| Citation: |
Ahmed Bchatnia, Abderrahmane Beniani, Boumediene Boukhari, Foued Mtiri. THEORETICAL AND NUMERICAL STABILITY OF THE BRESSE SYSTEM: EXPLORING FRACTIONAL DAMPING THROUGH TRADITIONAL AND NEURAL NETWORK APPROACHES[J]. Journal of Applied Analysis & Computation, 2025, 15(5): 2663-2694. doi: 10.11948/20240410
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THEORETICAL AND NUMERICAL STABILITY OF THE BRESSE SYSTEM: EXPLORING FRACTIONAL DAMPING THROUGH TRADITIONAL AND NEURAL NETWORK APPROACHES
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Abstract
This paper investigates the theoretical and numerical stability of the one-dimensional Bresse system with fractional damping terms in a bounded domain. We first establish the well-posedness of the system. Using the frequency domain approach and a theorem by Borichev and Tomilov, we derive the polynomial decay rate of the system. To validate these theoretical results, we propose a numerical scheme and compare its performance with the Fractional Physics-Informed Neural Network (fPINN). The comparative analysis highlights the effectiveness of traditional numerical methods and fPINNs in capturing the decay rate, offering new insights into the advancement of computational techniques for complex physical systems.
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References
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Ahmed Bchatnia, Abderrahmane Beniani, Boumediene Boukhari, Foued Mtiri. THEORETICAL AND NUMERICAL STABILITY OF THE BRESSE SYSTEM: EXPLORING FRACTIONAL DAMPING THROUGH TRADITIONAL AND NEURAL NETWORK APPROACHES[J]. Journal of Applied Analysis & Computation, 2025, 15(5): 2663-2694. doi: 10.11948/20240410
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Figure 1.
Piecewise Linear Interpolation Functions
$ h_i(x) $ $ (x_0, x_m) $ -
Figure 2.
Mesh of the domain
$ [0, L] \times [0, T] $ $ (x_i, t_j) $ -
Figure 3.
fPINNs to solve the problem (5) for calculate energy E(t).
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Figure 4.
Point cloud used for training the PINN and calculating the fractional derivative for each point
$ (x_i, t_j) $ -
Figure 5.
Energy by FEM for
$ T=10 $ -
Figure 6.
Energy by fPINN for
$ T=10 $ -
Figure 7.
Log Energy/t by FEM for
$ T=10 $ -
Figure 8.
Log Energy/t by fPINN for
$ T=10 $