Numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method

We study a numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method. Due to the lack of the monotonicity of the discretization coefficients of the variable-order fractional derivative in standard approximation schemes, existing numerical analysis techniques do not apply directly. By an approximate inversion technique, the proposed model is transformed as a second kind Volterra integral equation, based on which a collocation method under uniform or graded mesh is developed and analyzed. In particular, the error estimates improve the existing results by proving a consistent and sharper mesh grading parameter and characterizing the convergence rates in terms of the initial value of the variable order, which demonstrates its critical role in determining the smoothness of the solutions and thus the numerical accuracy.

[1]  G. Burton.  Sobolev Spaces , 2013 .

[2]  Hai-Wei Sun,et al.  Exponential-sum-approximation technique for variable-order time-fractional diffusion equations , 2021, Journal of Applied Mathematics and Computing.

[3]  Xiangcheng Zheng,et al.  An Optimal-Order Numerical Approximation to Variable-order Space-fractional Diffusion Equations on Uniform or Graded Meshes , 2020, SIAM J. Numer. Anal..

[4]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[5]  Xiangcheng Zheng,et al.  Approximate inversion for Abel integral operators of variable exponent and applications to fractional Cauchy problems , 2021, Fractional Calculus and Applied Analysis.

[6]  Xiangcheng Zheng,et al.  An Error Estimate of a Numerical Approximation to a Hidden-Memory Variable-Order Space-Time Fractional Diffusion Equation , 2020, SIAM J. Numer. Anal..

[7]  LinYumin,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007 .

[8]  Stefan Samko,et al.  Fractional integration and differentiation of variable order: an overview , 2012, Nonlinear Dynamics.

[9]  Roberto Garrappa,et al.  Variable-order fractional calculus: A change of perspective , 2021, Commun. Nonlinear Sci. Numer. Simul..

[10]  Fawang Liu,et al.  Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term , 2009, SIAM J. Numer. Anal..

[11]  Hu Sheng,et al.  On mean square displacement behaviors of anomalous diffusions with variable and random orders , 2010 .

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

[13]  Hongguang Sun,et al.  A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications , 2019, Fractional Calculus and Applied Analysis.

[14]  高等学校計算数学学報編輯委員会編.  高等学校計算数学学報 = Numerical mathematics , 1979 .

[15]  Xuan Zhao,et al.  Second-order approximations for variable order fractional derivatives: Algorithms and applications , 2015, J. Comput. Phys..

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

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

[18]  Carl F. Lorenzo,et al.  Variable Order and Distributed Order Fractional Operators , 2002 .

[19]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

[20]  Zhongqiang Zhang,et al.  A Generalized Spectral Collocation Method with Tunable Accuracy for Variable-Order Fractional Differential Equations , 2015, SIAM J. Sci. Comput..

[21]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[22]  Xiangcheng Zheng,et al.  Wellposedness and regularity of the variable-order time-fractional diffusion equations , 2019, Journal of Mathematical Analysis and Applications.

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

[24]  B. Ross,et al.  Integration and differentiation to a variable fractional order , 1993 .