Legendre–Galerkin method for the linear Fredholm integro-differential equations

Fathy, Mohamed; El-Gamel, Mohamed; El–Azab, M. S.

doi:10.1016/j.amc.2014.06.057

Cited by 27 publications

(18 citation statements)

References 24 publications

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“…Recently, many researchers used Legendre polynomials in different methods to construct various mathematical models. These methods can solve Lane-Emden type of differential equation [16], differential equation with second and fourth order [17], the equation of Cahn-Hilliard [18], integral equation of Fredholm type [19], Helmholtz equation [20], Volterra integral equations in the second kind [21], integral-differential of Fredholm type in linear form [22] and Abels integral equation [23]. Based on Legendre-Galerkin method, the pile flexural equation can be written as Ay(η) = f , where A is a differential operator.…”

Section: Introductionmentioning

confidence: 99%

Numerical Analysis of Laterally Loaded Long Piles in Cohesionless Soil

Abd-Elhamed¹,

Fathy²,

Abdelgaber³

2022

Computers, Materials &Amp; Continua

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The capability of piles to withstand horizontal loads is a major design issue. The current research work aims to investigate numerically the responses of laterally loaded piles at working load employing the concept of a beam-on-Winkler-foundation model. The governing differential equation for a laterally loaded pile on elastic subgrade is derived. Based on Legendre-Galerkin method and Runge-Kutta formulas of order four and five, the flexural equation of long piles embedded in homogeneous sandy soils with modulus of subgrade reaction linearly variable with depth is solved for both free-and fixed-headed piles. Mathematica, as one of the world's leading computational software, was employed for the implementation of solutions. The proposed numerical techniques provide the responses for the entire pile length under the applied lateral load. The utilized numerical approaches are validated against experimental and analytical results of previously published works showing a more accurate estimation of the response of laterally loaded piles. Therefore, the proposed approaches can maintain both mathematical simplicity and comparable accuracy with the experimental results.

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Section: Introductionmentioning

confidence: 99%

Numerical Analysis of Laterally Loaded Long Piles in Cohesionless Soil

Abd-Elhamed¹,

Fathy²,

Abdelgaber³

2022

Computers, Materials &Amp; Continua

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“…Many researchers used Legendre polynomials in different methods to construct various mathematical models. These methods can solve different models see ( [52]- [60]). The quantity of passings all throughout the planet has expanded significantly because of the spread of the new infection known as Covid (Coronavirus).…”

Section: Introductionmentioning

confidence: 99%

Legendre-Galerkin Method, Dynamical System, and Optimal Control for Solving Covid-19 Model

Mahdy

Lotfy

2021

Preprint

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The advancement of numerical demonstrating of irresistible illnesses is a key examination territory in different fields including the nature and the study of disease transmission. One point of these models is to comprehend the elements of conduct in irresistible infections. For the new strain of Covid (Coronavirus), there is no immunization to secure individuals and to forestall its spread up until now. All things being equal, control procedures related to medical services, for example, social separating, isolation, travel limitations, can be adjusted to control the pandemic of Coronavirus. This article reveals insights into the dynamical practices of nonlinear Coronavirus models dependent on strategy: the Legendre-Galerkin strategy. We summon a novel sign stream chart that is utilized to depict the Coronavirus model. Based on the Legendre-Galerkin method, the covid-19 model. Mathematica, as one of the world's leading computational software, was employed for the implementation of solutions. The proposed numerical techniques provide are excellent. Through our numerical investigations, it is uncovered that social removing between possibly tainted people who are conveying the infection and solid people can diminish or intrude on the spread of the infection. The mathematical reenactment results are insensible concurrence with the investigation forecasts. The free balance and dependability focus for the Coronavirus model is researched. Likewise, the presence of a consistently steady arrangement is demonstrate.

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“…In the substantial literature on the approximation of solutions of one-dimensional Fredholm integrodifferential equations (FIDEs), corresponding error analyses are notably scarce. For example, though the independent studies (in chronological order) [38,4,39,28,5,15,34,40,8,31,2,1,22,35] present diverse FIDE-solution techniques of varying degrees of efficiency and (disparate) accuracy, only [28,40,31,1] include a discussion of errors and, in even these cases, error analyses are limited (see summary in [21, §1]) to estimates of convergence rates: that is, the direct computation of theoretically predicted error bounds is almost entirely absent.…”

Section: Introductionmentioning

confidence: 99%

“…By contrast, the present work computes both Volterra andFredholm components of the VFIE to spectral accuracy and, moreover, determines explicit error bounds for||u − u N || using only the approximate derivative v N of the numerical solution u N . The error analysis is now presented for cases 1 and 2 given in(21) and(22) respectively.…”

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confidence: 99%