Open Access
Issue
ESAIM: M2AN
Volume 55, Number 5, September-October 2021
Page(s) 2211 - 2232
DOI https://doi.org/10.1051/m2an/2021045
Published online 13 October 2021
  1. R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Elsevier, San Diego (2003). [Google Scholar]
  2. M. Ainsworth and J. Oden, A posteriori error estimation in finite element analysis. Pure and Applied Mathematics, Wiley, New York (2000). [Google Scholar]
  3. R. Bagley and P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheology 27 (1983) 201–210. [Google Scholar]
  4. A. Bonfanti, J.L. Kaplan, G. Charras and A. Kabla, Fractional viscoelastic models for power-law materials. Soft Matter 16 (2020) 6002–6020. [PubMed] [Google Scholar]
  5. D. Boyadzhiev, H. Kiskinov, M. Veselinova and A. Zahariev, Stability analysis of linear distributed order fractional systems with distributed delays. Fract. Calc. Appl. Anal. 20 (2017) 914–935. [Google Scholar]
  6. M. Caputo and M. Fabrizio, The Kernel of the distributed order fractional derivatives with an application to complex materials. Fractal Fract. 1 (2017) 13. [Google Scholar]
  7. A.V. Chechkin, R. Gorenflo and I.M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66 (2002) 046129. [Google Scholar]
  8. A. Consiglio and F. Mainardi, On the evolution of fractional diffusive waves. Ricerche Mat. 70 (2021) 21–33. [Google Scholar]
  9. E. Cuesta, C. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75 (2006) 673–696. [Google Scholar]
  10. K. Diethelm and N.J. Ford, Numerical solution methods for distributed order differential equations. Fract. Calc. Appl. Anal. 4 (2001) 531–542. [Google Scholar]
  11. R. Du, A. Alikhanov and Z. Sun, Temporal second order difference schemes for the multi-dimensional variable-order time fractional sub-diffusion equations. Comput. Math. Appl. 79 (2020) 2952–2972. [Google Scholar]
  12. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics 19. American Mathematical Society, Rhode Island (1998). [Google Scholar]
  13. Z. Fang, H. Sun and H. Wang, A fast method for variable-order Caputo fractional derivative with applications to time-fractional diffusion equations. Comput. Math. Appl. 80 (2020) 1443–1458. [Google Scholar]
  14. N. Ford and M. Morgado, Distributed order equations as boundary value problems. Comput. Math. Appl. 64 (2012) 2973–2981. [Google Scholar]
  15. R. Gorenflo, Y. Luchko and Mirjana Stojanović, Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density. Fract. Calc. Appl. Anal. 16 (2013) 297–316. [Google Scholar]
  16. W. Hackbusch, Integral Equations: theory and Numerical Treatment. International series of numerical mathematics. Vol. 120, Birkhäuser Verlag, Basel (1995). [Google Scholar]
  17. J. Jia and H. Wang, A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains. Comput. Math. Appl. 73 (2018) 2031–2043. [Google Scholar]
  18. J. Jia, H. Wang and X. Zheng, A preconditioned fast finite element approximation to variable-order time-fractional diffusion equations in multiple space dimensions. Appl. Numer. Math. 163 (2021) 15–29. [Google Scholar]
  19. B. Jin, R. Lazarov, Z. Zhou, Two schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38 (2014) A146–A170. [Google Scholar]
  20. R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002). [Google Scholar]
  21. J. Li, F. Liu, L. Feng and I. Turner, A novel finite volume method for the Riesz space distributed-order advection-diffusion equation. Appl. Math. Model. 46 (2017) 536–553. [Google Scholar]
  22. B. Li, H. Luo and X. Xie, Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data. SIAM J. Numer. Anal. 57 (2019) 779–798. [Google Scholar]
  23. Z. Li, K. Fujishiro and G. Li, Uniqueness in the inversion of distributed orders in ultraslow diffusion equations. J. Comput. Appl. Math. 369 (2020) 112564. [Google Scholar]
