A convergence-preserving non-standard finite difference scheme for the solutions of singular Lane-Emden equations

Abstract

The derivation of numerical schemes for the solution of Lane-Emden equations requires meticulous consideration because they are highly nonlinear in nature; have singularity behaviors at the origin and in some cases, their exact solutions are known for only a few parameter ranges. This explains the reason for the failure of some existing methods in approximating such problems. Thus, in this research, a convergence-preserving Non-Standard Finite Difference Scheme (NSFDS) shall be derived for the solution of Lane-Emden equations. The main idea in this work is the approximation of both the linear and nonlinear terms non-locally and the renormalization/reformulation of the numerator and denominator functions of the Lane-Emden equations. The need for this approach came up due to some setbacks of existing methods where the exact solutions’ qualitative properties are not routinely transferred to the approximate solutions. The analysis of some properties of the NSFDS like convergence, dynamical consistency, monotone dependence on the initial value, and monotonicity of solutions as well as perturbation solution analysis shall be carried out. The NSFDS was then employed in solving some classes of singular initial value problems of the Lane-Emden type and the results obtained show that the proposed method is efficient and computationally cheap.