Analysis of $L1$-Galerkin FEMs for time-fractional nonlinear parabolic problems

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of $L1$-Galerkin finite element methods. The analysis of $L1$ methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality. In this paper, we establish such a fundamental inequality for the $L1$ approximation to the Caputo fractional derivative. In terms of the Gronwall type inequality, we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems. The theoretical results are illustrated by applying our proposed methods to three examples: linear Fokker-Planck equation, nonlinear Huxley equation and Fisher equation.

[1]  Hai-wei Sun,et al.  Three-Point Combined Compact Alternating Direction Implicit Difference Schemes for Two-Dimensional Time-Fractional Advection-Diffusion Equations , 2015 .

[2]  S. S. Ezz-Eldien,et al.  A new Jacobi spectral collocation method for solving 1+1 fractional Schrödinger equations and fractional coupled Schrödinger systems , 2014 .

[3]  Weihua Deng,et al.  Numerical algorithm for the time fractional Fokker-Planck equation , 2007, J. Comput. Phys..

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

[5]  CHANG-MING CHEN,et al.  Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation , 2012, Math. Comput..

[6]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[7]  Weihua Deng,et al.  Positivity and boundedness preserving schemes for space-time fractional predator-prey reaction-diffusion model , 2015, Comput. Math. Appl..

[8]  Changpin Li,et al.  Finite difference methods with non-uniform meshes for nonlinear fractional differential equations , 2016, J. Comput. Phys..

[9]  Wanrong Cao,et al.  Time-Splitting Schemes for Fractional Differential Equations I: Smooth Solutions , 2015, SIAM J. Sci. Comput..

[10]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[11]  Zhi-Zhong Sun,et al.  A compact finite difference scheme for the fractional sub-diffusion equations , 2011, J. Comput. Phys..

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

[13]  H. Srivastava,et al.  THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS. NORTH-HOLLAND MATHEMATICS STUDIES , 2006 .

[14]  Jingtang Ma,et al.  High-order finite element methods for time-fractional partial differential equations , 2011, J. Comput. Appl. Math..

[15]  Junying Cao,et al.  A high order schema for the numerical solution of the fractional ordinary differential equations , 2013, J. Comput. Phys..

[16]  Changpin Li,et al.  On the fractional Adams method , 2009, Comput. Math. Appl..

[17]  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..

[18]  Fawang Liu,et al.  Stability and convergence of an implicit numerical method for the non-linear fractional reaction–subdiffusion process , 2009 .

[19]  Ali H. Bhrawy,et al.  A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations , 2015, J. Comput. Phys..

[20]  Jiwei Zhang,et al.  Efficient implementation to numerically solve the nonlinear time fractional parabolic problems on unbounded spatial domain , 2016, J. Comput. Phys..

[21]  Barkai,et al.  From continuous time random walks to the fractional fokker-planck equation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  Mostafa Abbaszadeh,et al.  The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics , 2013 .

[23]  William McLean,et al.  Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation , 2009, Numerical Algorithms.

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

[25]  Bangti Jin,et al.  ON TWO SCHEMES FOR FRACTIONAL DIFFUSION AND DIFFUSION-WAVE EQUATIONS , 2014 .

[26]  I. Podlubny.  Fractional differential equations , 1998 .

[27]  Tridip Sardar,et al.  Revisited Fisher’s equation in a new outlook: A fractional derivative approach , 2015 .

[28]  B. Henry,et al.  The accuracy and stability of an implicit solution method for the fractional diffusion equation , 2005 .

[29]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[30]  Santos B. Yuste,et al.  An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations , 2004, SIAM J. Numer. Anal..

[31]  Fawang Liu,et al.  The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation , 2013, SIAM J. Sci. Comput..

[32]  Mehmet Merdan,et al.  Solutions of time-fractional reaction-diffusion equation with modified Riemann-Liouville derivative , 2012 .

[33]  Chengjian Zhang,et al.  A linear finite difference scheme for generalized time fractional Burgers equation , 2016 .

[34]  Hong Wang,et al.  A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations , 2012, SIAM J. Sci. Comput..

[35]  Hong Wang,et al.  Inhomogeneous Dirichlet Boundary-Value Problems of Space-Fractional Diffusion Equations and their Finite Element Approximations , 2014, SIAM J. Numer. Anal..