Convergence of solutions of discrete semi-linear space-time fractional evolution equations. arXiv:1910.07358
Preprint, arXiv:1910.07358 [math.AP] (2019).
Summary: Let \((-\Delta)_c^s\) be the realization of the fractional Laplace operator on the space of continuous functions \(C_0(\mathbb{R})\), and let \((-\Delta_h)^s\) denote the discrete fractional Laplacian on \(C_0(\mathbb{Z}_h)\), where \(0<s<1\) and \(\mathbb{Z}_h:=\{hj:\; j\in\mathbb{Z}\}\) is a mesh of fixed size \(h>0\). We show that solutions of fractional order semi-linear Cauchy problems associated with the discrete operator \((-\Delta_h)^s\) on \(C_0(\mathbb{Z}_h)\) converge to solutions of the corresponding Cauchy problems associated with the continuous operator \((-\Delta)_c^s\). In addition, we obtain that the convergence is uniform in \(t\) in compact subsets of \([0,\infty)\). We also provide numerical simulations that support our theoretical results.
MSC:
| 35R11 | Fractional partial differential equations |
| 47D07 | Markov semigroups and applications to diffusion processes |
| 37L05 | General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations |
| 49M25 | Discrete approximations in optimal control |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
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