• Original Articles •

### AN EFFICIENT NUMERICAL METHOD FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH TWO CAPUTO DERIVATIVES

Shuiping Yang1, Aiguo Xiao2

1. 1. Department of Mathematics, Huizhou University, Guangdong, 516007, China;
2. School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan, 411105, China
• Received:2014-09-05 Revised:2015-10-21 Online:2016-03-15 Published:2016-03-15
• Supported by:

This work is supported by projects from the National NSF of China (Grant No. 11501238, Grant No. 11401248), NSF of Guangdong Province (Grant No. 2014A-030313641), and NSF of Huizhou University (Grant No. hzuxl201420). The authors are very grateful to the reviewers for their useful comments.

Shuiping Yang, Aiguo Xiao. AN EFFICIENT NUMERICAL METHOD FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH TWO CAPUTO DERIVATIVES[J]. Journal of Computational Mathematics, 2016, 34(2): 113-134.

In this paper, we study the Hermite cubic spline collocation method with two parameters for solving a initial value problem (IVP) of nonlinear fractional differential equations with two Caputo derivatives. The convergence and nonlinear stability of the method are established. Some illustrative examples are provided to verify our theoretical results. The numerical results also indicate that the convergence order is min{4 ? α, 4 ? β}, where 0 < β < α < 1 are two parameters associated with the fractional differential equations. Mathematics subject classi cation: 65L20, 65L60, 34A08.

CLC Number:

