[Paper Review] Generalized Jacobi Functions and Their Applications to Fractional Differential Equations

Sheng Chen, Jie Shen+1 more|arXiv (Cornell University)|Jul 31, 2014

Fractional Differential Equations Solutions16 references6 citations

TL;DR

This paper introduces generalized Jacobi functions (GJFs) with tunable parameters to address the singularities and non-locality in fractional differential equations (FDEs). By constructing GJF-Petrov-Galerkin spectral methods, the authors achieve sparse, well-conditioned linear systems and spectral convergence rates dependent only on data smoothness, enabling high accuracy even for solutions with boundary singularities.

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.

Motivation & Objective

• Address the challenge of singular solutions and non-local fractional derivatives in FDEs, which hinder conventional numerical methods.
• Overcome the limitations of finite difference and finite element methods that produce dense, ill-conditioned matrices due to non-locality.
• Develop a spectral framework that matches the intrinsic singularity of FDE solutions using basis functions tailored to fractional calculus.
• Achieve optimal convergence rates independent of solution regularity, ensuring spectral accuracy when data are smooth.
• Extend the applicability of spectral methods to general-order FDEs, including high-order initial and boundary value problems previously unaddressed numerically.

Proposed method

• Define a new class of generalized Jacobi functions (GJFs) with two real parameters to capture boundary singularities inherent in FDE solutions.
• Establish fractional calculus properties of GJFs, including derivative formulas for Riemann-Liouville and Caputo derivatives.
• Construct GJF-Petrov-Galerkin spectral methods using weighted Sobolev spaces involving fractional derivatives.
• Choose parameters in GJFs to ensure the resulting linear systems are sparse and well-conditioned, enabling efficient solution.
• Derive error estimates in weighted Sobolev norms that depend only on the smoothness of the data, not on solution regularity.
• Leverage properties of Jacobi polynomials with real parameters, including derivative rules and recurrence relations, to enable stable computation.

Experimental results

Research questions

• RQ1 Can a new class of basis functions be constructed to naturally match the singular behavior of solutions to fractional differential equations?
• RQ2 How can spectral methods be designed to produce sparse and well-conditioned linear systems for FDEs despite their non-local nature?
• RQ3 What error estimates can be derived for GJF-based spectral approximations in weighted Sobolev spaces with fractional derivatives?
• RQ4 Can spectral convergence be achieved for FDEs with smooth data even when the solution is singular, by using appropriate basis functions?
• RQ5 To what extent can the proposed GJF framework be extended to general-order FDEs, including high-order initial and boundary value problems?

Key findings

• The proposed generalized Jacobi functions (GJFs) are intrinsically linked to fractional calculus and can be tuned via parameters to match solution singularities.
• GJF-Petrov-Galerkin methods lead to sparse and well-conditioned linear systems, significantly reducing computational cost compared to traditional spectral or finite difference methods.
• Error estimates show convergence rates dependent solely on the smoothness of the data, enabling spectral accuracy when data are analytic.
• For smooth data, exponential convergence is achieved despite the solution's singular nature, demonstrating the method's robustness.
• The framework successfully handles general-order fractional initial and boundary value problems, including high-order cases not previously treated numerically.
• Theoretical results are validated through approximation properties in weighted Sobolev spaces involving fractional derivatives, ensuring rigorous convergence analysis.