• A high order numerical method for solving nonlinear fractional differential equation with non-uniform meshes

      Fan, Lili; Yan, Yubin; University of Chester; Lvliang University (Springer Link, 2019-01-18)
      We introduce a high-order numerical method for solving nonlinear fractional differential equation with non-uniform meshes. We first transform the fractional nonlinear differential equation into the equivalent Volterra integral equation. Then we approximate the integral by using the quadratic interpolation polynomials. On the first subinterval $[t_{0}, t_{1}]$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{0}, t_{1}, t_{2}$ and in the other subinterval $[t_{j}, t_{j+1}], j=1, 2, \dots N-1$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{j-1}, t_{j}, t_{j+1}$. A high-order numerical method is obtained. Then we apply this numerical method with the non-uniform meshes with the step size $\tau_{j}= t_{j+1}- t_{j}= (j+1) \mu$ where $\mu= \frac{2T}{N (N+1)}$. Numerical results show that this method with the non-uniform meshes has the higher convergence order than the standard numerical methods obtained by using the rectangle and the trapzoid rules with the same non-uniform meshes.
    • High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

      Li, Zhiqiang; Liang, Zongqi; Yan, Yubin; Luliang University, P. R. China, Jimei University, P. R. China, University of Chester, UK (Springer Link, 2016-11-15)
      In this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order O(τ^(3−α) +h^2 ),0
    • Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth Data

      Yan, yubin; Wang, Yanyong; Yan, Yuyuan; Pani, Amiya K.; University of Chester, Lvliang University, Jimei University, Indian Institute of Technology Bombay (Springer Link, 2020-05-19)
      Two higher order time stepping methods for solving subdiffusion problems are studied in this paper. The Caputo time fractional derivatives are approximated by using the weighted and shifted Gr\"unwald-Letnikov formulae introduced in Tian et al. [Math. Comp. 84 (2015), pp. 2703-2727]. After correcting a few starting steps, the proposed time stepping methods have the optimal convergence orders $O(k^2)$ and $ O(k^3)$, respectively for any fixed time $t$ for both smooth and nonsmooth data. The error estimates are proved by directly bounding the approximation errors of the kernel functions. Moreover, we also present briefly the applicabilities of our time stepping schemes to various other fractional evolution equations. Finally, some numerical examples are given to show that the numerical results are consistent with the proven theoretical results.
    • On the behavior of the solutions for linear autonomous mixed type difference equation

      Yan, Yubin; Yenicerioglu, Ali Fuat; Pinelas, Sandra; University of Chester; Kocaeli University, Turkey; RUDN University, Russia (Springer Link, 2019-07-30)
      A class of linear autonomous mixed type difference equations is considered, and some new results on the asymptotic behavior and the stability are given, via a positive root of the corresponding characteristic equation.