A Discrete Grönwall Inequality with Applications to Numerical Schemes for Subdiffusion Problems

We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the beh...

[1]  Hermann Brunner,et al.  The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes , 1985 .

[2]  Zhi‐zhong Sun,et al.  A fully discrete difference scheme for a diffusion-wave system , 2006 .

[3]  Jiwei Zhang,et al.  A Second-Order Scheme with Nonuniform Time Steps for a Linear Reaction-Subdiffusion Problem , 2018, Communications in Computational Physics.

[4]  V. Thomée Galerkin Finite Element Methods for Parabolic Problems, Second Edition , 2006 .

[5]  Ivan G. Graham,et al.  Galerkin methods for second kind integral equations with singularities , 1982 .

[6]  Chuanju Xu,et al.  Error Analysis of a High Order Method for Time-Fractional Diffusion Equations , 2016, SIAM J. Sci. Comput..

[7]  Yubin Yan,et al.  An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data , 2018, SIAM J. Numer. Anal..

[8]  Masahiro Yamamoto,et al.  Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems , 2011 .

[9]  Jose L. Gracia,et al.  Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation , 2017, SIAM J. Numer. Anal..

[10]  A. Alikhanov A priori estimates for solutions of boundary value problems for fractional-order equations , 2010, 1105.4592.

[11]  Etienne Emmrich,et al.  Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator , 2009 .

[12]  Jiwei Zhang,et al.  Unconditional Convergence of a Fast Two-Level Linearized Algorithm for Semilinear Subdiffusion Equations , 2018, Journal of Scientific Computing.

[13]  Anatoly A. Alikhanov,et al.  A new difference scheme for the time fractional diffusion equation , 2014, J. Comput. Phys..

[14]  William McLean,et al.  A second-order accurate numerical method for a fractional wave equation , 2006, Numerische Mathematik.

[15]  R. Jackson Inequalities , 2007, Algebra for Parents.

[16]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[17]  Bangti Jin,et al.  An analysis of the L1 Scheme for the subdiffusion equation with nonsmooth data , 2015, 1501.00253.

[18]  Martin Stynes,et al.  Too much regularity may force too much uniqueness , 2016, 1607.01955.

[19]  Ying Zhao,et al.  A Weighted ADI Scheme for Subdiffusion Equations , 2016, J. Sci. Comput..

[20]  Ying Zhao,et al.  Stability of fully discrete schemes with interpolation-type fractional formulas for distributed-order subdiffusion equations , 2017, Numerical Algorithms.

[21]  Joakim Becker,et al.  A second order backward difference method with variable steps for a parabolic problem , 1998 .

[22]  Bangti Jin,et al.  Numerical Analysis of Nonlinear Subdiffusion Equations , 2017, SIAM J. Numer. Anal..

[23]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[24]  Zhi-Zhong Sun,et al.  A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications , 2014, J. Comput. Phys..