Numerical investigation of the dynamics for a normalized time-fractional diffusion equation

Chaeyoung Lee; Yunjae Nam; Minjoon Bang; Seokjun Ham; Junseok Kim; Chaeyoung Lee; Yunjae Nam; Minjoon Bang; Seokjun Ham; Junseok Kim

doi:10.3934/math.20241297

AIMS Mathematics

2024, Volume 9, Issue 10: 26671-26687. doi: 10.3934/math.20241297

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Numerical investigation of the dynamics for a normalized time-fractional diffusion equation

1.
Department of Mathematics, Kyonggi University, Suwon, 16227, Republic of Korea
2.
Program in Actuarial Science and Financial Engineering, Korea University, Seoul 02841, Republic of Korea
3.
Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea

Received: 05 July 2024 Revised: 02 September 2024 Accepted: 11 September 2024 Published: 13 September 2024
MSC : 35R11, 80M20, 39A14

In this study, we proposed a normalized time-fractional diffusion equation and conducted a numerical investigation of the dynamics of the proposed equation. We discretized the governing equation by using a finite difference method. The proposed normalized time-fractional diffusion equation features a different time scale compared to the conventional time-fractional diffusion equation. This distinct time scale provides an intuitive understanding of the fractional time derivative, which represents a weighted average of the temporal history of the time derivative. Furthermore, the sum of the weight function is one for all values of the fractional parameter and time. The primary advantage of the proposed model over conventional time-fractional equations is the unity property of the sum of the weight function, which allows us to investigate the effects of the fractional order on the evolutionary dynamics of time-fractional equations. To highlight the differences in performance between the conventional and normalized time-fractional diffusion equations, we have conducted several numerical experiments.
- normalized time-fractional diffusion equation,
- finite difference method,
- Gamma function
Citation: Chaeyoung Lee, Yunjae Nam, Minjoon Bang, Seokjun Ham, Junseok Kim. Numerical investigation of the dynamics for a normalized time-fractional diffusion equation[J]. AIMS Mathematics, 2024, 9(10): 26671-26687. doi: 10.3934/math.20241297

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Abstract

In this study, we proposed a normalized time-fractional diffusion equation and conducted a numerical investigation of the dynamics of the proposed equation. We discretized the governing equation by using a finite difference method. The proposed normalized time-fractional diffusion equation features a different time scale compared to the conventional time-fractional diffusion equation. This distinct time scale provides an intuitive understanding of the fractional time derivative, which represents a weighted average of the temporal history of the time derivative. Furthermore, the sum of the weight function is one for all values of the fractional parameter and time. The primary advantage of the proposed model over conventional time-fractional equations is the unity property of the sum of the weight function, which allows us to investigate the effects of the fractional order on the evolutionary dynamics of time-fractional equations. To highlight the differences in performance between the conventional and normalized time-fractional diffusion equations, we have conducted several numerical experiments.

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