Taiwanese Journal of Mathematics

Spectral Approximations for Nonlinear Fractional Delay Diffusion Equations with Smooth and Nonsmooth Solutions

Haiyu Liu, Shujuan Lü, and Hu Chen

Full-text: Open access

Abstract

A fully discrete scheme is proposed for the nonlinear fractional delay diffusion equations with smooth solutions, where the fractional derivative is described in Caputo sense with the order $\alpha$ ($0 \lt \alpha \lt 1$). The scheme is constructed by combining finite difference method in time and Legendre spectral approximation in space. Stability and convergence are proved rigorously. Moreover, a modified scheme is proposed for the equation with nonsmooth solutions by adding correction terms to the approximations of fractional derivative operator and nonlinear term. Numerical examples are carried out to support the theoretical analysis.

Article information

Source
Taiwanese J. Math., Volume 23, Number 4 (2019), 981-1000.

Dates
Received: 9 February 2018
Revised: 3 September 2018
Accepted: 9 September 2018
First available in Project Euclid: 18 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1563436877

Digital Object Identifier
doi:10.11650/tjm/180901

Mathematical Reviews number (MathSciNet)
MR3982070

Zentralblatt MATH identifier
07088956

Subjects
Primary: 65M12: Stability and convergence of numerical methods 65M06: Finite difference methods 65M70: Spectral, collocation and related methods 35R11: Fractional partial differential equations

Keywords
nonlinear fractional delay equations spectral method stability and convergence nonsmooth solutions correction terms

Citation

Liu, Haiyu; Lü, Shujuan; Chen, Hu. Spectral Approximations for Nonlinear Fractional Delay Diffusion Equations with Smooth and Nonsmooth Solutions. Taiwanese J. Math. 23 (2019), no. 4, 981--1000. doi:10.11650/tjm/180901. https://projecteuclid.org/euclid.twjm/1563436877


Export citation

References

  • A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys. 280 (2015), 424–438.
  • W. Cao, F. Zeng, Z. Zhang and G. E. Karniadakis, Implicit-explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions, SIAM J. Sci. Comput. 38 (2016), no. 5, A3070–A3093.
  • C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Scientific Computation, Springer-Verlag, Berlin, 2006.
  • H. Chen, S. Lü and W. Chen, Finite difference/spectral approximations for the distributed order time fractional reaction-diffusion equation on an unbounded domain, J. Comput. Phys. 315 (2016), 84–97.
  • X. Chen, F. Zeng and G. E. Karniadakis, A tunable finite difference method for fractional differential equations with non-smooth solutions, Comput. Methods Appl. Mech. Engrg. 318 (2017), 193–214.
  • L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci. 2003 (2003), no. 54, 3413–3442.
  • M. Dehghan and M. Abbaszadeh, Analysis of the element free Galerkin (EFG) method for solving fractional cable equation with Dirichlet boundary condition, Appl. Numer. Math. 109 (2016), 208–234.
  • K. Diethelm, J. M. Ford, N. J. Ford and M. Weilbeer, Pitfalls in fast numerical solvers for fractional differential equations, J. Comput. Appl. Math. 186 (2006), no. 2, 482–503.
  • N. J. Ford, J. Xiao and Y. Yan, A finite element method for time fractional partial differential equations, Fract. Calc. Appl. Anal. 14 (2011), no. 3, 454–474.
  • Z. Hao, K. Fan, W. Cao and Z. Sun, A finite difference scheme for semilinear space-fractional diffusion equations with time delay, Appl. Math. Comput. 275 (2016), 238–254.
  • J. Huang, Y. Tang, L. Vázquez and J. Yang, Two finite difference schemes for time fractional diffusion-wave equation, Numer. Algorithms 64 (2013), no. 4, 707–720.
  • B. Jin, R. Lazarov, J. Pasciak and Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA J. Numer. Anal. 35 (2015), no. 2, 561–582.
  • B. Jin, R. Lazarov and Z. Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Sci. Comput. 38 (2016), no. 1, A146–A170.
  • Y. Lenbury and D. V. Giang, Nonlinear delay differential equations involving population growth, Math. Comput. Modelling 40 (2004), no. 5-6, 583–590.
  • D. Li, H. Liao, W. Sun, J. Wang and J. Zhang, Analysis of $L1$-Galerkin FEMs for time-fractional nonlinear parabolic problems, arXiv:1612.00562.
  • X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal. 47 (2009), no. 3, 2108–2131.
  • Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), no. 2, 1533–1552.
  • C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal. 17 (1986), no. 3, 704–719.
  • F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in: Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), 291–348, CISM Courses and Lect. 378, Springer, Vienna, 1997.
  • S. Momani and Z. Odibat, Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. Math. Appl. 54 (2007), no. 7-8, 910–919.
  • K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
  • I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering 198, Academic Press, San Diego, CA, 1999.
  • M. Raberto, E. Scalas and F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study, Physica A 314 (2002), no. 1-4, 749–755.
  • P. Rahimkhani, Y. Ordokhani and E. Babolian, A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations, Numer. Algorithms 74 (2017), no. 1, 223–245.
  • F. A. Rihan, Computational methods for delay parabolic and time-fractional partial differential equations, Numer. Methods Partial Differential Equations 26 (2010), no. 6, 1556–1571.
  • A. Saadatmandi and M. Dehghan, A tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl. 62 (2011), no. 3, 1135–1142.
  • U. Saeed, M. Rehman and M. A. Iqbal, Modified Chebyshev wavelet methods for fractional delay-type equations, Appl. Math. Comput. 264 (2015), 431–442.
  • A. Si-Ammour, S. Djennoune and M. Bettayeb, A sliding mode control for linear fractional systems with input and state delays, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), no. 5, 2310–2318.
  • M. Stynes, E. O'Riordan and J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), no. 2, 1057–1079.
  • Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56 (2006), no. 2, 193–209.
  • S. Yaghoobi, B. P. Moghaddam and K. Ivaz, An efficient cubic spline approximation for variable-order fractional differential equations with time delay, Nonlinear Dynam. 87 (2017), no. 2, 815–826.
  • M. Yi and J. Huang, Wavelet operational matrix method for solving fractional differential equations with variable coefficients, Appl. Math. Comput. 230 (2014), 383–394.
  • M. Zayernouri, W. Cao, Z. Zhang and G. E. Karniadakis, Spectral and discontinuous spectral element methods for fractional delay equations, SIAM J. Sci. Comput. 36 (2014), no. 6, B904–B929.
  • F. Zeng, C. Li, F. Liu and I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput. 35 (2013), no. 6, A2976–A3000.
  • F. Zeng, I. Turner and K. Burrage, A stable fast time-stepping method for fractional integral and derivative operators, J. Sci. Comput. 77 (2018), no. 1, 283–307.
  • F. Zeng, Z. Zhang and G. E. Karniadakis, Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions, Comput. Methods Appl. Mech. Engrg. 327 (2017), 478–502.
  • M. Zheng, F. Liu, V. Anh and I. Turner, A high-order spectral method for the multi-term time-fractional diffusion equations, Appl. Math. Model. 40 (2016), no. 7-8, 4970–4985.