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.

中文翻译：

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求解分数延迟微分方程的分数阶斐波那契混合函数方法

本文的目的是提出一种寻找分数延迟微分方程近似解的有效方法。该技术基于块脉冲和分数阶斐波那契多项式的混合函数。首先，我们定义分数阶斐波那契多项式。接下来，使用分数阶的斐波那契多项式，我们引入了一组新的基函数。这些新函数称为分数阶斐波那契混合函数 (FFHF)，适用于平滑和分段平滑函数的逼近。获得了FFHF的黎曼-刘维尔积分运算矩阵和延迟运算矩阵。然后，使用这些矩阵和搭配方法，将问题简化为代数方程组。使用牛顿迭代法，我们解决了这个系统。提出了一些例子来测试本方法的效率和有效性。鉴于这些分数方程在许多现象的建模中的应用，这项工作的一个数值示例包括描述人口增长率的哈钦森模型。