Multi-term normalized time-fractional diffusion equations

Yongho Choi; Seungjae Lee; Junseok Kim; Yongho Choi; Seungjae Lee; Junseok Kim

doi:10.3934/era.2026297

Electronic Research Archive

2026, Volume 34, Issue 9: 6825-6845. doi: 10.3934/era.2026297

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Research article

Multi-term normalized time-fractional diffusion equations

1.
Department of Computer & Information Engineering, Daegu University, Gyeongsan-si, Gyeongsangbuk-do 38453, Republic of Korea
2.
Department of Mathematics, Korea University, Seoul 02841, Republic of Korea

Received: 23 June 2026 Revised: 23 July 2026 Accepted: 05 August 2026 Published: 12 August 2026

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-\alpha_{\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.
- multi-term normalized time-fractional,
- numerical method,
- Caputo derivative
Citation: Yongho Choi, Seungjae Lee, Junseok Kim. Multi-term normalized time-fractional diffusion equations[J]. Electronic Research Archive, 2026, 34(9): 6825-6845. doi: 10.3934/era.2026297

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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-\alpha_{\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.

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