A posteriori error analysis for a fractional differential equation

References

• D. Baffet and J.S. Hesthaven, High-order accurate local schemes for fractional differential equations, J. Sci. Comput., First online: 3 September, doi:10.1007/s10915-015-0089-1.
   Web of Science ®Google Scholar
• G.-H. Cao, H.-W. Sun, and Z.-Z. Sun, Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence, J. Comput. Phys. 280 (2015), pp. 510–528. doi: 10.1016/j.jcp.2014.09.033
   Web of Science ®Google Scholar
• Y. Chen, Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid, Adv. Comput. Math. 24 (2006), pp. 197–212. doi: 10.1007/s10444-004-7641-0
   Web of Science ®Google Scholar
• P. Das, Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems, J. Comput. Appl. Math. 290 (2015), pp. 16–25. doi: 10.1016/j.cam.2015.04.034
   Web of Science ®Google Scholar
• K. Diethlm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics Vol. 2004, Springer, Berlin, 2010.
   Google Scholar
• M.R. Eslahchi, M. Dehghan, and M. Parvizi, Application of the collocation method for solving nonlinear fractional integro-differential equations, J. Comput. Appl. Math. 257 (2014), pp. 105–128. doi: 10.1016/j.cam.2013.07.044
   Web of Science ®Google Scholar
• G.J. Fix and J.P. Roof, Least squares finite-element solution of a fractional order two-point boundary value problem, Comput. Math. Appl. 48 (2004), pp. 1017–1033. doi: 10.1016/j.camwa.2004.10.003
   Web of Science ®Google Scholar
• J.L. Gracia and M. Stynes, Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems, J. Comput. Appl. Math. 273 (2015), pp. 103–115. doi: 10.1016/j.cam.2014.05.025
   Web of Science ®Google Scholar
• J. Jia and H. Wang, Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions, J. Comput. Phys. 293 (2015), pp. 359–369. doi: 10.1016/j.jcp.2014.08.021
   Web of Science ®Google Scholar
• A.A. Kilbas, H.M. Srivastaval, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies Vol. 204, Elsevier, Amsterdam, 2006.
   Google Scholar
• M. Kolk, A. Pedas, and E. Tamme, Modified spline collocation for linear fractional differential equations, J. Comput. Appl. Math. 283 (2015), pp. 28–40. doi: 10.1016/j.cam.2015.01.021
   Web of Science ®Google Scholar
• N. Kopteva, Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem, SIAM J. Numer. Anal. 39 (2001), pp. 423–441. doi: 10.1137/S0036142900368642
   Web of Science ®Google Scholar
• N. Kopteva, Maximum norm a posteriori error estimates for a 1D singularly perturbed semilinear reaction-diffusion problem, IMA J. Numer. Anal. 27 (2007), pp. 576–592. doi: 10.1093/imanum/drl020
   Web of Science ®Google Scholar
• N. Kopteva, Maximum norm a posteriori error estimate for a 2D singularly perturbed semilinear reaction-diffusion problem, SIAM J. Numer. Anal. 46 (2008), pp. 1602–1618. doi: 10.1137/060677616
   Web of Science ®Google Scholar
• T. Linss, Analysis of a system of singularly perturbed convection-diffusion equations with strong coupling, SIAM J. Numer. Anal. 47 (2009), pp. 1847–1862. doi: 10.1137/070683970
   Web of Science ®Google Scholar
• T. Linss, A posterior error estimation for a singularly perturbed problem with two small parameters, Int. J. Numer. Anal. Model. 7 (2010), pp. 491–506.
   Web of Science ®Google Scholar
• 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), pp. 12–20.
   Web of Science ®Google Scholar
• Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl. 351 (2009), pp. 218–223. doi: 10.1016/j.jmaa.2008.10.018
   Web of Science ®Google Scholar
• X. Ma and C. Huang, Spectral collocation method for linear fractional integro-differential equations, Appl. Math. Model. 38 (2014), pp. 1434–1448. doi: 10.1016/j.apm.2013.08.013
   Web of Science ®Google Scholar
• W. McLean and K. Mustapha, Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation, Numer. Algorithms 52 (2009), pp. 69–88. doi: 10.1007/s11075-008-9258-8
   Web of Science ®Google Scholar
• M.M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (2006), pp. 80–90. doi: 10.1016/j.apnum.2005.02.008
   Web of Science ®Google Scholar
• S. Momani and Z. Odibat, Numerical solutions of the space-time fractional advection-dispersion equation, Numer. Methods Partial Differential Equations 24 (2008), pp. 1416–1429. doi: 10.1002/num.20324
   Web of Science ®Google Scholar
• D.A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl. 56 (2008), pp. 1138–1145. doi: 10.1016/j.camwa.2008.02.015
   Web of Science ®Google Scholar
• K. Mustapha and W. McLean, Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation, Numer. Algorithms 56 (2011), pp. 159–184. doi: 10.1007/s11075-010-9379-8
   Web of Science ®Google Scholar
• H.K. Pang and H.W. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys. 231 (2012), pp. 693–703. doi: 10.1016/j.jcp.2011.10.005
   Web of Science ®Google Scholar
• A. Pedas and E. Tamme, Piecewise polynomial collocation for linear boundary value problems of fractional differential equations, J. Comput. Appl. Math. 236 (2012), pp. 3349–3359. doi: 10.1016/j.cam.2012.03.002
   Web of Science ®Google Scholar
• A. Pedas and E. Tamme, Numerical solution of nonlinear fractional differential equations by spline collocation methods, J. Comput. Appl. Math. 255 (2014), pp. 216–230. doi: 10.1016/j.cam.2013.04.049
   Web of Science ®Google Scholar
• A. Pedas and E. Tamme, Spline collocation for nonlinear fractional boundary value problems, Appl. Math. Comput. 244 (2014), pp. 502–513.
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
   Google Scholar
• M. Stynes and J.L. Gracia, A finite difference method for a two-point boundary value problem with a Caputo fractional derivative, IMA J. Numer. Anal. 35 (2015), pp. 689–721. doi: 10.1093/imanum/dru011
   Web of Science ®Google Scholar
• M. Stynes and J.L. Gracia, Boundary layers in a two-point boundary value problem with a Caputo fractional derivative, Comput. Methods Appl. Math. 15 (2015), pp. 79–95. doi: 10.1515/cmam-2014-0024
   Web of Science ®Google Scholar
• L.-N. Sun, Y.-J. Wu, and A.-L. Yang, Uniform convergence analysis of finite difference approximations for advection-reaction-diffusion problem on adaptive grids, Int. J. Comput. Math. 88 (2011), pp. 3292–3307. doi: 10.1080/00207160.2011.591928
   Web of Science ®Google Scholar
• C. Tadjeran, M.M. Meerschaert, and H.P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006), pp. 205–213. doi: 10.1016/j.jcp.2005.08.008
   Web of Science ®Google Scholar
• Y. Zhang, A finite difference method for fractional partial differential equation, Appl. Math. Comput. 215 (2009), pp. 524–529.
   Web of Science ®Google Scholar
• P. Zhuang, F. Liu, V. Anh, and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal. 47 (2009), pp. 1760–1781. doi: 10.1137/080730597
   Web of Science ®Google Scholar