An iterative finite difference method for approximating the two-branched solution of Bratu's problem

In this paper, we propose an iterative finite difference (IFD) scheme to simultaneously approximate both branches of a two-branched solution to the one-dimensional Bratu's problem. We first introduce a transformation to convert Bratu's problem into a simpler one. The transformed nonlinear ordinary differential equation is discretized using the Newton–Raphson–Kantorovich approximation in function space. The convergence of the sequence of approximations is proved to be quadratic. Then, we apply the classical finite difference method to approximate the sequence of approximations. The proposed new scheme has two main advantages. First, it produces accurate numerical solutions with low computational cost. Second, it is able to compute the two branches of the solution of Bratu's problem, even for small values of the transition parameter λ where the numerical computation of the upper branch of the solution becomes challenging. Numerical examples are provided to show the efficiency and accuracy of the proposed scheme.