Department of Mathematics, University of Karachi, Karachi 75270, Pakistan.
Department of Humanities & Social Sciences, Bahria University, Karachi 75260, Pakistan.
J Adv Res. 2020 Dec 3;32:109-118. doi: 10.1016/j.jare.2020.11.015. eCollection 2021 Sep.
The fusion of fractional order differential equations and fuzzy numbers has been widely used in modelling different engineering and applied sciences problems. In addition to these, the Allee effect, which is of high importance in field of biology and ecology, has also shown great contribution among other fields of sciences to study the correlation between density and the mean fitness of the subject.
The present paper is intended to measure uncertain dynamics of an economy by restructuring the Cobb-Douglas paradigm of the renowned Solow-Swan model. The purpose of study is further boosted innovatively by subsuming the perception of logistic growth with Allee effect in the dynamics of physical capital and labor force.
Fractional order derivative and neutrosophic fuzzy (NF) theory are applied on the parameters of the Cobb-Douglas equation. Distinctively, cogitating fractional order derivative to study the change at each fractional stage; single-valued triangular neutrosophic fuzzy numbers (SVTNFN) to cope the uncertain situations; logistic growth function with Allee effect to analyze the factors in natural way, are the significant and novel features of this endeavor.
The incorporation of the aforementioned theories and effects in the Cobb-Douglas equation, resulted in producing maximum sustainable capital investment and maximum capacity of labor force. The solutions in intervals located different possible solutions for different membership degrees, which accumulated the uncertain circumstances of a country.
Explicitly, these notions add new facts and figures not only in the dynamical study of capital and labor, which has been overlooked in classical models, but also left the door open for discussion and implementation on classical models of different fields.
分数阶微分方程和模糊数的融合已经广泛应用于建模不同的工程和应用科学问题。除此之外，在生物学和生态学领域非常重要的阿利效应，在其他科学领域也对研究密度和主体平均适应性之间的相关性做出了重要贡献。
本文旨在通过重构著名的索洛-斯旺模型的柯布-道格拉斯范例来衡量经济的不确定动态。本研究的目的通过在物质资本和劳动力的动态中纳入对数增长与阿利效应的感知进一步得到创新性的增强。
分数阶导数和 neutrosophic 模糊（NF）理论应用于柯布-道格拉斯方程的参数上。特别地，通过考虑分数阶导数来研究每个分数阶阶段的变化；通过单值三角 neutrosophic 模糊数（SVTNFN）来应对不确定情况；通过带有阿利效应的对数增长函数来自然地分析因素，是这项努力的显著和新颖的特点。
将上述理论和效应纳入柯布-道格拉斯方程中，导致产生了最大可持续资本投资和最大劳动力容量。解决方案位于不同可能的解决方案的不同区间，为不同的隶属度积累了不确定情况。
明确地，这些概念不仅在经典模型中被忽视的资本和劳动力的动态研究中增加了新的事实和数字，而且为不同领域的经典模型的讨论和实施打开了大门。