  24. C. Lorenzo and T. Hartley, Variable order and distributed order fractional operators. Nonlinear Dyn. 29 (2002) 57–98. [Google Scholar]
  25. Y. Luchko and F. Francesco, Cauchy and signaling problems for the time-fractional diffusion-wave equation. J. Vib. Acoust. 136 (2014) 051008. [Google Scholar]
  26. R.L. Magin, H. Karani, S. Wang and Y. Liang, Fractional order complexity model of the diffusion signal decay in MRI. Mathematics 7 (2019) 348. [Google Scholar]
  27. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: an Introduction to Mathematical Models. World Scientific (2010). [Google Scholar]
  28. S. Mashayekhi and M. Razzaghi, Numerical solution of distributed order fractional differential equations by hybrid functions. J. Comput. Phys. 315 (2016) 169–181. [Google Scholar]
  29. W. McLean and K. Mustapha, A second-order accurate numerical method for a fractional wave equation. Numer. Math. 105 (2007) 481–510. [Google Scholar]
  30. M.M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus. De Gruyter Studies in Mathematics (2011). [Google Scholar]
  31. I. Podlubny, Fractional Differential Equations. Academic Press (1999). [Google Scholar]
  32. S. Patnaik and F. Semperlotti, Application of variable- and distributed-order fractional operators to the dynamic analysis of nonlinear oscillators. Nonlinear Dyn. 100 (2020) 561–580. [Google Scholar]
  33. K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J Math. Anal. Appl. 382 (2011) 426–447. [Google Scholar]
  34. J. Suzuki, Y. Zhou, M. D’Elia and M. Zayernouri, A thermodynamically consistent fractional visco-elasto-plastic model with memory-dependent damage for anomalous materials. Comput. Meth. Appl. Mech. Engrg. 373 (2021) 113494. [Google Scholar]
  35. M. Samiee, E. Kharazmi, M. Meerschaert, M. Zayernouri, A unified Petrov-Galerkin spectral method and fast solver for distributed-order partial differential equations. Commun. Appl. Math. Comput. 1 (2020) 1–30. [Google Scholar]
  36. T. Sandev, R. Metzler and A. Chechkin, From continuous time random walks to the generalized diffusion equation. Fract. Calc. Appl. Anal. 21 (2018) 10–28. [Google Scholar]
  37. P. Spanos and G. Malara, Nonlinear random vibrations of beams with fractional derivative elements. J. Eng. Mech. 140 (2014) 04014069. [Google Scholar]
  38. M. Stojanović and R. Gorenflo, Nonlinear two-term time fractional diffusion-wave problem. Nonlinear Anal-Real 11 (2010) 3512–3523. [Google Scholar]
  39. M. Stynes, E. O’Riordan and J.L. Gracia, Error analysis of a finite difference method on graded mesh for a time-fractional diffusion equation. SIAM J Numer. Anal. 55 (2017) 1057–1079. [Google Scholar]
  40. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics 1054. Springer-Verlag, New York (1984). [Google Scholar]
  41. H. Wang and X. Zheng, Wellposedness and regularity of the variable-order time-fractional diffusion equations. J. Math. Anal. Appl. 475 (2019) 1778–1802. [Google Scholar]
  42. X. Zheng and H. Wang, An error estimate of a numerical approximation to a hidden-memory variable-order space-time fractional diffusion equation. SIAM J. Numer. Anal. 58 (2020) 2492–2514. [Google Scholar]
  43. X. Zheng and H. Wang, The unique identification of variable-order fractional wave equations. Z. Angew. Math. Phys. 72 (2021) 100. [Google Scholar]
  44. X. Zheng and H. Wang, A hidden-memory variable-order fractional optimal control model: analysis and approximation. SIAM J. Control Optim. 59 (2021) 1851–1880. [Google Scholar]
  45. X. Zheng and H. Wang, Analysis and discretization of a variable-order fractional wave equation. Commun. Nonlinear Sci. 104 (2022) 106047. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you