Solution and stability of a set of PTH order linear differential equations with periodic coefficients via Chebyshev polynomials

S. C. Sinha, Eric Butcher

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

Chebyshev polynomials are utilized to obtain solutions of a set of p th order linear differential equations with periodic coefficients. For this purpose, the operational matrix of differentiation associated with the shifted Chebyshev polynomials of the first kind is derived. Utilizing the properties of this matrix, the solution of a system of differential equations can be found by solving a set of linear algebraic equations without constructing the equivalent integral equations. The Floquet Transition Matrix (FTM) can then be computed and its eigenvalues (Floquet multipliers) subsequently analyzed for stability. Two straightforward methods, the differential state space formulation and the differential direct formulation, are presented and the results are compared with those obtained from other available techniques. The well-known Mathieu equation and a higher order system are used as illustrative examples.

Original languageEnglish (US)
Pages (from-to)165-190
Number of pages26
JournalMathematical Problems in Engineering
Volume2
Issue number2
DOIs
StatePublished - 1996
Externally publishedYes

Fingerprint

Periodic Coefficients
Chebyshev Polynomials
Linear differential equation
Differential equations
Polynomials
Floquet multipliers
Mathieu Equation
Operational Matrix
Formulation
Transition Matrix
System of Differential Equations
Algebraic Equation
Linear equation
Integral Equations
State Space
Higher Order
Eigenvalue
Linear equations
Integral equations

Keywords

  • Differentiation operational matrix
  • Floquest transition matrix(FTM)
  • Floquet multiplier
  • Periodic systems
  • Shifted Chebyshev palynomials of the first kind
  • Stability

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)

Cite this

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abstract = "Chebyshev polynomials are utilized to obtain solutions of a set of p th order linear differential equations with periodic coefficients. For this purpose, the operational matrix of differentiation associated with the shifted Chebyshev polynomials of the first kind is derived. Utilizing the properties of this matrix, the solution of a system of differential equations can be found by solving a set of linear algebraic equations without constructing the equivalent integral equations. The Floquet Transition Matrix (FTM) can then be computed and its eigenvalues (Floquet multipliers) subsequently analyzed for stability. Two straightforward methods, the differential state space formulation and the differential direct formulation, are presented and the results are compared with those obtained from other available techniques. The well-known Mathieu equation and a higher order system are used as illustrative examples.",
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AU - Sinha, S. C.

AU - Butcher, Eric

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N2 - Chebyshev polynomials are utilized to obtain solutions of a set of p th order linear differential equations with periodic coefficients. For this purpose, the operational matrix of differentiation associated with the shifted Chebyshev polynomials of the first kind is derived. Utilizing the properties of this matrix, the solution of a system of differential equations can be found by solving a set of linear algebraic equations without constructing the equivalent integral equations. The Floquet Transition Matrix (FTM) can then be computed and its eigenvalues (Floquet multipliers) subsequently analyzed for stability. Two straightforward methods, the differential state space formulation and the differential direct formulation, are presented and the results are compared with those obtained from other available techniques. The well-known Mathieu equation and a higher order system are used as illustrative examples.

AB - Chebyshev polynomials are utilized to obtain solutions of a set of p th order linear differential equations with periodic coefficients. For this purpose, the operational matrix of differentiation associated with the shifted Chebyshev polynomials of the first kind is derived. Utilizing the properties of this matrix, the solution of a system of differential equations can be found by solving a set of linear algebraic equations without constructing the equivalent integral equations. The Floquet Transition Matrix (FTM) can then be computed and its eigenvalues (Floquet multipliers) subsequently analyzed for stability. Two straightforward methods, the differential state space formulation and the differential direct formulation, are presented and the results are compared with those obtained from other available techniques. The well-known Mathieu equation and a higher order system are used as illustrative examples.

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