Multi-term normalized time-fractional diffusion equations

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Abstract

This paper proposes a new multi-term normalized time-fractional diffusion equation. The proposed model extends the normalized time-fractional derivative to include several fractional orders. It keeps the normalization property and uses weighting coefficients to control short- and long-term memory effects. An implicit finite difference method is developed, and the tridiagonal linear system is solved by the Thomas algorithm. The proposed algorithm is unconditionally stable and converges with a temporal accuracy of 2 − α_max and second-order spatial accuracy. Numerical convergence tests agree with the theoretical results. Numerical examples show that the fractional orders and weighting coefficients control the diffusion rate, memory strength, and frequency-dependent decrease. Additionally, the computational results show that the proposed model is more flexible than the single-term normalized equation for describing anomalous diffusion with multiple memory scales. Finally, a simple two-dimensional extension of the proposed model is presented. MATLAB codes for both the single-term and multi-term normalized time-fractional diffusion models are provided in the Appendix.

Keywords

multi-term normalized time-fractional / numerical method / Caputo derivative

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