Normalized time-fractional diffusion equation on curved surfaces

초록

In this paper, we study a normalized time-fractional diffusion equation on curved surfaces and propose a numerical method on triangulated meshes for its approximation. The Laplace-Beltrami operator is discretized by a cotangent-type formula on surface vertices, and the normalized time-fractional derivative is approximated by a history-dependent weighted scheme. Based on these discretizations, we derive a fully discrete explicit update formula and apply it to diffusion problems on spherical surfaces. Numerical experiments show that the proposed method captures the effect of the fractional order on surface diffusion and produces results consistent with the classical case when the order approaches one. For long-time simulations, an exponential-sum fast-history strategy is applied to lower the cost of the temporal nonlocal term.

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