In the present work, a generalized fractional integral operational matrix is derived by using classical Legendre wavelets. Then, a numerical scheme based on this operational matrix and Lagrange multipliers is proposed for solving variational problems with fractional order. This approach has been applied on some illustrative examples. The results obtained for these examples demonstrate that the suggested technique is efficient for solving variational problems with fractional order and gives a very perfect agreement with the exact solution. The results are depicted in graphical maps and data tables. The integral square error, maximum absolute error, and order of convergence have been evaluated to analyze the precision of the suggested method. The present scheme provides better and comparable results with some other existing approaches available in the literature.
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.
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