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Kavooci, Zahra, Ghanbari, Kazem, Mirzaei, Hanif. (1402). Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval‎. سامانه مدیریت نشریات علمی, 12(2), 226-235. doi: 10.22034/cmde.2023.54630.2275
Zahra Kavooci; Kazem Ghanbari; Hanif Mirzaei. "Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval‎". سامانه مدیریت نشریات علمی, 12, 2, 1402, 226-235. doi: 10.22034/cmde.2023.54630.2275
Kavooci, Zahra, Ghanbari, Kazem, Mirzaei, Hanif. (1402). 'Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval‎', سامانه مدیریت نشریات علمی, 12(2), pp. 226-235. doi: 10.22034/cmde.2023.54630.2275
Kavooci, Zahra, Ghanbari, Kazem, Mirzaei, Hanif. Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval‎. سامانه مدیریت نشریات علمی, 1402; 12(2): 226-235. doi: 10.22034/cmde.2023.54630.2275

Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval‎

Computational Methods for Differential Equations
مقاله 3، دوره 12، شماره 2، خرداد 2024، صفحه 226-235    اصل مقاله (361.34 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2023.54630.2275
نویسندگان
Zahra Kavooci1؛ Kazem Ghanbari* 1، 2؛ Hanif Mirzaei1
1Faculty of Sciences, Sahand University of Technology, Tabriz, Iran.
2School of Mathematics and Statistics, Carleton University, Ottawa, Canada.
چکیده
Most of fractional differential equations are considered on a fixed interval. In this paper, we consider a typical fractional differential equation on a symmetric interval $[-\alpha,\alpha]$, where $\alpha$ is the order of fractional derivative. For a positive real number α we prove that the solutions are  $T_{n,\alpha}(x)=(\alpha+x)^\frac{1}{2}Q_{n,\alpha}(x)$ where $Q_{n,\alpha}(x)$ produce a family of orthogonal polynomials with respect to the weight function$w_\alpha(x)=(\frac{\alpha+x}{\alpha-x})^{\frac{1}{2}}$ on $[-\alpha,\alpha]$. For integer case $\alpha = 1 $, we show that these polynomials coincide with classical Chebyshev polynomials of the third kind. Orthogonal properties of the solutions lead to practical results in determining solutions of some fractional differential equations. 
کلیدواژه‌ها
Orthogonal polynomials؛ Fractional Chebyshev differential equation؛ Riemann-Liouville and Caputo derivatives
مراجع
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