JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2015, Vol. 50 ›› Issue (02): 67-74.doi: 10.6040/j.issn.1671-9352.0.2014.078

Previous Articles     Next Articles

Wavelets method for solving system of fractional differential equations and the convergence analysis

CHEN Yi-ming, KE Xiao-hong, HAN Xiao-ning, SUN Yan-nan, LIU Li-qing   

  1. College of Science, Yanshan University, Qinhuangdao 066004, Hebei, China
  • Received:2014-03-05 Revised:2014-11-03 Online:2015-02-20 Published:2015-01-27

PDF (PC)

1114

Abstract: The Legendre wavelets defined by the shifted Legendre polynomial is used to solve the numerical solution of the system of fractional differential equations with variable coefficient.The convergence analysis is presented to show that this method is correct for solving the fractional differential equations. Finally, three numerical examples are given to demonstrate the feasibility and efficiency of this method.

Key words: Legendre wavelets, operational matrix, convergence analysis, system of fractional differential equations, shifted Legendre polynomial

CLC Number: 

  • O241.8
[1] SUN H, ABDELWAHAB A, ONARAL B. Linear approximation of transfer function with a pole of fractional order[J]. IEEE Transactions on Automatic Control, 1984, 29:441-444.
[2] KOELLER R C. Applications of fractional calculus to the theory of viscoelasticity[J]. Journal of Applied Mechanics, 1984, 51:299-307.
[3] DEBNATH L. Recent applications of fractional calculus to science and engineering[J]. International Journal of Mathematics and Mathematical Sciences, 2003, 54:3413-3442.
[4] GRIGORENKOL, GRIGORENKO E. Chaotic dynamics of the fractional Lorenz system[J]. Physical Review Letters, 2003, 91(3):034101.
[5] YUSTE S B. Weighted average finite difference methods for fractional diffusion equations[J]. Computational Physics, 2006, 216:264-274.
[6] MOMANI S M, ODIBAT Z. Numerical approach to differential equations of fractional order[J]. Journal of Computational and Applied Mathematics, 2007: 96-110.
[7] LI Xinxiu. Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method[J]. Communications in Nonlinear Science and Numerical Simulation, 2012, 17:3934-3946.
[8] WU Guocheng, LEE E W M. Fractional variational iteration method and its application[J]. Physics Letters A, 2010, 374(25):2506-2509.
[9] MOMANI S M, ODIBAT Z. Homotopy Perturbation method for nonlinear partial differential equations of fractional order[J]. Physic Letters A, 2007, 365:315-350.
[10] REHMAN Mu, KHAN R A. The Legendre wavelet method for solving fractional differential equations[J]. Communications in Nonlinear Science and Numerical Simulation, 2011, 16:4163-4173.
[11] LI Yuanlu. Solving a nonlinear fractional differential equation using Chebyshev wavelets[J]. Communications in Nonlinear Science and Numerical Simulation, 2010, 15:2284-2292.
[12] MATIGNON D. Stability results of fractional differential equations with applications to control processing[C]. Proceeding of IMACS, IEEE-SMC, Lille,France, 1996: 963-968.
[13] TAVAZOEI M S, HAERI M. A note on the stability of fractional order systems[J]. Mathematics and Computers in Simulation, 2009, 79(5):1566-1576.
[14] ODIBAT Z. Analytic study on linear systems of fractional differential equations[J]. Computers and Mathematics with Applications, 2010, 59:1171-1183.
[15] DAFTARDAR-GEJJI V, JAFARI H. Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives[J]. Journal of Mathematical Analysis and Applications, 2007, 328(2):1026-1033.
[16] BONILLA B, RIVERO M, TRUJILLO J J. On systems of linear fractional differential equations with constant coefficients[J]. Applied Mathematics and Computation, 2007, 187(1):68-78.
[17] 高芳,江卫华.分数阶微分方程组边值问题解的存在性[J].数学的实践与认识,2013,43(10):254-260. GAO Fang, JIANG Weihua. Existenceof solutions for boundary value problem of fractional differential equation system[J]. Mathematics in Practice and Theory, 2013, 43(10):254-260.
[1] WANG Yang, ZHAO Yan-jun, FENG Yi-fu. On successive-overrelaxation acceleration of MHSS iterations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(8): 61-65.
[2] ZHANG Jian-song1, NIU Hai-feng2. A new streamline-diffusion mixed finite element method for compressible  miscible displacement problem in porous medium [J]. J4, 2011, 46(12): 6-12.
[3] . A high accuracy finite volume element method based on cubic spline interpolation for twopoint boundary value problems [J]. J4, 2009, 44(2): 45-51.
[4] GUO Hui,LIN Chao . A least-squares mixed finite element procedure with the method of
characteristics for convection-dominated Sobolev equations [J]. J4, 2008, 43(9): 45-50 .
[5] GUO Hui .

A least-squares mixed finite element procedure with the method of
characteristics for convection-dominated diffusion equations

[J]. J4, 2008, 43(8): 6-10 .
[6] ZHANG Jian-song,YANG Dan-ping* . Numerical analysis for the thermistor problem with mixed boundary conditions [J]. J4, 2007, 42(8): 1-08 .
[7] ZHANG Jian-song and YANG Dan-ping . A fully-discrete splitting positive definite mixed element scheme finite for compressible miscible displacement in porous media [J]. J4, 2006, 41(1): 1-10 .
[8] CHENG Ai-jie and LU Ning . Galerkin method for controlledrelease and spread of solute in porous media [J]. J4, 2006, 41(1): 16-20 .
[9] ZHANG Jian-song and YANG Dan-ping . A fullydiscrete splitting positive definite mixed element scheme finite for compressible miscible displacement in porous media [J]. J4, 2006, 41(1): 1-10 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd