Chebyshev spectral methods for multi-order fractional neutral pantograph equations

Ezz-Eldien, S. S.; Wang, Y.; Abdelkawy, M. A.; Zaky, Mahmoud A.; Aldraiweesh, Ahmed; Machado, J.A.T.

doi:10.1007/s11071-020-05728-x

Cited by 49 publications

(17 citation statements)

References 30 publications

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“…The pantograph differential equations have many applications in the area of theory [2], cell-growth biological based models [3] and control system [4]. The pantograph differential equations have been solved by many techniques, some of them are intelligent networks [5], Chebyshev spectral scheme [6], spectral tau scheme [7], multidimensional homotopy optimal asymptotic scheme [8], Genocchi operational based matrix scheme [9], least-squares-Epsilon-Ritz scheme [10], Taylor operation scheme [11], Galerkin multi-wavelets scheme [12], heuristic computing approach [13], Sinc numerical scheme [14], Laplace transform scheme [15], spectral collocation approach [16], multistep block method [17], Legendre Tau computational scheme [18] and Euler-Maruyama scheme [19]. This singular study is considered very significant due to its extensive applications in radiators cooling, dusty fluids, classical/quantum-based mechanics, models of gas cloud and galaxies [20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Design of Morlet Wavelet Neural Network for Solving a Class of Singular Pantograph Nonlinear Differential Models

et al. 2021

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The aim of this study is to design a layer structure of feed-forward artificial neural networks using the Morlet wavelet activation function for solving a class of pantograph differential Lane-Emden models. The Lane-Emden pantograph differential equation is one of the important kind of singular functional differential model. The numerical solutions of the singular pantograph differential model are presented by the approximation capability of the Morlet wavelet neural networks (MWNNs) accomplished with the strength of global and local search terminologies of genetic algorithm (GA) and interior-point algorithm (IPA), i.e., MWNN-GAIPA. Three different problems of the singular pantograph differential models have been numerically solved by using the optimization procedures of MWNN-GAIPA. The correctness of the designed MWNN-GAIPA is observed by comparing the obtained results with the exact solutions. The analysis for 3, 6 and 60 neurons are also presented to check the stability and performance of the designed scheme. Moreover, different statistical analysis using forty number of trials is presented to check the convergence and accuracy of the proposed MWNN-GAIPA scheme.

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Section: Introductionmentioning

confidence: 99%

Design of Morlet Wavelet Neural Network for Solving a Class of Singular Pantograph Nonlinear Differential Models

et al. 2021

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“…For example, in References [ 7 , 8 , 9 ], a homotopy approach and power series are developed for solving linear MDEs and coinciding of the estimated solution with the exact solution is investigated. In Reference [ 10 ], the spectral tau method is studied and the convergence of the presented approach is investigated by

norm. In Reference [ 11 ], by obtaining the fractional integral of Taylor wavelets in the sense of Riemann-Liouville definition, an estimated solution is presented for fractional MDEs.…”

Section: Introductionmentioning

confidence: 99%

Machine Learning for Modeling the Singular Multi-Pantograph Equations

Mosavi

Shokri

Mansor

et al. 2020

Entropy

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In this study, a new approach to basis of intelligent systems and machine learning algorithms is introduced for solving singular multi-pantograph differential equations (SMDEs). For the first time, a type-2 fuzzy logic based approach is formulated to find an approximated solution. The rules of the suggested type-2 fuzzy logic system (T2-FLS) are optimized by the square root cubature Kalman filter (SCKF) such that the proposed fineness function to be minimized. Furthermore, the stability and boundedness of the estimation error is proved by novel approach on basis of Lyapunov theorem. The accuracy and robustness of the suggested algorithm is verified by several statistical examinations. It is shown that the suggested method results in an accurate solution with rapid convergence and a lower computational cost.

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“…During last decades, orthogonal functions have been applied extensively for solving different classes of problems. The major reason for this is that solving the main problem turns into solving a simple algebraic system [11][12][13]. We remind that the Chebyshev polynomials can be effectively utilized for approximating any sufficiently differentiable function.…”

Section: Introductionmentioning

confidence: 99%

Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations

et al. 2020

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In this study, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method. For this purpose, we derive a V-O fractional integration operational matrix (OM) for CWs and use it in our method. In the established scheme, we approximate the unknown functions by CWs with unknown coefficients and reduce the problem to an algebraic system. In this way, we simplify the computation of nonlinear terms by obtaining some new results for CWs. Finally, we demonstrate the applicability of the presented algorithm by solving a few numerical examples.

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Chebyshev spectral methods for multi-order fractional neutral pantograph equations

Cited by 49 publications

References 30 publications

Design of Morlet Wavelet Neural Network for Solving a Class of Singular Pantograph Nonlinear Differential Models

Design of Morlet Wavelet Neural Network for Solving a Class of Singular Pantograph Nonlinear Differential Models

Machine Learning for Modeling the Singular Multi-Pantograph Equations

Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations

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