A Pell-Lucas approximation to solve the Abel equation of the second kind

Abstract

In this paper, a collocation method based on the Pell-Lucas polynomials is presented for the numerical solution of the Abel equation of the second kind. Abel equation of the second kind corresponds to a nonlinear differential equation. As a first step, the matrix forms of the Pell-Lucas polynomials and the assumed solution form are constructed. Then, the first derivative of solution, the nonlinear terms and the initial condition are introduced in matrix forms. By using the evenly spaced collocation points and the matrix relations, nonlinear problem is transformed to a system of the nonlinear algebraic equations. Consequently, the solution of this system gives the coefficients of the assumed approximate solution. In addition, the error analysis is done. Accordingly, an upper bound of the errors is stated. Besides, error estimation technique and the residual improvement technique are presented by using residual function. Moreover, the method is applied to two examples and the comparisons are made with the results of other methods in tables to show the computational efficiency of present method. The code of method and the presented graphics are constituted by using Matlab program.