200
Views
12
CrossRef citations to date
0
Altmetric
Original Articles

A backward Euler difference scheme for the integro-differential equations with the multi-term kernels

ORCID Icon, & ORCID Icon
Pages 1254-1267 | Received 26 Oct 2018, Accepted 07 Apr 2019, Published online: 08 May 2019

References

  • A. Cascone, C. D'Apice, B. Piccoli, and L. Rarità, Circulation of car traffic in congested urban areas, Commun. Math. Sci. 6 (2008), pp. 765–784. doi: 10.4310/CMS.2008.v6.n3.a12
  • H. Chen and D. Xu, A compact difference scheme for an evolution equation with a weakly singular kernel, Numer. Math. Theory Methods Appl. 5 (2012), pp. 559–572. doi: 10.4208/nmtma.2012.m11032
  • H. Chen, C. Chen, and D. Xu, A second-order fully discrete difference scheme for a partial integro-differential equation, Math. Numer. Sinica 28 (2006), pp. 141–154. (in Chinese)
  • H. Chen, S. Gan, D. Xu, and Q. Liu, A second order BDF compact difference scheme for fractional order Volterra equations, Int. J. Comput. Math. 93 (2016), pp. 1140–1154. doi: 10.1080/00207160.2015.1021695
  • A. Cutolo, B. Piccoli, and L. Rarità, An upwind-Euler scheme for an ODE-PDE model of supply chains, SIAM J. Comput. 33 (2011), pp. 1669–1688. doi: 10.1137/090767479
  • M. Dehghan, M. Safarpoor, and M. Abbaszadeh, Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations, J. Comput. Appl. Math. 290 (2015), pp. 174–195. doi: 10.1016/j.cam.2015.04.037
  • K.B. Hannsgen and R.L. Wheeler, Uniform L1 behavior in classes of integrodifferential quations with completely monotonic kernels, SIAM J. Math. Anal. 15 (1984), pp. 579–594. doi: 10.1137/0515044
  • X. Hu and L. Zhang, A compact finite difference scheme for the fourth-order fractional diffusion-wave system, Comput. Phys. Commun. 182 (2011), pp. 1645–1650. doi: 10.1016/j.cpc.2011.04.013
  • X. Hu and L. Zhang, On finite difference methods for fourth-order fractional diffusion-wave and sub-diffusion systems, Appl. Math. Comput. 218 (2012), pp. 5019–5034.
  • X. Hu and L. Zhang, A new implicit compact difference scheme for the fourth-order fractional diffusion-wave system, Int. J. Comput. Math. 91 (2014), pp. 2215–2231. doi: 10.1080/00207160.2013.871000
  • B. Jin, R. Lazarov, Y. Liu, and Z. Zhou, The Galerkin finite element method for multi-term time-fractional diffusion equations, J. Comput. Phys. 281 (2015), pp. 825–843. doi: 10.1016/j.jcp.2014.10.051
  • F. Liu, M.M. Meerschaert, R.J. McGough, P. Zhuang, and Q. Liu, Numerical methods for solving the multi-term time-fractional wave-diffusion equations, Fract. Calc. Appl. Anal. 15 (2013), pp. 9–25.
  • J.C. Lopez-Marcos, A difference scheme for a nonlinear partial integro-differential equation, SIAM J. Numer. Anal. 27 (1990), pp. 20–31. doi: 10.1137/0727002
  • C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal. 17 (1986), pp. 704–719. doi: 10.1137/0517050
  • R. Manzo, B. Piccoli, and L. Rarità, Optimal distribution of traffic flows in emergency cases, European J. Appl. Math. 23 (2012), pp. 515–535. doi: 10.1017/S0956792512000071
  • R.D. Noren, Uniform L1 behavior for the solution of a Volterra equation with a parameter, SIAM J. Math. Anal. 19 (1988), pp. 270–286. doi: 10.1137/0519020
  • R.D. Noren, Uniform L1 behavior in classes of integro-differential equations with convex kernels, J. Integral Equations Appl. 1 (1988), pp. 385–399. doi: 10.1216/JIE-1988-1-3-385
  • A.K. Pani and R.K. Sinha, On the backward Euler method for time dependent parabolic integro-differential equation with nonsmooth initial data, J. Integral Equations Appl. 19 (1998), pp. 219–249. doi: 10.1216/jiea/1181074222
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • G.M.M. Reddy and R.K. Sinha, Ritz-Volterra reconstructions and a posteriori error analysis of finite element method for parabolic integro-differential equations, IMA J. Numer. Anal. 35 (2015), pp. 341–371. doi: 10.1093/imanum/drt059
  • M. Renardy, M.J. Hrusa, and J.A. Nohel, Mathematics Problem in Viscoelasticity, Longman, London, 1987.
  • M. Slodicka, Numerical solution of a parabolic equation with a weakly singular positive-type memory term, Electron. J. Differential Equations 1997 (1997), pp. 1–12.
  • T. Tang, A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math. 11 (1993), pp. 309–319. doi: 10.1016/0168-9274(93)90012-G
  • D. Xu, The global behavior of time discretization for an abstract Volterra equation in Hilbert space, Calcolo 34 (1997), pp. 71–104.
  • D. Xu, The time discretization in classes of integro-differential equations with completely monotonic kernels: Weighted asymptotic stability, Sci. China Math. 56 (2013), pp. 395–424. doi: 10.1007/s11425-012-4410-2
  • D. Xu, The time discretization in classes of integro-differential equations with completely monotonic kernels: Weighted asymptotic convergence, Numer. Methods Partial Differential Equations 32 (2016), pp. 896–935. doi: 10.1002/num.22035
  • D. Xu, Numerical asymptotic stability for the integro-differential equations with the multi-term kernels, Appl. Math. Comput. 309 (2017), pp. 107–132.
  • Y. Zhang, Z. Sun, and H. Wu, Error estimates of Crank-Nicolson type difference schemes for the sub-diffusion equation, SIAM J. Numer. Anal. 49 (2011), pp. 2302–2322. doi: 10.1137/100812707

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.