[Paper Review] Spectral method for substantial fractional differential equations

Can Huang, Qingshuo Song+1 more|arXiv (Cornell University)|Aug 26, 2014

Fractional Differential Equations Solutions17 references5 citations

TL;DR

This paper introduces a non-polynomial spectral Petrov-Galerkin method and associated collocation technique for substantial fractional differential equations (FDEs) of order $\nu$ and $1+\nu$ with $0<\nu<1$. By extending generalized Laguerre polynomials to $\alpha \leq -1$ and carefully selecting trial and test spaces, the method yields diagonal, well-conditioned linear systems and achieves high convergence rates through parameter tuning in basis selection.

ABSTRACT

In this paper, a non-polynomial spectral Petrov-Galerkin method and associated collocation method for substantial fractional differential equations (FDEs) are proposed, analyzed, and tested. We extend a class of generalized Laguerre polynomials to form our basis. By a proper scaling of trial basis and test basis, our Petrov-Galerkin method results in a diagonal and thus well-conditioned linear systems for both fractional advection equation and fractional diffusion equation. In the meantime, we construct substantial fractional differential collocation matrices and provide explicit forms for both type of equations. Moreover, the proposed method allows us to adjust a parameter in basis selection according to different given data to maximize the convergence rate. This fact has been proved in our error analysis and confirmed in our numerical experiments.

Motivation & Objective

• To develop a high-order numerical method for substantial fractional differential equations, which are less explored than standard FDEs.
• To address the challenge of ill-conditioned linear systems in spectral methods for FDEs by constructing well-conditioned, diagonal systems through basis and space selection.
• To extend generalized Laguerre polynomials to $\alpha \leq -1$ while preserving polynomial structure, enabling accurate approximation of non-polynomial solutions.
• To provide explicit construction of substantial fractional differential collocation matrices for both advection and diffusion equations.
• To enable adjustable convergence rates via a tunable parameter in the basis, validated through error analysis and numerical experiments.

Proposed method

• The method employs a non-polynomial spectral Petrov-Galerkin framework using a basis formed by $e^{-\sigma x} x^{\alpha} L_n^{\alpha}(x)$, where $L_n^{\alpha}(x)$ are generalized Laguerre polynomials extended to $\alpha \leq -1$.
• Trial and test spaces are constructed with different scaling to ensure the resulting linear system is diagonal and well-conditioned, minimizing condition number growth.
• The method leverages properties of generalized Laguerre polynomials and their fractional derivatives to enable exact computation of substantial fractional derivatives in the weak form.
• Explicit expressions for substantial fractional differential collocation matrices are derived for both fractional advection and diffusion equations.
• A parameter in the basis is adjusted to maximize convergence rate, with theoretical justification provided via error analysis.
• The approach is validated through rigorous error analysis and numerical experiments demonstrating high convergence rates and well-conditioning.

Experimental results

Research questions

• RQ1 Can a spectral Petrov-Galerkin method be designed for substantial FDEs that yields diagonal, well-conditioned linear systems?
• RQ2 How can generalized Laguerre polynomials be extended to $\alpha \leq -1$ while preserving polynomial structure and enabling accurate approximation?
• RQ3 What is the impact of a tunable parameter in the basis on the convergence rate of the spectral method for substantial FDEs?
• RQ4 Can explicit collocation matrices be constructed for substantial fractional derivatives in both advection and diffusion equations?
• RQ5 How does the condition number of the resulting linear system scale with the polynomial degree in this new framework?

Key findings

• The proposed Petrov-Galerkin method results in diagonal linear systems due to careful selection of trial and test spaces, significantly improving conditioning and computational efficiency.
• The method achieves high convergence rates, with the rate maximized by tuning a parameter in the basis, as proven in error analysis and confirmed numerically.
• The extension of generalized Laguerre polynomials to $\alpha \leq -1$ produces a polynomial result for all real $\alpha$, differing fundamentally from prior extensions.
• Explicit forms of substantial fractional differential collocation matrices are derived for both fractional advection and diffusion equations, enabling efficient collocation-based computation.
• The condition number of the system grows slowly due to diagonalization, contrasting with the $\mathcal{O}(N^{2\nu})$ growth seen in standard spectral collocation methods.
• Numerical experiments confirm the theoretical convergence rates and demonstrate the effectiveness of parameter tuning in enhancing accuracy.