Pell-Lucas Collocation Method to Solve Second-Order Nonlinear Lane-Emden Type Pantograph Differential Equations

Abstract

In this article, we present a collocation method for second-order nonlinear Lane-Emden type pantograph differential equations under intial conditions. According to the method, the solution of the problem is sought depending on the Pell-Lucas polynomials. The Pell-Lucas polynomials are written in matrix form based on the standard bases. Then, the solution form and its the derivatives are also written in matrix forms. Next, a transformation matrix is constituted for the proportion delay of the solution form. By using the matrix form of the solution, the nonlinear term in the equation is also expressed in matrix form. By using the obtained matrix forms and equally spaced collocation points, the problem is turned into an algebraic system of equations. The solution of this system gives the coefficient matrix in the solution form. In addition, the error estimation and the residual improvement technique are also presented. All presented methods are applied to three examples. The results of applications are presented in tables and graphs. In addition, the results are compared with the results of other methods in the literature.