Neural minimization methods (NMM) for solving variable order fractional delay differential equations (FDDEs) with simulated annealing (SA)

Shaikh, Amber; Jamal, Moshtagh; Hanif, Fozia; Khan, M. Sadiq Ali; Inayatullah, Syed

doi:10.1371/journal.pone.0223476

Cited by 2 publications

(2 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Subsequently, in Ref. [150] the authors found the numerical solution of delayed FDE based on the application of neural minimization using Chebyshev simulated annealing ANN and Legendre simulated annealing ANN. Chebyshev and Legendre polynomials were used with SA to reduce mean square error and get more accurate numerical approximations.…”

Section: Simulated Annealing Algorithm (Sa)mentioning

confidence: 99%

Artificial neural networks: a practical review of applications involving fractional calculus

Viera-Martin

Gómez-Aguilar

Solís-Pérez

et al. 2022

Eur. Phys. J. Spec. Top.

View full text Add to dashboard Cite

In this work, a bibliographic analysis on artificial neural networks (ANNs) using fractional calculus (FC) theory has been developed to summarize the main features and applications of the ANNs. ANN is a mathematical modeling tool used in several sciences and engineering fields. FC has been mainly applied on ANNs with three different objectives, such as systems stabilization, systems synchronization, and parameters training, using optimization algorithms. FC and some control strategies have been satisfactorily employed to attain the synchronization and stabilization of ANNs. To show this fact, in this manuscript are summarized, the architecture of the systems, the control strategies, and the fractional derivatives used in each research work, also, the achieved goals are presented. Regarding the parameters training using optimization algorithms issue, in this manuscript, the systems types, the fractional derivatives involved, and the optimization algorithm employed to train the ANN parameters are also presented. In most of the works found in the literature where ANNs and FC are involved, the authors focused on controlling the systems using synchronization and stabilization. Furthermore, recent applications of ANNs with FC in several fields such as medicine, cryptographic, image processing, robotic are reviewed in detail in this manuscript. Works with applications, such as chaos analysis, functions approximation, heat transfer process, periodicity, and dissipativity, also were included. Almost to the end of the paper, several future research topics arising on ANNs involved with FC are recommended to the researchers community. From the bibliographic review, we concluded that the Caputo derivative is the most utilized derivative for solving problems with ANNs because its initial values take the same form as the differential equations of integer-order.

show abstract

Section: Simulated Annealing Algorithm (Sa)mentioning

confidence: 99%

Artificial neural networks: a practical review of applications involving fractional calculus

Viera-Martin

Gómez-Aguilar

Solís-Pérez

et al. 2022

Eur. Phys. J. Spec. Top.

View full text Add to dashboard Cite

show abstract

“…In recent years, neural architecture-based approximation schemes have been used to solve FDEs, ODEs, PDEs, and delay differential equations (DDEs) [31][32][33][34][35][36][37][38]. In 2013, Lefik [39] illustrated that an ANN performs the numerical representation of the inverse relation.…”

Section: Introductionmentioning

confidence: 99%

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

Verma,

Sumelka,

Yadav

2023

Symmetry

View full text Add to dashboard Cite

This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods.

show abstract

Neural minimization methods (NMM) for solving variable order fractional delay differential equations (FDDEs) with simulated annealing (SA)

Cited by 2 publications

References 43 publications

Artificial neural networks: a practical review of applications involving fractional calculus

Artificial neural networks: a practical review of applications involving fractional calculus

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

Contact Info

Product

Resources

About