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This paper is concerned with deriving an operational matrix of fractional-order integration of Fibonacci polynomials. As an application of this matrix, a spectral algorithm for solving some fractional-order initial value problems is exhibited and implemented. The key idea for obtaining the suggested spectral numerical solutions for these equations is actually based on utilizing the developed Fibonacci operational matrix along with the application of tau method in order to reduce the fractional-order differential equation with its initial conditions into a system of linear algebraic equations in the unknown expansion coefficients which can be efficiently solved. Some illustrative examples are included aiming to ascertain the efficiency and applicability of the presented algorithm. The numerical results reveal that the proposed algorithm is easy and applicable.

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