An efficient 6th-order compact difference scheme with error estimation for nonlocal Lane–Emden equation

Abstract

In this manuscript, we present a new and efficient 6th-order compact difference method for solving the nonlocal Lane–Emden equation. This method effectively addresses singular-type problems without the need to modify or remove the singularities. To achieve this, we construct a uniform mesh across the domain and develop a new sixth-order discrete method that approximates the derivatives, transforming the differential equation into a system of equations. Use Newton or any iterative technique to obtain the numerical solution of the system of equations. Our new scheme efficiently handles the singularity at $t=0$. Additionally, we conduct a mathematical analysis of the method’s consistency, stability, error bounds, and convergence rate. We also include several numerical problems from the existing literature to demonstrate the accuracy, efficiency, and applicability of the proposed scheme. Also, compare the numerical approximations with the existing recent techniques. The newly proposed scheme offers 6th-order accuracy using a small-size matrix and delivers better numerical results than the existing methods.