On the numerical solution of the Lane-Emden, Bratu and Troesch equations.

Abstract

Many engineering and physics problems are modelled using differential equations, which may be highly nonlinear and difficult to solve analytically. Numerical techniques are often used to obtain approximate solutions. In this study, we consider the solution of three nonlinear ordinary differential equations; namely, the initial value Lane-Emden equation, the boundary value Bratu equation, and the boundary value Troesch problem. For the Lane- Emden equation, a comparison is made between the accuracy of solutions using the finite difference method and the multi-domain spectral quasilinearization method along with the exact solution. We found that the multi-domain spectral quasilinearization method gave a better solution. For the Bratu problem, a comparison is made between the spectral quasilinearization method and the higher-order spectral quasilinearization method. The higher-order spectral quasilinearization method gave more accurate results. The Troesch problem is solved using the higher-order spectral quasilinearization method and the finite difference method. The solutions obtained are compared in terms of accuracy. Overall, the higher-order spectral quasilinearization method and multi-domain spectral quasilinearization method gave the accurate solutions, making these two methods to be the most reliable for these three problems.