An artificial neural network approach for a class of time-fractional diffusion and diffusion-wave equations

Yinlin Ye; Hongtao Fan; Yajing Li; Ao Huang; Weiheng He; Yinlin Ye; Hongtao Fan; Yajing Li; Ao Huang; Weiheng He

doi:10.3934/nhm.2023047

Networks and Heterogeneous Media

2023, Volume 18, Issue 3: 1083-1104. doi: 10.3934/nhm.2023047

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An artificial neural network approach for a class of time-fractional diffusion and diffusion-wave equations

College of Science, Northwest A & F University, Yangling, Shanxi 712100, China

Received: 13 October 2022 Revised: 12 March 2023 Accepted: 13 March 2023 Published: 30 March 2023

In this paper, the artificial neural network method is applied to solve the time-fractional diffusion and diffusion-wave equations. This method combines Taylor series and neural network method, and uses the terms of different power terms of Taylor series as neurons. An error function is given to update the weights of the proposed neural network. In addition, in order to balance the contributions of different error terms in the error function, we propose an adaptive weight adjustment method. In the end, four numerical examples are given to demonstrate the effectiveness of proposed method in solving the time-fractional diffusion and diffusion-wave equations.
- fractional diffusion equation,
- fractional diffusion-wave equation,
- artificial neural network method,
- Taylor series,
- adaptive weight adjustment
Citation: Yinlin Ye, Hongtao Fan, Yajing Li, Ao Huang, Weiheng He. An artificial neural network approach for a class of time-fractional diffusion and diffusion-wave equations[J]. Networks and Heterogeneous Media, 2023, 18(3): 1083-1104. doi: 10.3934/nhm.2023047

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Abstract

In this paper, the artificial neural network method is applied to solve the time-fractional diffusion and diffusion-wave equations. This method combines Taylor series and neural network method, and uses the terms of different power terms of Taylor series as neurons. An error function is given to update the weights of the proposed neural network. In addition, in order to balance the contributions of different error terms in the error function, we propose an adaptive weight adjustment method. In the end, four numerical examples are given to demonstrate the effectiveness of proposed method in solving the time-fractional diffusion and diffusion-wave equations.

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