Fractional-order Fibonacci-hybrid functions approach for solving fractional delay differential equations

The aim of the current paper is to propose an efficient method for finding the approximate solution of fractional delay differential equations. This technique is based on hybrid functions of block-pulse and fractional-order Fibonacci polynomials. First, we define fractional-order Fibonacci polynomials. Next, using Fibonacci polynomials of fractional-order, we introduce a new set of basis functions. These new functions are called fractional-order Fibonacci-hybrid functions (FFHFs) which are appropriate for the approximation of smooth and piecewise smooth functions. The Riemann–Liouville integral operational matrix and delay operational matrix of the FFHFs are obtained. Then, using these matrices and collocation method, the problem is reduced to a system of algebraic equations. Using Newton’s iterative method, we solve this system. Some examples are proposed to test the efficiency and effectiveness of the present method. Given the application of these kinds of fractional equations in the modeling of many phenomena, a numerical example of this work includes the Hutchinson model which describes the rate of population growth.