An efficient numerical approach for solving three-point Lane-Emden-Fowler boundary value problem

For three-point Lane-Emden-Fowler boundary value problems (LEFBVPs), we propose two robust algorithms consisting of Bernstein and shifted Chebyshev polynomials coupled with the collocation technique. The first algorithm uses the Bernstein collocation method with uniform collocation points, while the second is based on the shifted Chebyshev collocation method with roots as its collocation points. In both algorithms, the equivalent integral equations of three-point LEFBVPs are constructed. Then the approximation and collocation approach are used to generate a system of nonlinear equations. Then we implement Newton's method to solve this system. The current method is different from the traditional method as we do not require to approximate u ' and u '' appearing in the problem. It reduces not only the computational time but also the truncation error. The existence of a unique solution is discussed. Error analysis of both algorithms is demonstrated. A well known property of the proposed techniques is that it provides efficient solution for a few collocation points. The numerical results verify the same. (c) 2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

Automated summary

Two robust algorithms based on Bernstein and shifted Chebyshev polynomials coupled with the collocation technique are proposed for solving three-point Lane-Emden-Fowler boundary value problems (LEFBVPs). The algorithms construct equivalent integral equations of the problems and utilize approximation and collocation approach to generate a system of nonlinear equations, which is then solved by Newton's method. Unlike traditional methods, this approach avoids the need to approximate the derivatives u' and u'', resulting in reduced computational time and truncation error. Numerical results demonstrate the efficiency of the proposed techniques with a few collocation points.