Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations. (English) Zbl 07678011
Summary: 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 \(_4F_3(1)\). 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.
MSC:
| 11B83 | Special sequences and polynomials |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 34A08 | Fractional ordinary differential equations |
| 34K06 | Linear functional-differential equations |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
Chebyshev polynomials; duplication formula; fractional differential equations; hypergeometric functions; tau methodSoftware:
SumToolsReferences:
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