Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods

•A framework for solving multi-delay fractional differential equations is proposed.•Fractional delay differential equations with irrational delays are discretized.•The method possesses spectral convergence with efficient computational time.•The convergence, error estimates, and numerical stability o...

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Bibliographic Details
Published in:Applied Mathematical Modelling 2018-04, Vol.56, p.424-448
Main Authors: Dabiri, Arman, Butcher, Eric A.
Format: Article
Language:English
Subjects:
Chebyshev approximation
Collocation methods
Computing time
Convergence
Convergence and error estimates
Delay differential equation
Differential equations
Mathematical analysis
Multi-order fractional differential equation
Numerical analysis
Numerical method
Numerical methods
Numerical stability
Spectra
Spectral method
Spectrum analysis
Studies
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Summary:•A framework for solving multi-delay fractional differential equations is proposed.•Fractional delay differential equations with irrational delays are discretized.•The method possesses spectral convergence with efficient computational time.•The convergence, error estimates, and numerical stability of the method are studied.•Several illustrative practical examples show the advantages of the method. This paper discusses a general framework for the numerical solution of multi-order fractional delay differential equations (FDDEs) in noncanonical forms with irrational/rational multiple delays by the use of a spectral collocation method. In contrast to the current numerical methods for solving fractional differential equations, the proposed framework can solve multi-order FDDEs in a noncanonical form with incommensurate orders. The framework can also solve multi-order FDDEs with irrational multiple delays. Next, the framework is enhanced by the fractional Chebyshev collocation method in which a Chebyshev operation matrix is constructed for the fractional differentiation. Spectral convergence and small computational time are two other advantages of the proposed framework enhanced by the fractional Chebyshev collocation method. In addition, the convergence, error estimates, and numerical stability of the proposed framework for solving FDDEs are studied. The advantages and computational implications of the proposed framework are discussed and verified in several numerical examples.
ISSN:0307-904X
1088-8691
0307-904X
DOI:10.1016/j.apm.2017.12.012