Non-optimal fourth-order and optimal sixth-order B-spline collocation methods for Lane-Emden boundary value problems

Abstract

In the present work, the authors describe two B-spline collocation methods for a class of nonlinear singular Lane-Emden type equations which describe several phenomena in theoretical physics and astrophysics. The first method is based on non-optimal quintic B-spline collocation approach, while the second method is based on optimal quintic B-spline collocation approach. We note that the former method yields fourth-order convergent approximation to the solution of the second-order boundary value problems. To obtain an optimal convergence of order six to the solution of the same problem, the later method is designed by perturbing the original problem. Convergence results for both methods are established via Green's function approach. Four nonlinear examples are provided to illustrate the suggested methods and to verify their theoretical rates of convergence. It is observed that the first method yields non-optimal fourth order accuracy and the second method exhibits optimal sixth order accuracy for the solution of the problem. Moreover, numerical results appear to be higher accurate when compared to other methods.