Efficient modified Chebyshev differentiation matrices for fractional differential equations

Arman Dabiri, Eric Butcher

Research output: Contribution to journalArticle

36 Citations (Scopus)

Abstract

This paper compares several fractional operational matrices for solving a system of linear fractional differential equations (FDEs) of commensurate or incommensurate order. For this purpose, three fractional collocation differentiation matrices (FCDMs) based on finite differences are first proposed and compared with Podlubny's matrix previously used in the literature, after which two new efficient FCDMs based on Chebyshev collocation are proposed. It is shown via an error analysis that the use of the well-known property of fractional differentiation of polynomial bases applied to these methods results in a limitation in the size of the obtained Chebyshev-based FCDMs. To compensate for this limitation, a new fast spectrally accurate FCDM for fractional differentiation which does not require the use of the gamma function is proposed. Then, the Schur-Pade and Schur decomposition methods are implemented to enhance and improve numerical stability. Therefore, this method overcomes the previous limitation regarding the size limitation. In several illustrative examples, the convergence and computation time of the proposed FCDMs are compared and their advantages and disadvantages are outlined.

Original languageEnglish (US)
Pages (from-to)284-310
Number of pages27
JournalCommunications in Nonlinear Science and Numerical Simulation
Volume50
DOIs
StatePublished - Sep 1 2017

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Differentiation (calculus)
Fractional Differential Equation
Chebyshev
Differential equations
Fractional
Collocation
Schur Decomposition
Operational Matrix
Convergence of numerical methods
Polynomial Basis
Gamma function
Numerical Stability
Error analysis
Decomposition Method
Linear differential equation
Error Analysis
Finite Difference
Polynomials
Decomposition

Keywords

  • Chebyshev differentiation matrix
  • Fractional differential equation
  • Numerical stability
  • Spectral collocation method

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Applied Mathematics

Cite this

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abstract = "This paper compares several fractional operational matrices for solving a system of linear fractional differential equations (FDEs) of commensurate or incommensurate order. For this purpose, three fractional collocation differentiation matrices (FCDMs) based on finite differences are first proposed and compared with Podlubny's matrix previously used in the literature, after which two new efficient FCDMs based on Chebyshev collocation are proposed. It is shown via an error analysis that the use of the well-known property of fractional differentiation of polynomial bases applied to these methods results in a limitation in the size of the obtained Chebyshev-based FCDMs. To compensate for this limitation, a new fast spectrally accurate FCDM for fractional differentiation which does not require the use of the gamma function is proposed. Then, the Schur-Pade and Schur decomposition methods are implemented to enhance and improve numerical stability. Therefore, this method overcomes the previous limitation regarding the size limitation. In several illustrative examples, the convergence and computation time of the proposed FCDMs are compared and their advantages and disadvantages are outlined.",
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AU - Dabiri, Arman

AU - Butcher, Eric

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N2 - This paper compares several fractional operational matrices for solving a system of linear fractional differential equations (FDEs) of commensurate or incommensurate order. For this purpose, three fractional collocation differentiation matrices (FCDMs) based on finite differences are first proposed and compared with Podlubny's matrix previously used in the literature, after which two new efficient FCDMs based on Chebyshev collocation are proposed. It is shown via an error analysis that the use of the well-known property of fractional differentiation of polynomial bases applied to these methods results in a limitation in the size of the obtained Chebyshev-based FCDMs. To compensate for this limitation, a new fast spectrally accurate FCDM for fractional differentiation which does not require the use of the gamma function is proposed. Then, the Schur-Pade and Schur decomposition methods are implemented to enhance and improve numerical stability. Therefore, this method overcomes the previous limitation regarding the size limitation. In several illustrative examples, the convergence and computation time of the proposed FCDMs are compared and their advantages and disadvantages are outlined.

AB - This paper compares several fractional operational matrices for solving a system of linear fractional differential equations (FDEs) of commensurate or incommensurate order. For this purpose, three fractional collocation differentiation matrices (FCDMs) based on finite differences are first proposed and compared with Podlubny's matrix previously used in the literature, after which two new efficient FCDMs based on Chebyshev collocation are proposed. It is shown via an error analysis that the use of the well-known property of fractional differentiation of polynomial bases applied to these methods results in a limitation in the size of the obtained Chebyshev-based FCDMs. To compensate for this limitation, a new fast spectrally accurate FCDM for fractional differentiation which does not require the use of the gamma function is proposed. Then, the Schur-Pade and Schur decomposition methods are implemented to enhance and improve numerical stability. Therefore, this method overcomes the previous limitation regarding the size limitation. In several illustrative examples, the convergence and computation time of the proposed FCDMs are compared and their advantages and disadvantages are outlined.

KW - Chebyshev differentiation matrix

KW - Fractional differential equation

KW - Numerical stability

KW - Spectral collocation method

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