Fractional-Order Chelyshkov Collocation Method for Solving Systems of Fractional Differential Equations

Ahmed, A.; Al-Ahmary, T. A.

doi:10.1155/2022/4862650

Cited by 4 publications

(6 citation statements)

References 44 publications

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“…ht contains the input x t , ht is added to the hidden state, Θ is the hadamard product of the matrix [29,30], and then the update memory phase is expressed as follows.…”

Section: Grumentioning

confidence: 99%

Prediction of anaerobic digestion performance by quantum convolutional reconstruction gated recurrent neural network*

Hou,

Che,

et al. 2024

Phys. Scr.

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Methane as a renewable energy source has become a hot topic in recent years. Methane is a bioenergy source produced during the anaerobic digestion of organic waste, and the anaerobic digestion process must be monitored and controlled to produce the required amount of methane in a stable manner. Mathematical modeling is used to simulate digester operation to predict the biogas production from anaerobic digestion, to avoid reactor loading or performance degradation, and to ensure efficient operation of the system. In this paper, a Quantum Convolutional Reconstruction Gated Recurrent Neural Network is proposed. The original data features are extracted by convolutional neural network to reduce the dimensionality and retain the information, the parameterized quantum circuit is integrated in the gating recurrent unit, and the quantum reset gate and quantum update gate are constructed. The information extracted by the Convolution Neural networks is input into the quantum gated recurrent neural network, and the quantum storage unit integrates the information into the hidden layer state, thus processing the hidden layer state information more efficiently. The experimental results show that the prediction accuracy of the A Quantum Convolution Reconstructed Gated Recurrent Neural Network is improved from 81.95% to 88.21%, and the MAE value is reduced from 54.53% to 37.38%.

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“…ht contains the input x t , ht is added to the hidden state, Θ is the hadamard product of the matrix [29,30], and then the update memory phase is expressed as follows.…”

Section: Grumentioning

confidence: 99%

Prediction of anaerobic digestion performance by quantum convolutional reconstruction gated recurrent neural network*

Hou,

Che,

et al. 2024

Phys. Scr.

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show abstract

“…In this section, inspired by Ahmed, and Al-Ahmary [4] and Ahmed et al [5], we intend to define the FSCPs on the arbitrary interval [a, b]. To do this, we consider the mapping κ α : [a, b] −→ [0, 1] as:…”

Section: Fscps On the Interval [A B]mentioning

confidence: 99%

“…, N continuous and differentiable over the entire domain such that x, W i (x), W i x (x), W i xx (x) satisfy the equations system (6) at every point on the domain. Considering the solution of the problem (4)…”

mentioning

confidence: 99%

“…Also, the fractional-order hybrid of block-pulse functions and Bernoulli polynomials in Postavaru, and Toma [45] was considered for solving two-dimensional fractional order equations. Fractional-order Chelyshkov collocation method (FCCM) was applied for solving systems of fractional differential equations in [4]. This method was also used for finding numerical solution of fractional variational and optimal control problems in [5].…”

mentioning

confidence: 99%

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Generalized model of stochastic common property fishery differential game: Numerical study

Nikooeinejad,

Heydari

2024

Preprint

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In a generalized model of stochastic common property differential game problem, we have a more flexible framework that extends the model to handle the natural growth, price and cost functions for a wide range of parameters. Unlike previous studies that appeared in the literature, instead of keeping the model as parsimonious as possible, we consider the nonlinear resource stock dynamics and nonquadratic payoff functions, focusing on monopoly, duopoly and oligopoly games with a feedback information structure. It is observed that the choice of model parameters has an important influence on the nature and analytical behavior of solution of the nonlinear HJB PDEs (ODEs) and consequently on Nash equilibrium strategies for finite- (infinite-) horizon planning. We can not derive a closed-form solution for this class of generalized models. Hence, this paper presents a collocation method based on the fractional-order shifted Chelyshkov polynomials to solve the nonlinear Hamilton-Jacobi-Bellman PDEs (ODEs) in the generalized stochastic common property differential game problem with finite- (infinite-) horizon. Numerical results show that the choice of different parameters in the approximate solutions based on the fractional-order shifted Chelyshkov polynomials greatly affect the performance of the collocation method. Also, the results show that the firms' optimal extraction plans are strongly influenced by the natural growth, price and cost functions with respect to different parameters.

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“…In the work [28], the asymptotic stability is studied via linearization, and in [29,30], for retarded and neutral systems with distributed delays, respectively, the preservation of the asymptotic stability of linear systems is studied under nonlinear perturbation. For numerical aspects, we refer to [31,32].…”

Section: Introductionmentioning

confidence: 99%

On Stability Criteria Induced by the Resolvent Kernel for a Fractional Neutral Linear System with Distributed Delays

2023

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We consider an initial problem (IP) for a linear neutral system with distributed delays and derivatives in Caputo’s sense of incommensurate order, with different kinds of initial functions. In the case when the initial functions are with bounded variation, it is proven that this IP has a unique solution. The Krasnoselskii’s fixed point theorem, a very appropriate tool, is used to prove the existence of solutions in the case of the neutral systems. As a corollary of this result, we obtain the existence and uniqueness of a fundamental matrix for the homogeneous system. In the general case, without additional assumptions of boundedness type, it is established that the existence and uniqueness of a fundamental matrix lead existence and uniqueness of a resolvent kernel and vice versa. Furthermore, an explicit formula describing the relationship between the fundamental matrix and the resolvent kernel is proven in the general case too. On the base of the existence and uniqueness of a resolvent kernel, necessary and sufficient conditions for the stability of the zero solution of the homogeneous system are established. Finally, it is considered a well-known economics model to describe the dynamics of the wealth of nations and comment on the possibilities of application of the obtained results for the considered systems, which include as partial case the considered model.

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Fractional-Order Chelyshkov Collocation Method for Solving Systems of Fractional Differential Equations

Cited by 4 publications

References 44 publications

Prediction of anaerobic digestion performance by quantum convolutional reconstruction gated recurrent neural network*

Prediction of anaerobic digestion performance by quantum convolutional reconstruction gated recurrent neural network*

Generalized model of stochastic common property fishery differential game: Numerical study

On Stability Criteria Induced by the Resolvent Kernel for a Fractional Neutral Linear System with Distributed Delays

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