A Chebyshev functions method for solving linear and nonlinear fractional differential equations based on Hilfer fractional derivative
محورهای موضوعی : Numerical analysisM. H. Derakhshan 1 , A. Aminataei 2
1 - Faculty of Mathematics, K. N. Toosi University of Technology, P. O. Box: 16765-3381, Tehran, Iran
2 - Faculty of Mathematics, K. N. Toosi University of Technology, P. O. Box: 16765-3381, Tehran, Iran
کلید واژه: Operational matrix, Chebyshev polynomials, fractional-order differential equation, Hilfer fractional derivative,
چکیده مقاله :
The theory of derivatives and integrals of fractional in fractional calculus have found enormousapplications in mathematics, physics and engineering so for that reason we needan efficient and accurate computational method for the solution of fractional differential equations.This paper presents a numerical method for solving a class of linear and nonlinear multi-order fractional differential equations with constant coefficients subject to initial conditions based on the fractional order Chebyshev functions that this function is defined as follows:\begin{equation*}\overline{T}_{i+1}^{\alpha}(x)=(4x^{\alpha}-2)\overline{T}_{i}^{\alpha}(x)\overline{T}_{i-1}^{\alpha}(x),\,i=0,1,2,\ldots,\end{equation*}where $\overline{T}_{i+1}^{\alpha}(x)$ can be defined by introducing the change of variable $x^{\alpha},\,\alpha>0$, on the shifted Chebyshevpolynomials of the first kind. This new method is an adaptation of collocationmethod in terms of truncated fractional order Chebyshev Series. To do this method, a new operational matrix of fractional order differential in the Hilfer sense for the fractional order Chebyshev functions is derived. By using this method we reduces such problems to those ofsolving a system of algebraic equations thus greatly simplifying the problem. At the end of this paper, several numerical experiments are given to demonstrate the efficiency and accuracy of the proposed method.
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