This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of fractional Bernstein polynomials. The oscillation equation describes electrical circuits and exhibits a wide range of nonlinear dynamical behaviors. The proposed variable order model is of current interest in a lot of application areas in engineering and applied sciences. The purpose of this study is to analyze the behavior of the fractional force-free and forced oscillation equations under the variable-order fractional operator. The basic idea behind using the approximation technique is that it converts the proposed model into non-linear algebraic equations with the help of collocation nodes for easy computation. Different cases of the proposed model are examined under the selected variable order parameters for the first time in order to show the precision and performance of the mentioned scheme. The dynamic behavior and results are presented via tables and graphs to ensure the validity of the mentioned scheme. Further, the behavior of the obtained solutions for the variable order is also depicted. From the calculated results, it is observed that the mentioned scheme is extremely simple and efficient for examining the behavior of nonlinear random (constant or variable) order fractional models occurring in engineering and science.

翻译：本文提出了一种基于分数阶Bernstein小波的近似技术，用于变阶分数阶振动方程的数值模拟。该分数阶Bernstein小波通过分数阶Bernstein多项式构造。振动方程可描述电路系统并展现丰富的非线性动力学行为。所提出的变阶模型在工程与应用科学的众多领域具有重要研究价值。本研究旨在分析变阶分数阶算子作用下无外力与受迫振动方程的行为特征。该近似技术的核心思想是通过配置节点将所提模型转化为非线性代数方程组，便于计算求解。为验证该方法的精度与性能，首次针对选定的变阶参数探讨了模型的不同情形。通过表格与图表呈现动态行为与结果，以确认方法的有效性。此外，还展示了变阶情况下所得解的行为特征。计算结果表明，该方法在研究工程与科学中非线性随机（常/变阶）分数阶模型行为时具有极高的简便性与高效性。