Method of weighted residuals as applied to nonlinear differential equations

Baluch, Mohammed H.; Mohsen, M. F. N.; Ali, Amjid

doi:10.1016/0307-904x(83)90135-x

Cited by 8 publications

(3 citation statements)

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“…where ( ) R x is a measure of error called the Residual [23,29]. Multiplying (3.2) by an arbitrary weight function ( ) u x and integrating over the domain to obtain…”

Section: Galerkin Weighted Residual Methodsmentioning

confidence: 99%

Entropy Generation Analysis of a Reactive MHD Third Grade Fluid in a Cylindrical Pipe with Radially Applied Magnetic Field and Hall Current

Aiyesimi

Jiya

Bolarin

et al. 2019

ARJOM

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The combined effects of chemical reaction, radially applied magnetic field and Hall effect on entropy generation of a steady third grade magnetohydrodynamic fluid flowing through a uniformly circular pipe was studied. The governing equations are presented and the resulting non-linear dimensionless equations are solved numerically using Galerkin Weighted Residual Method. The velocity, temperature and concentration profile were obtained and utilized in computing the entropy number. A parametric study of germane parameters involved are presented graphically and discussed. It was observed that irreversibility due to heat transfer dominates the flow compared to fluid friction and Hall parameter inhibits the Bejan number while Magnetic parameter enhances the Bejan number.

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“…where ( ) R x is a measure of error called the Residual [23,29]. Multiplying (3.2) by an arbitrary weight function ( ) u x and integrating over the domain to obtain…”

Section: Galerkin Weighted Residual Methodsmentioning

confidence: 99%

Entropy Generation Analysis of a Reactive MHD Third Grade Fluid in a Cylindrical Pipe with Radially Applied Magnetic Field and Hall Current

Aiyesimi

Jiya

Bolarin

et al. 2019

ARJOM

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“…This residual is obtained from the approximate solution of the equation of motion (Eq. (1)); as follows [Rezaiee-Pajand and Karimi-Rad (2014); Baluch et al (1983); Wang and Au (2004)]:…”

Section: Proposed Methodsmentioning

confidence: 99%

Application of Weight Functions in Nonlinear Analysis of Structural Dynamics Problems

Ghassemieh

Gholampour

Massah

2016

Int. J. Comput. Methods

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This paper presents a weighted residual method with several weight functions for solving differential equation of motion in nonlinear structural dynamics problems. Order of variation of acceleration is assumed to be quadratic in each time step in which polynomial of displacement would contain five unknown coefficients. Five equations are required for determination of these coefficients in each time step. These equations are obtained from initial conditions, satisfying equation of motions at both ends, and weighted residual integration. In this study, four procedures are considered for weight function to be used in the weighted residual integration as; unit weight function, Petrov-Galerkin's weight function, least square weight function, and collocation weight function. Due to higher order of acceleration in the proposed method, the results indicate better and more accurate responses. Among the tested functions, the unit weighted function method demonstrated to be non-dissipative and its numerical dispersion showed to be clearly less than the common Newmark's linear acceleration method. Also critical time step duration in stability investigation for weighted function procedure showed to be larger than the critical time step duration obtained by other methods used in the nonlinear structural dynamics problems.

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“…The first consists of expressing the unknown solution as a sum of known (pre-selected) basis or expansion functions with unknown expansion coefficients. The second step determines these coefficients such that the approximation is as close as possible to the accurate solution [11], [17], [18].…”

Section: Approximation In Inner Product Space and The Mwrmentioning

confidence: 99%

A Unified View of Computational Electromagnetics

Chen

Wang

Hoefer

2022

IEEE Trans. Microwave Theory Techn.

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This article presents a unified description of numerical methods for solving electromagnetic field problems. Traditionally, these methods are considered to be independent and alternative ways of solving Maxwell's equations or equations derived therefrom. However, they all are projective approximations of the unknown solution by known expansion or basis functions with unknown amplitude coefficients that are determined using the so-called method of weighted residuals (MWR). This common feature forms the proposed unifying framework that may serve both as a systematic introduction to computational electromagnetics and as a way to provide insight into the nature, strengths, and limitations of the various algorithms used by present and future field solvers. For convenience, the fundamental equations and concepts of electromagnetics are summarized in order to facilitate the demonstration of the unified approach. Our aim is to provide students, designers, and researchers with a framework for developing new numerical algorithms, to guide users in the selection and application of electromagnetic design tools, and to foster informed engineering judgment. This article also serves as a review and summary of earlier theoretical works reported in different places and at different times.

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Method of weighted residuals as applied to nonlinear differential equations

Cited by 8 publications

References 1 publication

Entropy Generation Analysis of a Reactive MHD Third Grade Fluid in a Cylindrical Pipe with Radially Applied Magnetic Field and Hall Current

Entropy Generation Analysis of a Reactive MHD Third Grade Fluid in a Cylindrical Pipe with Radially Applied Magnetic Field and Hall Current

Application of Weight Functions in Nonlinear Analysis of Structural Dynamics Problems

A Unified View of Computational Electromagnetics

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