Application of the singular value decomposition to the numerical computation of the coefficients of amplitude equations and normal forms

Kangping Chen, Daniel D. Joseph

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

The Fredholm alternative is a standard procedure by which one generates the coefficients of amplitude equations and normal forms. The alternative requires that the inhomogeneous terms in the underlying system of differential equations, which contain the unknown coefficients, be orthogonal to the independent eigenvectors spanning the null space of the adjoint system of differential equations. The numerical computation of the adjoint eigenvectors and their application to solvability is frequently difficult and inefficient. Typically the underlying system of the inhomogeneous differential equation is discretized and solved as an inhomogeneous matrix-valued eigenvalue problem. We find that the solvability conditions which lead to values of the unknown coefficients are conviniently and economically computed by application of the singular value decomposition directly to the matrix formulation, avoiding completely the computation of an adjoint system of differential equations.

Original languageEnglish (US)
Pages (from-to)425-430
Number of pages6
JournalApplied Numerical Mathematics
Volume6
Issue number6
DOIs
StatePublished - 1990
Externally publishedYes

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Amplitude Equations
Singular value decomposition
System of Differential Equations
Numerical Computation
Normal Form
Adjoint System
Differential equations
Eigenvector
Coefficient
Eigenvalues and eigenfunctions
Fredholm Alternative
Unknown
Solvability Conditions
Null Space
Eigenvalue Problem
Solvability
Differential equation
Formulation
Alternatives
Term

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Modeling and Simulation

Cite this

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