An efficient technique based on Green's function for solving two-point boundary value problems and its convergence analysis

This study proposes an accurate approximation to the solution of second-order nonlinear two-point boundary value problems, including the well-known Bratu problem, using an iterative technique based on Green's function. The approach relies on constructing an equivalent integral representation of the problem incorporating Green's function. The proposed methodology provides a reliable approximate solution and takes just a few iterations to achieve good accuracy. The mathematical formulation is further supported by discussing in detail the convergence analysis of this approach. Different numerical examples are used to check the robustness and effectiveness of the scheme. The numerical testing for nonlinear problems with nonlinear boundary conditions demonstrates that the proposed method outperforms other existing methods, including the finite element method, the finite volume method, the finite difference method, the B-spline method, and the Adomain's decomposition method.