Transforming Controlled Duffing Oscillator to Optimization Schemes Using New Symmetry-Shifted G(t)-Polynomials

Abstract

This work introduces and studies the important properties of a special class of new symmetry-shifted Gt-olynomials (NSSG). Such polynomials have a symmetry property over the interval [−2, 0], with Gn−2,00=−1nGn−2,0(−2). An explicit formulation of an NSSG operational matrixwas constructed, which served as a powerful tool for obtaining the desired numerical solutions. Then, a modified direct computational algorithm was suggested for solving the controlled Duffing oscillator problem. The idea behind the proposed algorithm is based on using symmetry basis functions, which are important and have real-world applications in physics and engineering. The original controlled Duffing oscillator problem was transformed into a nonlinear quadratic programming problem. Finally, numerical experiments are presented to validate our theoretical results. The numerical results emphasize that the modified proposed approach reaches the desired value of the performance index, with a few computations and with the minimum order of the NSSG basis function when compared with the other existing method, which is an important factor to consider when choosing the appropriate method in other mathematical and engineering applications.