Abstract
In this paper, we solve a system of fractional differential equations within a fractional derivative involving the Mittag-Leffler kernel by using the spectral methods. We apply the Chebyshev polynomials as a base and obtain the necessary operational matrix of fractional integral using the Clenshaw–Curtis formula. By applying the operational matrix, we obtain a system of linear algebraic equations. The approximate solution is computed by solving this system. The regularity of the solution investigated and a convergence analysis is provided. Numerical examples are provided to show the effectiveness and efficiency of the method.
Highlights
1 Introduction Despite the long history of fractional calculus in the field of mathematics, a large amount of real world applications of this field has appeared mainly during the last decades. This type of calculus has become so wide that almost no branch of science and engineering cannot be found without fractional calculus and a lot of books have been written in these regards
We recall that the Riemann–Liouville definition entails physically unacceptable initial conditions [1]; for the Liouville–Caputo fractional derivative, the initial conditions are expressed in terms of integer-order derivatives having direct physical significance [1, 5]
A few years ago Caputo and Fabrizio [6] have opened the following subject of debate within the mathematical community: is it possible to describe all nonlocal phenomena within the same basic kernels, namely the power kernel involved within the definition of Riemann–Liouville derivative and some other few basic fractional derivatives
Summary
Despite the long history of fractional calculus in the field of mathematics, a large amount of real world applications of this field has appeared mainly during the last decades. Numerical solution of the system (1) using operational matrix spectral methods based on Chebyshev polynomials is very important. We use this formula to obtain the operational matrix of the fractional integration.
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