Fast and precise spectral method for solving pantograph type Volterra integro-differential equations

Ezz-Eldien, S. S.; Doha, E. H.

doi:10.1007/s11075-018-0535-x

Cited by 52 publications

(22 citation statements)

References 47 publications

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“…One of the most useful and important types of orthogonal functions is Jacobi polynomials. There are some special cases of this orthogonal polynomial family such as Legendre, the four kinds of Chebyshev, and Gegenbauer polynomials . The Jacobi polynomials have been used to solve various problems in different fields of science.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Rasanan

Bajalan

Parand

et al. 2019

Math Methods in App Sciences

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By the rapid growth of available data, providing data-driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive-decision-making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg-Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.

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Section: Preliminaries and Notationsmentioning

confidence: 99%

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Rasanan

Bajalan

Parand

et al. 2019

Math Methods in App Sciences

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“…Although there are numerous studies in literature on cooling of electronics components using heat sink [20][21][22][23][24][25][26][27], an in-depth examination and review of the past works indicate to the best of our knowledge that FGM has not been applied in heat sinks except in [20]. Recent works on the applications of the non-Fourier heat transfer in extended surfaces, transient analysis of fins under dehumidification, and other optimization studies as well as the recent advancements in Legendre polynomials have been well presented [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]. us, considering the significant capabilities and benefits of FGM coupled with established enhanced heat transfer characteristics of porous fins.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Transient Thermal Modeling and Analysis of a Convective-Radiative Fin with Functionally Graded Material in a Magnetic Environment

Sobamowo

Oguntala

Yinusa

2019

Modelling and Simulation in Engineering

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Nonlinear transient thermal analysis of a convective-radiative fin with functionally graded materials (FGMs) under the influence of magnetic field is presented in this study. The developed nonlinear thermal models of linear, quadratic, and exponential variation of thermal conductivity are solved approximately and analytically using the differential transformation method (DTM). In order to verify the accuracies of the nonlinear solutions, exact analytical solutions are also developed with the aids of Bessel, Legendre, and modified Bessel functions. Good agreements are established between the exact and the approximate analytical solutions. In the parametric studies, effects of heat enhancement capacity of fin with functionally graded material as compared to fin with homogeneous material are investigated. Also, influence of the Lorentz force and radiative heat transfer on the thermal performance of the fin are analyzed. From the results, it is shown that increase in radiative and magnetic field parameters as well as the in-homogeneity index improve the thermal performance of the fin. Also, the transient responses reveal that the FGM fin with quadratic-law and exponential-law function shows the slowest and fasted thermal responses, respectively. This study will provide a very good platform for the design and optimization of an improved heat transfer enhancement in thermal systems, where the surrounding fluid is influenced by a magnetic field.

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“…Many real-life phenomena have been described using fractional differential equations (FDEs), such as viscoelasticy (Koeller 1984), continuum mechanics (Carpinteri and Mainardi 1997), optimal control (Bhrawy and Ezz-Eldien 2016), hydrologic modeling (Benson et al 2013), variational problems (Ezz-Eldien 2016), fluid mechanics (Kulish and Lage 2002), finance (Jiang et al 2012), and others (Gaul et al 1991;Ferreira et al 2008; Ezz-Eldien and El-Kalaawy 2018; Dzielinski et al 2010). Searching for numerical techniques for approximating the solutions of FDEs has been strongly considered in the last decades, such as the radial basis functions method (Hosseini et al 2014), the Haar wavelet method , the fractional finite volume method (Liu et al 2014), the Adomian decomposition method (Khodabakhshi et al 2014), the operational method Kim et al 2014), and so on (Heydari et al 2013;Bhrawy and Zaky 2017;Ezz-Eldien and Doha 2018;Stern et al 2014). As a spectral approach for solving numerically some kinds of differential equations, the tau method is classified as one of the most important used methods due to its high convergence and accuracy.…”

Section: Introductionmentioning

confidence: 99%

On solving fractional logistic population models with applications

Ezz-Eldien

2018

Comp. Appl. Math.

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Fast and precise spectral method for solving pantograph type Volterra integro-differential equations

Cited by 52 publications

References 47 publications

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Nonlinear Transient Thermal Modeling and Analysis of a Convective-Radiative Fin with Functionally Graded Material in a Magnetic Environment

On solving fractional logistic population models with applications

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