An optimization method for solving fractional oscillation equation

Abstract

This paper seeks to present an optimization method to estimate the solutions of nonlinear oscillation equations of fractional order. The mentioned method is based on Bernstein polynomials (Bps). In the presented numerical approach, the operational matrices of the ordinary and fractional derivatives of Bernstein polynomials are utilized to estimate the solution of the model under the study. In this technique, the unknown function is expanded in terms of Bps. By using the residual function and its 2-norm, the problem under consideration is converted into a constrained nonlinear optimization one. So that, the constraint equations are obtained from the given initial conditions and the object function is obtained from the residual function. Finally, we obtain the unknown coefficients optimally by a set of unknown Lagrange multipliers. The main advantage of this approach is that it reduces such problems to those optimization problems, which greatly simplifies them and also leads to obtain a good approximate solution for them. The accuracy and efficiency of the presented method are supported by some examples. At the end, we compare the numerical results with other results.

Keywords

• Optimization method
• Operational matrix
• Oscillation equations
• Bernstein polynomials