Müntz Spectral Methods for the Time-Fractional Diffusion Equation

Abstract

In this paper, we propose and analyze a fractional spectral method for the time-fractional diffusion equation (TFDE). The main novelty of the method is approximating the solution by using a new class of generalized fractional Jacobi polynomials (GFJPs), also known as Müntz polynomials. We construct two efficient schemes using GFJPs for TFDE: one is based on the Galerkin formulation and the other on the Petrov–Galerkin formulation. Our theoretical or numerical investigation shows that both schemes are exponentially convergent for general right-hand side functions, even though the exact solution has very limited regularity (less than {H^{1}}). More precisely, an error estimate for the Galerkin-based approach is derived to demonstrate its spectral accuracy, which is then confirmed by numerical experiments. The spectral accuracy of the Petrov–Galerkin-based approach is only verified by numerical tests without theoretical justification. Implementation details are provided for both schemes, together with a series of numerical examples to show the efficiency of the proposed methods.

Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview

“…Hence, recently, there have been immense interest in developing schemes that can take care of the solution singularity directly. In the context of spacetime formulations, singularity enriched trial and/or test spaces, e.g., generalized Jacobi polynomials [13] (including Jacobi poly-fractonomials [108]) and Müntz polynomials [32], are extremely promising and have demonstrated very encouraging numerical results. However, the rigorous convergence analysis of such schemes can be very challenging, and is mostly missing for nonsmooth problem data.…”

A Müntz-Collocation Spectral Method for Weakly Singular Volterra Integral Equations

“…Lemma 2.4 (see [16,17]). The fractional Jacobi polynomials J α,β,λ n ∞ n=0 satisfy the following singular Sturm-Liouville problem:…”

“…This paper aims at designing, developing, and testing a fractional Jacobi spectral method for the weakly singular VIEs, which has the capability to capture the limited regular solution in a more efficient way. The new method will make use of the fractional Jacobi polynomial J α,β,λ N +1 (x), recently introduced in [16,17] to deal with some singular problems. The advantage of the proposed approach is that the exponential convergence can be guaranteed for solutions, which are smooth after the variable change x → x 1/λ for suitable parameter λ.…”

A Petrov–Galerkin spectral method for the numerical simulation and analysis of fractional anomalous diffusion

“…and the semi-analytical methods such as the series expansion, Adomian decomposition, and homotopy techniques can only provide the exact solution for some simple cases or may result in series solutions with slow convergence, classical numerical methods have been extended or modified for the numerical simulation of fractional differential equations and fractional PDEs such as finite element, [1][2][3] finite difference [4][5][6] meshless, 7 and spectral methods. [8][9][10][11][12] The fractional derivatives are used successfully in providing mathematical models for the diffusion process in disordered media and the transport in intracellular crowded environments, where the spread of particles is described in a nonlinear power-law form, namely, the fractional anomalous diffusion. 13,14 This kind of phenomenon appears in different contexts such as the movement of lipids on model membranes, proteins in the nucleoplasm, solute transport in porous media, self-propulsion of radioactive colloidal particles, etc.…”