Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks

Zúñiga-Aguilar, C. J.; Ugalde, Hector M. Romero; Gómez-Aguilar, J. F.; Escobar-Jiménez, R.F.; Valtierra-Rodríguez, Martín

doi:10.1016/j.chaos.2017.06.030

Cited by 95 publications

(37 citation statements)

References 54 publications

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“…In addition, the neural network for solving ODEs and PDEs has been discussed a lot and still has some potential to work on. The recent research articles such as [63][64][65][66] have stud- ied using neural network method to solve several fractional differential equations (FDEs).…”

Section: Discussionmentioning

confidence: 99%

A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

2018

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This paper develops a Legendre neural network method (LNN) for solving linear and nonlinear ordinary differential equations (ODEs), system of ordinary differential equations (SODEs), as well as classic Emden-Fowler equations. The Legendre polynomial is chosen as a basis function of hidden neurons. A single hidden layer Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials. The improved extreme learning machine (IELM) algorithm is used for network weights training when solving algebraic equation systems, and several algorithm steps are summed up. Convergence was analyzed theoretically to support the proposed method. In order to demonstrate the performance of the method, various testing problems are solved by the proposed approach. A comparative study with other approaches such as conventional methods and latest research work reported in the literature are described in detail to validate the superiority of the method. Experimental results show that the proposed Legendre network with IELM algorithm requires fewer neurons to outperform the numerical algorithm in the latest literature in terms of accuracy and execution time.

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Section: Discussionmentioning

confidence: 99%

A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

2018

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“…Suppose that the approximate solution to (27) is y = ω T φ(x) + b, the original optimal problem is described as follows: min ω,b,e i J(ω, e) = 1 2 ω T ω + 1 2 γ e T e (28) subject to…”

Section: Linear Ordinary Differential Equations For Multi-point Boundmentioning

confidence: 99%

“…Neural network, which is one of machine intelligence techniques, has universal function approximation capabilities [20][21][22], and the solution obtained from the neural network is differentiable and in closed analytic form. Neural network has been widely used for solving ordinary differential equations [23,24], partial differential equations [25][26][27], fractional differential equations [28][29][30], and integro-differential equations [31,32]. Chakraverty and Mall [33] analyzed a regression-based neural network algorithm to solve two-point boundary value problems of fourth-order linear ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

The LS-SVM algorithms for boundary value problems of high-order ordinary differential equations

Yin

et al. 2019

Adv Differ Equ

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This paper introduces the improved LS-SVM algorithms for solving two-point and multi-point boundary value problems of high-order linear and nonlinear ordinary differential equations. To demonstrate the reliability and powerfulness of the improved LS-SVM algorithms, some numerical experiments for third-order, fourth-order linear and nonlinear ordinary differential equations with two-point and multi-point boundary conditions are performed. The idea can be extended to other complicated ordinary differential equations.

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“…Many numerical algorithms to solve the differential equations of fractional-order (FDE) have been proposed [29][30][31]. We will use the widely-used method, which is modified based on the Adams-Bashforth-Moulton predictor-corrector scheme, to solve time-delayed differential equations of fractional-order (FDDE) [31]. The method is described below.…”

Section: Numerical Algorithmmentioning

confidence: 99%

Dynamics Analysis of a New Fractional-Order Hopfield Neural Network with Delay and Its Generalized Projective Synchronization

Xie

2018

Entropy

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In this paper, a new three-dimensional fractional-order Hopfield-type neural network with delay is proposed. The system has a unique equilibrium point at the origin, which is a saddle point with index two, hence unstable. Intermittent chaos is found in this system. The complex dynamics are analyzed both theoretically and numerically, including intermittent chaos, periodicity, and stability. Those phenomena are confirmed by phase portraits, bifurcation diagrams, and the Largest Lyapunov exponent. Furthermore, a synchronization method based on the state observer is proposed to synchronize a class of time-delayed fractional-order Hopfield-type neural networks.

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Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks

Cited by 95 publications

References 54 publications

A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

The LS-SVM algorithms for boundary value problems of high-order ordinary differential equations

Dynamics Analysis of a New Fractional-Order Hopfield Neural Network with Delay and Its Generalized Projective Synchronization

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