Convergence of solutions of discrete semi-linear space-time fractional evolution equations

10/16/2019

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by   Harbir Antil, et al.

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Let (-Δ)_c^s be the realization of the fractional Laplace operator on the space of continuous functions C_0(R), and let (-Δ_h)^s denote the discrete fractional Laplacian on C_0(Z_h), where 0<s<1 and Z_h:={hj: j∈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 (-Δ_h)^s on C_0(Z_h) converge to solutions of the corresponding Cauchy problems associated with the continuous operator (-Δ)_c^s. In addition, we obtain that the convergence is uniform in t in compact subsets of [0,∞). We also provide numerical simulations that support our theoretical results.