Fractional neural network models for nonlinear Riccati systems

Lodhi, Sadia; Manzar, Muhammad Anwaar; Raja, Muhammad Asif Zahoor

doi:10.1007/s00521-017-2991-y

Cited by 78 publications

(22 citation statements)

References 71 publications

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“…(6) via Meyer wavelet neural networks (MWNN) optimized with global search efficacy of genetic algorithms (GAs) and sequential quadratic programming (SQP), i.e., MWNN-GASQP. The solvers based on meta-heuristic intelligent computing have been extensively applied for the analysis of linear/nonlinear, singular/non-singular systems using neural networks optimized with evolutionary/swarming-based computing schemes (Lodhi 2019;Raja et al 2017a;Bukhari 2020;Waseem 2020;Ahmad 2020Ahmad ,2019. Some recent applications of the evolutionary/swarming-based numerical computing are Painlevé equation-based models in random matrix theory (Raja et al 2018a), nonlinear prey-predator models (Umar 2019), Bagley-Torvik systems in fluid mechanics.…”

Section: Problem Statement and Related Workmentioning

confidence: 99%

“…The present contribution is basically the extension of the fractional neural network proposed earlier used exponential function as an activation function for hidden neurons (Lodhi 2019;Raja et al 2017a), while, here in the proposed investigation, we replaced the said activation function with Meyer wavelet kernel (availability or ease for calculation of fraction derivative) to present fractional Meyer wavelet neural networks (MWNNs). As reported in (Raja et al 2017a), the presented fractional MWNNs is a kind of unsupervised fractional neural networks and training of the weights are conducted with integrated heuristics of global and local search methodologies.…”

Section: Novelty and Contributionmentioning

confidence: 99%

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FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane–Emden system

et al. 2020

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In the present study, a novel fractional Meyer neuro-evolution-based intelligent computing solver (FMNEICS) is presented for numerical treatment of doubly singular multi-fractional Lane-Emden system (DSMF-LES) using combined heuristics of Meyer wavelet neural networks (MWNN) optimized with global search efficacy of genetic algorithms (GAs) and sequential quadratic programming (SQP), i.e., MWNN-GASQP. The design of novel FMNE-ICS for DSMF-LES is presented after derivation from standard Lane-Emden equation, and the singular points and shape factors along with fractional-order terms are analyzed. The MWNN modeling strength is used to represent the system model DSMF-LES in the meansquared error-based merit function and optimization of the networks is carried out with integrated optimization ability of GASQP. The verification, validation, and perfection of the FMNEICS for three different cases of DSMF-LES are established through comparative studies from reference solutions on convergence, robustness, accuracy, and stability measures. Moreover, the observations through the statistical analysis further authenticate the worth of proposed fractional MWNN-GASQP-based stochastic solver. Keywords Multi-fractional Lane-Emden model • Multi-singular systems • Artificial neural networks • Meyer wavelet neural networks • Sequential quadratic programming • Genetic algorithms Mathematics Subject Classification 34-XX • 34A08 • 34A34 • 82B31

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Section: Problem Statement and Related Workmentioning

confidence: 99%

Section: Novelty and Contributionmentioning

confidence: 99%

FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane–Emden system

et al. 2020

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“…Here, α represents the fractional order with α > 0 and α ∈ R, v(t) is the solution, and r (t) is the forcing term, the constant c n denotes the initial conditions, while p(t) and q(t) are variable coefficients of linear and nonlinear terms, respectively. (Lodhi et al 2019;Raja et al 2015) Consider the following nonlinear quadratic Riccati fractional differential equation:…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Raja et al (2015) presented a new computational intelligence technique using artificial neural networks (ANNs) and sequential quadratic programming (SQP) to solve the nonlinear quadratic Riccati differential equations of fractional order. Lodhi et al (2019) employed a feed-forward artificial FrNN to find the approximate solutions of nonlinear systems of Riccati equations with the arbitrary order and developed the energy function of the system. Jaber and Ahmad (2018) applied residual power series (RPS) method to generate the solution for the nonlinear time-fractional Navier-Stokes equation in two dimensions in the form of a rapidly convergent series.…”

Section: Introductionmentioning

confidence: 99%

Fractional generalised homotopy analysis method for solving nonlinear fractional differential equations

Saratha¹,

Bagyalakshmi

Krishnan

2020

Comp. Appl. Math.

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This paper presents a novel hybrid technique which is developed by incorporating the fractional derivatives in the generalised integral transform method. Homotopy analysis method is combined with fractional generalised integral transform method to solve the fractional order nonlinear differential equations. The performance of the proposed method is analysed by solving various categories of nonlinear fractional differential equations like Navier Stokes's model and Riccatti equations, etc. Unlike the other analytical methods, the hybrid method provides a better way to control the convergence region of the obtained series solution through an auxiliary parameter h. Furthermore, as proposed in this paper, the 'Fractional Generalised Homotopy Analysis Method' along with the several examples reveal that this method can be effectively used as a tool for solving various kinds of nonlinear fractional differential equations.

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“…Several numerical methods have been used to solve other problems. For example, artificial neural networks (ANN) and sequential quadratic programming (SQP) investigated for solution of non-linear quadratic fractional Riccati differential equation Raja et al (2015), Lodhi et al (2015). In Raja et al (2016), an effective numerical method was developed for the approximation of fractional of the Bagley-Torvik equations using fractional neural networks (FNN) optimized with interior point algorithms (IPA).…”

Section: Introductionmentioning

confidence: 99%

A reliable numerical approach for analyzing fractional variational problems with subsidiary conditions

Sayevand

Rostami

2019

Comp. Appl. Math.

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This paper obtains the necessary optimality conditions for a new class of isoperimetric fractional variational problems depending on indefinite integrals (IFVPDI). A new condition is added to the IFVPDI and a modified direct numerical approach based on the Epsilon-Ritz method and polynomial basis functions is applied. The optimization problem is reduced to the problem of optimizing a real value function. The convergence and error analysis of the proposed approach are also assessed. Illustrative examples are included demonstrating the validity and applicability of the method.

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Fractional neural network models for nonlinear Riccati systems

Cited by 78 publications

References 71 publications

FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane–Emden system

FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane–Emden system

Fractional generalised homotopy analysis method for solving nonlinear fractional differential equations

A reliable numerical approach for analyzing fractional variational problems with subsidiary conditions

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