  D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204(1996), 609-625. J. Dixon and S. Mckee, Weakly singular discrete Gronwall inequalities, Z. Angew Math. Mech., 66:11 (1986), 535-544. K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electron. T. Numer. Ana., 5 (1997), 1-6. K. Diethelm, G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorthms, 16 (1997), 231-253. K. Diethelm, N.J. Ford and A. D. Freed, A predictor-Corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3-22. K. Diethelm, N.J. Ford, Numerical solution of the Bagley-Torvik equation, BIT., 42 (2002), 490-507. J.H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol., 15:2 (1999), 86-90. C.P. Li., A. Chen, J.-J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation, J. Comput. Phys., 230 (2011), 3352-3368. F. Liu, V. Anh and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2004), 209-219. F. Liu, P. Zhuang, V. Anh, I. Turner and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput., 191 (2007), 12-20. C. Lubich, Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comput., 41 (1983), 87-102. C. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comput., 45 (1985), 463-469. C. Lubich, Discretized fractional calculus, SIAM J. Mathe. Anal., 17 (1986), 704-719. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. A. Pedas and E. Tamme, Spline collocation methods for linear multi-term fractional differential equations, J. Comput. Appl. Math., 236 (2011), 167-176. A. Pedas and E. Tamme, On the convergence of spline collocation methods for solving fractional differential equations, J. Comput. Appl. Math., 235 (2011), 3502-3514. M. Weilbeer, Efficient Numerical Methods for Fractional Differential Equations and their Analytical Background, Papierflieger, 2006. V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of systems with long-range interaction, Commun. Nonlinear Sci. Numer. Simul., 11:8 (2006), 885-898. V. V. Uchaikin and R. T. Sibatov, Fractional theory for transport in disordered semiconductors, Commun. Nonlinear Sci. Numer. Simul., 13:4 (2008), 715-727. S.P. Yang and A.G. Xiao, Cubic spline collocation method for fractional differential equations, J. Appl. Math., Volume 2013 (2013), Article ID 864025, 20 pages. X. Zhao, Z. Sun and G.E. Karniadakis, Second-order approximations for variable order fractional derivatives: Algorithms and applications, J. Comput. Phys., 293 (2015), 184-200.
  Mohammad Tanzil Hasan, Chuanju Xu. HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY [J]. Journal of Computational Mathematics, 2020, 38(4): 580-605.  Jie Chen, Zhengkang He, Shuyu Sun, Shimin Guo, Zhangxin Chen. EFFICIENT LINEAR SCHEMES WITH UNCONDITIONAL ENERGY STABILITY FOR THE PHASE FIELD MODEL OF SOLID-STATE DEWETTING PROBLEMS [J]. Journal of Computational Mathematics, 2020, 38(3): 452-468.  Li Cai, Ye Sun, Feifei Jing, Yiqiang Li, Xiaoqin Shen, Yufeng Nie. A FULLY DISCRETE IMPLICIT-EXPLICIT FINITE ELEMENT METHOD FOR SOLVING THE FITZHUGH-NAGUMO MODEL [J]. Journal of Computational Mathematics, 2020, 38(3): 469-486.  Liying Zhang, Jing Wang, Weien Zhou, Landong Liu, Li Zhang. CONVERGENCE ANALYSIS OF PARAREAL ALGORITHM BASED ON MILSTEIN SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONS [J]. Journal of Computational Mathematics, 2020, 38(3): 487-501.  Yu Du, Haijun Wu, Zhimin Zhang. SUPERCONVERGENCE ANALYSIS OF THE POLYNOMIAL PRESERVING RECOVERY FOR ELLIPTIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS [J]. Journal of Computational Mathematics, 2020, 38(1): 223-238.  Gerald Gamrath, Ambros Gleixner, Thorsten Koch, Matthias Miltenberger, Dimitri Kniasew, Dominik Schlögel, Alexander Martin, Dieter Weninger. TACKLING INDUSTRIAL-SCALE SUPPLY CHAIN PROBLEMS BY MIXED-INTEGER PROGRAMMING [J]. Journal of Computational Mathematics, 2019, 37(6): 866-888.  Oleg Burdakov, Yuhong Dai, Na Huang. STABILIZED BARZILAI-BORWEIN METHOD [J]. Journal of Computational Mathematics, 2019, 37(6): 916-936.  Zhouhong Wang, Yuhong Dai, Fengmin Xu. A ROBUST INTERIOR POINT METHOD FOR COMPUTING THE ANALYTIC CENTER OF AN ILL-CONDITIONED POLYTOPE WITH ERRORS [J]. Journal of Computational Mathematics, 2019, 37(6): 843-865.  Ernest K. Ryu, Wotao Yin. PROXIMAL-PROXIMAL-GRADIENT METHOD [J]. Journal of Computational Mathematics, 2019, 37(6): 778-812.  Liaoyuan Zeng, Yuhong Dai, Yakui Huang. CONVERGENCE RATE OF GRADIENT DESCENT METHOD FOR MULTI-OBJECTIVE OPTIMIZATION [J]. Journal of Computational Mathematics, 2019, 37(5): 689-703.  Lin Chen. STABILITY OF THE STOCHASTIC θ-METHOD FOR SUPER-LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY [J]. Journal of Computational Mathematics, 2019, 37(5): 704-720.  Yubo Yang, Heping Ma. A LINEAR IMPLICIT L1LEGENDRE GALERKIN CHEBYSHEV COLLOCATION METHOD FOR GENERALIZED TIME-AND SPACE-FRACTIONAL BURGERS EQUATION [J]. Journal of Computational Mathematics, 2019, 37(5): 629-644.  Houchao Zhang, Dongyang Shi. SUPERCONVERGENCE ANALYSIS FOR TIME-FRACTIONAL DIFFUSION EQUATIONS WITH NONCONFORMING MIXED FINITE ELEMENT METHOD [J]. Journal of Computational Mathematics, 2019, 37(4): 488-505.  Jue Wang, Qingnan Zeng. A FOURTH-ORDER COMPACT AND CONSERVATIVE DIFFERENCE SCHEME FOR THE GENERALIZED ROSENAU-KORTEWEG DE VRIES EQUATION IN TWO DIMENSIONS [J]. Journal of Computational Mathematics, 2019, 37(4): 541-555.  Jianfei Huang, Yue Zhao, Sadia Arshad, Kuangying Li, Yifa Tang. ALTERNATING DIRECTION IMPLICIT SCHEMES FOR THE TWO-DIMENSIONAL TIME FRACTIONAL NONLINEAR SUPER-DIFFUSION EQUATIONS [J]. Journal of Computational Mathematics, 2019, 37(3): 297-315.
Viewed
Full text

Abstract