Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

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.

Automated summary

This article presents an approximation technique using fractional order Bernstein wavelets for numerical simulations of fractional oscillation equations with variable order. The equations describe electrical circuits exhibiting various nonlinear dynamical behaviors. The proposed variable order model has current interest in engineering and applied sciences. To analyze the behavior of the equations under variable-order fractional operator, the proposed model is converted into nonlinear algebraic equations using collocation nodes. Different cases of the model are examined to demonstrate the precision and performance of the method. The results confirm the simplicity and efficiency of the scheme for studying nonlinear random order fractional models in engineering and science.