• Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes

      Green, Charles Wing Ho; email: 1604518@chester.ac.uk; Liu, Yanzhi; email: 39036@llu.edu.cn; Yan, Yubin; orcid: 0000-0002-5686-5017; email: y.yan@chester.ac.uk (MDPI, 2021-10-27)
      We consider the predictor-corrector numerical methods for solving Caputo–Hadamard fractional differential equations with the graded meshes logtj=loga+logtNajNr, j=0, 1, 2, …, N with a≥1 and r≥1, where loga=logt0logt1⋯logtN=logT is a partition of [logt0, logT]. We also consider the rectangular and trapezoidal methods for solving Caputo–Hadamard fractional differential equations with the non-uniform meshes logtj=loga+logtNaj(j+1)N(N+1), j=0, 1, 2, …, N. Under the weak smoothness assumptions of the Caputo–Hadamard fractional derivative, e.g., DCHa, tαy(t)∉C1[a, T] with α∈(0, 2), the optimal convergence orders of the proposed numerical methods are obtained by choosing the suitable graded mesh ratio r≥1. The numerical examples are given to show that the numerical results are consistent with the theoretical findings.
    • Spatial Discretization for Stochastic Semi-Linear Subdiffusion Equations Driven by Fractionally Integrated Multiplicative Space-Time White Noise

      Wang, Junmei; orcid: ; email: 39041@llhc.edu.cn; Hoult, James; orcid: ; email: 1405193@chester.ac.uk; Yan, Yubin; orcid: 0000-0002-5686-5017; email: y.yan@chester.ac.uk (MDPI, 2021-08-12)
      Spatial discretization of the stochastic semi-linear subdiffusion equations driven by fractionally integrated multiplicative space-time white noise is considered. The nonlinear terms f and σ satisfy the global Lipschitz conditions and the linear growth conditions. The space derivative and the fractionally integrated multiplicative space-time white noise are discretized by using the finite difference methods. Based on the approximations of the Green functions expressed by the Mittag–Leffler functions, the optimal spatial convergence rates of the proposed numerical method are proved uniformly in space under some suitable smoothness assumptions of the initial value.