaces, namely, u ∈ H m ω, * (Λ).…”

“…Recently, some efficient spectral/spectral-element DG methods for a class of one-dimensional FPDEs with constantcoefficients and one-sided fractional derivatives have been proposed in [31,33] by using eigenfunctions of fractional Sturm-Liouville problems as basis functions. Related spectral algorithms and their rigorous error analyses have been established in [3]. However, these results can not extended to more general FPDEs with two-sided fractional derivatives and variable coefficients.…”

“…On the other hand, the main disadvantage of global methods such as spectral methods is no longer an issue for fractional PDEs. Moreover, for fractional PDEs with constant coefficients, spectral methods with suitable basis functions can even result in sparse linear systems [31,3]. The main purpose of this paper is to develop some efficient spectral algorithms to solve a class of fractional elliptic equations.…”

Super-Exponentially Convergent Parallel Algorithm for Eigenvalue Problems with Fractional Derivatives

“…In [9], Chen, Shen and Wang consider a spectral approximation of fractional differential equations (FDEs). A new class of generalized Jacobi functions (GJFs) is defined, which are the eigenfunctions of some fractional Jacobi-type differential operator and can serve as natural basis functions for properly designed spectral methods for FDEs.…”

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

“…(2) Use some nonpolynomial (or singular) basis functions or collocation spectral methods to capture the singularity of the solutions of (1.1), see [1], [5], [13], [14], [34], [27], [59], [63], [67]. (3) Separate the solution into two parts: smooth and nonsmooth parts.…”