Pell-Lucas collocation method for solving a class of second order nonlinear differential equations with variable delays

Abstract

: In this study, the approximate solution of the nonlinear differential equation with variable delays is investigated by means of a collocation method based on the truncated Pell-Lucas series. In the first stage of the method, the assumed solution form (the truncated Pell-Lucas polynomial solution) is expressed in the matrix form of the standard bases. Next, the matrix forms of the necessary derivatives, the nonlinear terms, and the initial conditions are written. Then, with the help of the equally spaced collocation points and these matrix relations, the problem is reduced to a system of nonlinear algebraic equations. Finally, the obtained system is solved by using MATLAB. The solution of this system gives the coefficient matrix in the assumed solution form. Moreover, the error analysis is performed. Accordingly, two theorems about the upper limit of the errors and the error estimation are given and these theorems are proven. In addition, the residual improvement technique is presented. The presented methods are applied to three examples. The obtained results are displayed in tables and graphs. Also, the obtained results are compared with the results of other methods in the literature. All results in this study have been calculated by using MATLAB.