Numerical investigation and deep learning approach for fractal–fractional order dynamics of Hopfield neural network model

Avcı, İbrahim; Lort, Hüseyin; Tatlıcıoğlu, Buğce E.

doi:10.1016/j.chaos.2023.114302

Cited by 6 publications

(1 citation statement)

References 27 publications

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“…The history of this topic can be found in [1][2][3][4][5]. The interdisciplinary applications of fractional calculus span an impressive array of fields, expanding beyond bioengineering, biology, chaotic systems, control theory, economics, electrochemistry, finance, quantum mechanics, optics, oncology, physics, rheology, social sciences, viscoelasticity, and so on [6][7][8][9][10][11][12][13][14][15][16]. This expansive scope underscores the versatility and profound impact of fractional calculus in addressing intricate challenges across diverse scientific and engineering domains.…”

Section: Introductionmentioning

confidence: 99%

Spectral collocation with generalized Laguerre operational matrix for numerical solutions of fractional electrical circuit models

Avcı

2024

Mathematical Modelling and Numerical Simulation With Applications

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In this paper, we introduce a pioneering numerical technique that combines generalized Laguerre polynomials with an operational matrix of fractional integration to address fractional models in electrical circuits. Specifically focusing on Resistor-Inductor ($RL$), Resistor-Capacitor ($RC$), Resonant (Inductor-Capacitor) ($LC$), and Resistor-Inductor-Capacitor ($RLC$) circuits within the framework of the Caputo derivative, our approach aims to enhance the accuracy of numerical solutions. We meticulously construct an operational matrix of fractional integration tailored to the generalized Laguerre basis vector, facilitating a transformation of the original fractional differential equations into a system of linear algebraic equations. By solving this system, we obtain a highly accurate approximate solution for the electrical circuit model under consideration. To validate the precision of our proposed method, we conduct a thorough comparative analysis, benchmarking our results against alternative numerical techniques reported in the literature and exact solutions where available. The numerical examples presented in our study substantiate the superior accuracy and reliability of our generalized Laguerre-enhanced operational matrix collocation method in effectively solving fractional electrical circuit models.

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Section: Introductionmentioning

confidence: 99%

Spectral collocation with generalized Laguerre operational matrix for numerical solutions of fractional electrical circuit models

Avcı

2024

Mathematical Modelling and Numerical Simulation With Applications

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Dynamic analysis of a class of fractional‐order dry friction oscillators

Si,

Xie,

Zhao

et al. 2024

Math Methods in App Sciences

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This article investigates a class of Duffing nonlinear dynamic systems with fractional‐order dry friction and conducts in‐depth research on the stability, chaotic characteristics, and erosion of the safety basin of this system; the results are verified through numerical simulation. First, the average method is used to approximate the amplitude–frequency relationship of the system, and the accuracy of the analytical results is verified through numerical experiments. Second, the Melnikov method is used to obtain the conditions for the system to enter chaos in the Smale horseshoe sense, and the Melnikov curve is drawn for further verification. Then, bifurcation diagrams are drawn for the changes in various parameters in the system, with a focus on analyzing the influence of friction factors on chaotic bifurcation. By applying the definition and calculation principle of the maximum Lyapunov exponent, and drawing and utilizing the maximum Lyapunov exponent graph, the chaotic state that the system enters under different parameters is more clearly defined. Finally, the evolution law of the safety basin under various parameter changes, especially dry friction changes, is analyzed, and the erosion and bifurcation mechanism of the safety basin is studied. Comparing with the bifurcation diagram, it reveals that chaos primarily contributes to the erosion of the safety basin.

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A Physics informed neural network approach for solving time fractional Black-Scholes partial differential equations

Nuugulu,

Patidar,

Tarla

2024

Optim Eng

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We present a novel approach for solving time fractional Black-Scholes partial differential equations (tfBSPDEs) using Physics Informed Neural Network (PINN) approach. Traditional numerical methods are faced with challenges in solving fractional PDEs due to the non-locality and non-differentiability nature of fractional derivative operators. By leveraging the ideas of Riemann sums and the refinement of tagged partitions of the time domain, we show that fractional derivatives can directly be incorporated into the loss function when applying the PINN approach to solving tfBSPDEs. The approach allows for the simultaneous learning of the underlying process dynamics and the involved fractional derivative operator without a need for the use of numerical discretization of the fractional derivatives. Through some numerical experiments, we demonstrate that, the PINN approach is efficient, accurate and computationally inexpensive particularly when dealing with high frequency and noisy data. This work augments the understanding between advanced mathematical modeling and machine learning techniques, contributing to the body of knowlege on the advancement of accurate derivative pricing models.

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Numerical investigation and deep learning approach for fractal–fractional order dynamics of Hopfield neural network model

Cited by 6 publications

References 27 publications

Spectral collocation with generalized Laguerre operational matrix for numerical solutions of fractional electrical circuit models

Spectral collocation with generalized Laguerre operational matrix for numerical solutions of fractional electrical circuit models

Dynamic analysis of a class of fractional‐order dry friction oscillators

A Physics informed neural network approach for solving time fractional Black-Scholes partial differential equations

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