Fractional calculus can be used to fully describe numerous real-world situations in a wide range of scientific disciplines, including natural science, social science, electrical, chemical, and mechanical engineering, economics, statistics, weather forecasting, and particularly biomedical engineering. Different types of derivatives can be useful when solving various fractional calculus problems. In this study, we suggested a single step modified one parameter family of the Caputo-type fractional iterative method. Convergence analysis shows that the proposed family of methods' order of convergence is $\vartheta +1$. To determine the error equation of the proposed technique, the computer algebra system CAS-Maple is employed. To illustrate the accuracy, validity, and usefulness of the proposed technique, we consider a few real-world applications from the fields of civil and chemical engineering. In terms of residual error, computational time, computational order of convergence, efficiency, and absolute error, the test examples' acquired numerical results demonstrate that the newly proposed algorithm performs better than the other classical fractional iterative scheme already existing in the literature. Using the computer program Mathematia 9.0, we compare the draw basins of attraction of the suggested fractional numerical algorithm to those of the currently used fractional iterative methods for the graphical analysis. The graphical results show how quickly the newly developed fractional method converges, confirming its supremacy to other techniques.

分数阶微积分可用于全面描述各种科学学科中的众多现实情况，包括自然科学、社会科学、电气、化学和机械工程、经济学、统计学、天气预报，尤其是生物医学工程。在解决各种分数阶微积分问题时，不同类型的导数可能很有用。在这项研究中，我们建议单步修改 Caputo 型分数阶迭代方法的一个参数族。收敛性分析表明，所提出的方法族的收敛顺序是$\theta +\mathrm{1个}$. 为了确定所提出技术的误差方程，采用了计算机代数系统 CAS-Maple。为了说明所提出技术的准确性、有效性和实用性，我们考虑了土木和化学工程领域的一些实际应用。在残差、计算时间、计算收敛阶数、效率和绝对误差方面，测试实例获得的数值结果表明，新提出的算法比文献中已有的其他经典分数迭代方案性能更好。使用计算机程序 Mathematia 9.0，我们将建议的分数阶数值算法的吸引力盆地与目前用于图形分析的分数阶迭代方法的吸引力盆地进行比较。