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

Abstract

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.

New Stability Results for Abstract Fractional Differential Equations with Delay and Non-Instantaneous Impulses

“…Therefore, we guarantee the existence of a distinct mild solution defined on the interval [−1, 3] for the given problem (12). In conclusion, condition (H 5 ) is fulfilled by Z (ϑ) =…”

“…In recent years, there has been significant progress in both ordinary and partial fractional differential equations. For more details on the applications of fractional calculus, the reader is directed to the books of Abbas et al [1][2][3], Herrmann [4], Hilfer [5], Kilbas et al [6], Samko et al [7], and Zhou [8] and papers [9][10][11][12][13][14][15]. In [16,17], Benchohra et al demonstrated the existence, uniqueness, and stability results for various classes of problems with different conditions and some form of extension of the well-known Hilfer fractional derivative, which unifies the Riemann-Liouville and Caputo fractional derivatives.…”