Weighted Residual Methods

Weighted residual method (WRM) is an approximation technique in which solution of differential equation is approximated by linear combination of trial or shape functions having unknown coefficients. The approximate solution is then substituted in the governing differential equation resulting in erro...

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Published in:Advanced Numerical and Semi-Analytical Methods for Differential Equations pp. 31 - 43
Main Authors: Chakraverty, Snehashish, Mahato, Nisha, Karunakar, Perumandla, Dilleswar Rao, Tharasi
Format: Book Chapter
Language:English
Published: United States Wiley 2019
John Wiley & Sons, Incorporated
John Wiley & Sons, Inc
Edition:1
Subjects:
boundary value problems
collocation method
Galerkin method
least‐square method
ordinary differential equations
subdomain method
weighted residual methods
Boundary conditions
Shape
IEEE Sections
Force
Method of moments
Standards
ISBN:9781119423423, 1119423422
Online Access:Get full text
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Abstract Weighted residual method (WRM) is an approximation technique in which solution of differential equation is approximated by linear combination of trial or shape functions having unknown coefficients. The approximate solution is then substituted in the governing differential equation resulting in error or residual. Finally, in the WRM the residual is forced to vanish at average points or made as small as possible depending on the weight function in order to find the unknown coefficients. The authors illustrate various WRMs, viz. collocation, subdomain, least‐square, and Galerkin methods applied for solving ordinary differential equations subject to boundary conditions. They also check the efficiency of various WRMs by comparing the solution obtained using collocation, subdomain, least‐square, and Galerkin methods for the boundary value problems. Lastly, the authors present few exercise problems for self‐validation.
AbstractList Weighted residual method (WRM) is an approximation technique in which solution of differential equation is approximated by linear combination of trial or shape functions having unknown coefficients. The approximate solution is then substituted in the governing differential equation resulting in error or residual. Finally, in the WRM the residual is forced to vanish at average points or made as small as possible depending on the weight function in order to find the unknown coefficients. The authors illustrate various WRMs, viz. collocation, subdomain, least‐square, and Galerkin methods applied for solving ordinary differential equations subject to boundary conditions. They also check the efficiency of various WRMs by comparing the solution obtained using collocation, subdomain, least‐square, and Galerkin methods for the boundary value problems. Lastly, the authors present few exercise problems for self‐validation.
Weighted residual method (WRM) is an approximation technique in which solution of differential equation is approximated by linear combination of trial or shape functions having unknown coefficients. The approximate solution is then substituted in the governing differential equation resulting in error or residual. Finally, in the WRM the residual is forced to vanish at average points or made as small as possible depending on the weight function in order to find the unknown coefficients. The authors illustrate various WRMs, viz. collocation, subdomain, least‐square, and Galerkin methods applied for solving ordinary differential equations subject to boundary conditions. They also check the efficiency of various WRMs by comparing the solution obtained using collocation, subdomain, least‐square, and Galerkin methods for the boundary value problems. Lastly, the authors present few exercise problems for self‐validation.
Author Karunakar, Perumandla
Dilleswar Rao, Tharasi
Mahato, Nisha
Chakraverty, Snehashish
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2019 John Wiley & Sons, Inc.
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Keywords Boundary conditions
Shape
IEEE Sections
Force
Method of moments
Standards
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Lindgren (c03-cit-0006) 2009
Baluch, Mohsen, Ali (c03-cit-0004) 1983; 7
Gerald, Wheatley (c03-cit-0001) 2004
Finlayson (c03-cit-0002) 2013; 73
Locker (c03-cit-0005) 1971; 154
References_xml – year: 2017
  ident: c03-cit-0003
  article-title: Weighted Residual Methods: Principles, Modifications and Applications
– volume: 7
  start-page: 362
  issue: 5
  year: 1983
  end-page: 365
  ident: c03-cit-0004
  article-title: Method of weighted residuals as applied to nonlinear differential equations
  publication-title: Applied Mathematical Modelling
– volume: 154
  start-page: 57
  year: 1971
  end-page: 68
  ident: c03-cit-0005
  article-title: The method of least squares for boundary value problems
  publication-title: Transactions of the American Mathematical Society
– year: 2004
  ident: c03-cit-0001
  article-title: Applied Numerical Analysis
– volume: 73
  year: 2013
  ident: c03-cit-0002
  article-title: The Method of Weighted Residuals and Variational Principles
– year: 2009
  ident: c03-cit-0006
  article-title: From Weighted Residual Methods to Finite Element Methods
– year: 2011
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  article-title: A First Course in the Finite Element Method
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Snippet Weighted residual method (WRM) is an approximation technique in which solution of differential equation is approximated by linear combination of trial or shape...
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StartPage 31
SubjectTerms boundary value problems
collocation method
Galerkin method
least‐square method
ordinary differential equations
subdomain method
weighted residual methods
Title Weighted Residual Methods
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