Neural network method for solving nonlinear fractional advection-diffusion equation with spatiotemporal variable-order

Hai-Dong Qu, Xuan Liu, Xin Lu, Mati ur Rahman and Zi-Hang She

Chaos, Solitons & Fractals, 2022, vol. 156, issue C

Abstract: In this article, neural network method (NNM) is presented to solve the spatiotemporal variable-order fractional advection-diffusion equation with a nonlinear source term. The network is established by using shifted Legendre orthogonal polynomials with adjustable coefficients. According to the properties of variable fractional derivative, the loss function of neural network is deduced theoretically. Assume that the source function satisfies the Lipschitz hypothesis, the reasonable range for learning rate is discussed in details. Then neural networks are trained repeatedly on the training set to reduce the loss functions for two numerical examples. Numerical results show that the neural network method is better than the finite difference method in solving some nonlinear variable fractional order problems. Finally, several graphs and some numerical analysis are given to confirm our conclusions.

Keywords: Variable-order fractional derivative; Neural network method; Fractional advection-diffusion equation (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (5)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:156:y:2022:i:c:s0960077922000674

DOI: 10.1016/j.chaos.2022.111856

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