In this paper, a modified reproducing kernel algorithm is proposed to solve a class of quadratic and cubic logistic equations with Caputo-Fabrizio fractional derivative in Hilbert space. These equations are the generalization of Verhulst’s model which describes population growth taking in account that individuals will compete for limited resources. A novel reproducing kernel function is constructed to create an orthogonal system and to calculate the analytical and approximate solutions in the desirable Sobolev space. The stability, convergence, and complexity of the proposed approach are discussed. Furthermore, the effects of the Caputo-Fabrizio fractional derivatives are studied in solving the population growth model comparing with those of the classical Caputo derivatives. The main motivation for using the proposed technique is high accuracy and low computational cost compared to other existing methods especially when involving fractional differentiation operators. In this orientation, the effectiveness, applicability, and feasibility of this technique are verified by numerical examples. In a numerical viewpoint, the obtained results indicate that the suggested intelligent method has many advantages in accuracy and stability using the new Caputo-Fabrizio derivative.

针对希尔伯特空间中的Caputo-Fabrizio分数阶导数，提出了一种改进的再生核算法来求解一类二次和三次对数方程。这些方程式是Verhulst模型的概括，该模型描述了人口增长，并考虑到个人将竞争有限的资源。构建了一种新颖的再生核函数，以创建正交系统并计算所需Sobolev空间中的解析解和近似解。讨论了所提出方法的稳定性，收敛性和复杂性。此外，与经典Caputo衍生物相比，研究了Caputo-Fabrizio分式衍生物在解决人口增长模型中的作用。与其他现有方法相比，使用提出的技术的主要动机是高精度和低计算成本，尤其是在涉及分数微分算符时。在这种情况下，通过数值实例验证了该技术的有效性，适用性和可行性。从数值角度来看，所得结果表明，使用新型Caputo-Fabrizio衍生物，所建议的智能方法在准确性和稳定性方面具有许多优势。