Numerical Solutions of Duffing Equations Involving Linear Integral with Shifted Chebyshev Polynomials

Abstract

The purpose of this study is to give a shifted Chebyshev polynomial approximation for the solution of Duffing-van der Pol equation involving linear integral term (DEILI). For this purpose, a new Chebyshev collocation method is introduced. This method is based on taking the truncated shifted Chebyshev expansion of the function. This method based on first taking the truncated Chebyshev series of the solution function in the DEILI and then, transforms DEILI and given conditions into a matrix equation and then, we have the system of nonlinear algebraic equation using collocation points. Then, solving the system of algebraic equations we have the coefficients of the truncated Chebyshev series. In addition, examples that illustrate the pertinent features of the method are presented, and the results of study are discussed.

Bu çalışmanın amacı linear terim içeren Duffing-van der Pol denkleminin shifted Chebyshev polinomları yardımı ile yaklaşık çözümlerini sunmaktır. Bu amaçla Chebyshev sıralama metodu verilmiştir. Metodun ana karekteristiği verilen denklemi kesilmiş Chebyshev serisinin katasyılarının içeren bir denklem sistemine indirgemesidir. Bu sistem çözülerek kesilmiş Chebyshev serisinin katsayıları bulunur. Dolayısıyla yaklaşık çözüm elde edilir. Ayrıca, metodun uygulanabilirlini göstermek için örnekler sunulmuştur.