A Modified Generalized Laguerre Spectral Method for Fractional Differential Equations on the Half Line

Băleanu, Dumitru; Bhrawy, A. H.; Taha, T. M.

doi:10.1155/2013/413529

Cited by 28 publications

(23 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, we do not include such a discussion here due to the limit on the length of the paper. In the literature, spectral methods for FODEs on the half line have been investigated using the generalized Laguerre polynomials; see, e.g., [6] for Laguerre spectral methods and, e.g., [25] for Laguerre spectral collocation methods, without taking into account the endpoint singularity.…”

Section: Introductionmentioning

confidence: 99%

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

Zhang¹,

Zeng²,

Karniadakis³

2015

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

We present optimal error estimates for spectral Petrov-Galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval. We also develop Laguerre spectral Petrov-Galerkin methods and collocation methods for fractional equations on the half line. Numerical results confirm the error estimates. Introduction.Numerical methods for fractional differential equations have been investigated for decades; see, e.g., [8,12,33]. However, spectral methods for these equations have a rather short history since the solutions usually have some singular lower-order derivatives. The recent investigation of spectral methods for these singular problems is motivated by the following facts. First, all numerical methods for these problems are nonlocal as spectral methods are. For example, in finite difference methods, see, e.g., [33], and in finite element methods, see, e.g., [13,18], data at almost all grids or all elements are exploited to approximate the integral operators at one grid or element. These methods are still nonlocal, even when a "fixed memory principle" is applied to reduce the use of all data; see, e.g., [15,19]. Second, spectral methods lead to high-order accuracy when the solutions are smooth; see, e.g., [7,24] for integerorder differential equations. For fractional differential equations with weakly singular kernels, solutions can be smooth even when some inputs of the equations are singular.Consider the following fractional ordinary differential equation (FODE) over the interval I = (−1, 1):

show abstract

Section: Introductionmentioning

confidence: 99%

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

Zhang¹,

Zeng²,

Karniadakis³

2015

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…In this section, we concisely point out some definitions of the VO-F operators [8,12,17]. We then collect some important properties of the modified generalized Laguerre polynomials [4]. Assume that u(x) = 0 for x < 0.…”

Section: Preliminaries and Fundamentalsmentioning

confidence: 99%

New Recursive Approximations for Variable-Order Fractional Operators With Applications

Zaky

Doha

Taha

et al. 2018

Mathematical Modelling and Analysis

View full text Add to dashboard Cite

To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation. In this paper, we develop Laguerre spectral collocation methods for solving variable-order fractional initial value problems on the half line. Specifically, we derive three-term recurrence relations to efficiently calculate the variable-order fractional integrals and derivatives of the modified generalized Laguerre polynomials, which lead to the corresponding fractional differentiation matrices that will be used to construct the collocation methods. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation methods.

show abstract

“…In virtue of the difficulty for looking for the exact solutions of FPDEs, more and more scholars try to seek the numerical solutions by different numerical methods. These numerical methods mainly cover finite element (FE) methods [1][2][3][4][5][6][7][8][9][10][11][12][13], mixed finite element (MFE) methods [14,15], finite difference (FD) methods , finite volume (element) methods [25,39,40], (local) discontinuous Galerkin (L)DG methods [41][42][43], spectral methods [44][45][46][47][48][49] and so on.…”

Section: Introductionmentioning

confidence: 99%

Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem

Yang

et al. 2015

Computers & Mathematics with Applications

145

View full text Add to dashboard Cite

Available online xxxx Keywords:Time-fractional fourth-order reaction-diffusion problem Finite element method Finite difference scheme Caputo-fractional derivative Unconditional stability Optimal a priori error analysis a b s t r a c tIn this article, a finite difference/finite element algorithm, which is based on a finite difference approximation in time direction and finite element method in spatial direction, is presented and discussed to cast about for the numerical solutions of a time-fractional fourth-order reaction-diffusion problem with a nonlinear reaction term. To avoid the use of higher-order elements, the original problem with spatial fourth-order derivative need to be changed into a second-order coupled system by introducing an intermediate variable σ = ∆u. Then the fully discrete finite element scheme is formulated by using a finite difference approximation for time fractional and integer derivatives and finite element method in spatial direction. The unconditionally stable result in the norm, which just depends on initial value and source item, is derived. Some a priori estimates of L 2 -norm with optimal order of convergence O(∆ 2−α t +h m+1 ), where ∆ t and h are time step length and space mesh parameter, respectively, are obtained. To confirm the theoretical analysis, some numerical results are provided by our method.

show abstract

A Modified Generalized Laguerre Spectral Method for Fractional Differential Equations on the Half Line

Cited by 28 publications

References 35 publications

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

New Recursive Approximations for Variable-Order Fractional Operators With Applications

Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem

Contact Info

Product

Resources

About