Spectral approximations to the fractional integral and derivative

被引：146

作者：

Li, Changpin ^{[1
]}

Zeng, Fanhai ^{[1
]}

Liu, Fawang ^{[2
]}

机构：

[1] Shanghai Univ, Dept Math, Shanghai 200444, Peoples R China

[2] Queensland Univ Technol, Sch Math Sci, Brisbane, Qld 4001, Australia

来源：

FRACTIONAL CALCULUS AND APPLIED ANALYSIS | 2012年 / 15卷 / 03期

基金：

中国国家自然科学基金;

关键词：

fractional integral; Caputo derivative; spectral approximation; Jacobi polynomials; DIFFERENTIAL-EQUATIONS; DIFFUSION EQUATIONS; ALGORITHMS; CALCULUS; SYSTEMS;

D O I：

10.2478/s13540-012-0028-x

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

引用

页码：383 / 406

页数：24

共 32 条

[1]

Canuto C., 2006, SPECRAL METHODS FUND

[2] Finite element formulations for transient dynamic analysis in structural systems with viscoelastic treatments containing fractional derivative models [J].

Cortes, Fernando ;

Elejabarrieta, Maria Jesus .

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 2007, 69 (10) :2173-2195

[3] Algorithms for the fractional calculus: A selection of numerical methods [J].

Diethelm, K ;

Ford, NJ ;

Freed, AD ;

Luchko, Y .

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 2005, 194 (6-8) :743-773

[4] A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order [J].

Doha, E. H. ;

Bhrawy, A. H. ;

Ezz-Eldien, S. S. .

COMPUTERS & MATHEMATICS WITH APPLICATIONS, 2011, 62 (05) :2364-2373

[5] Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations [J].

Doha, E. H. ;

Bhrawy, A. H. ;

Ezz-Eldien, S. S. .

APPLIED MATHEMATICAL MODELLING, 2011, 35 (12) :5662-5672

[6] The numerical solution of linear multi-term fractional differential equations: systems of equations [J].

Edwards, JT ;

Ford, NJ ;

Simpson, AC .

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2002, 148 (02) :401-418

[7] A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations [J].

Esmaeili, Shahrokh ;

Shamsi, M. .

COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, 2011, 16 (09) :3646-3654

[8] Application of fractional calculus to fluid mechanics [J].

Kulish, VV ;

Lage, JL .

JOURNAL OF FLUIDS ENGINEERING-TRANSACTIONS OF THE ASME, 2002, 124 (03) :803-806

[9] FINITE DIFFERENCE METHODS FOR FRACTIONAL DIFFERENTIAL EQUATIONS [J].

Li, Changpin ;

Zeng, Fanhai .

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 2012, 22 (04)

[10] Numerical approaches to fractional calculus and fractional ordinary differential equation [J].

Li, Changpin ;

Chen, An ;

Ye, Junjie .

JOURNAL OF COMPUTATIONAL PHYSICS, 2011, 230 (09) :3352-3368

← 1 2 3 4 →