Journal
INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION
Volume 23, Issue 7-8, Pages 1253-1268Publisher
WALTER DE GRUYTER GMBH
DOI: 10.1515/ijnsns-2020-0124
Keywords
Chebyshev polynomials; duplication formula; fractional differential equations; hypergeometric functions; tau method
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This paper introduces an explicit formula for approximating the fractional derivatives of Chebyshev polynomials of the first-kind in the Caputo sense. It is applied to a spectral solution of a certain type of fractional delay differential equations using an explicit Chebyshev tau method. The efficiency and accuracy of the proposed algorithm are demonstrated through numerical results.
This paper presents an explicit formula that approximates the fractional derivatives of Chebyshev polynomials of the first-kind in the Caputo sense. The new expression is given in terms of a terminating hypergeometric function of the type F-4(3)(1). The integer derivatives of Chebyshev polynomials of the first-kind are deduced as a special case of the fractional ones. The formula will be applied for obtaining a spectral solution of a certain type of fractional delay differential equations with the aid of an explicit Chebyshev tau method. The shifted Chebyshev polynomials of the first-kind are selected as basis functions and the spectral tau method is employed for obtaining the desired approximate solutions. The convergence and error analysis are discussed. Numerical results are presented illustrating the efficiency and accuracy of the proposed algorithm.
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