A two-branched numerical solution of the two-dimensional Bratu's problem

We intend to numerically solve the famous two-dimensional Bratu's problem through developing a new iterative finite difference algorithm. The proposed scheme is capable of accurately approximating both branches of the solution of the considered problem. We first introduce a new transformation of Bratu's problem conserving the solution bifurcated behavior. Then, using Newton-Raphson-Kantorovich approximation in function space, we develop an iterative finite difference method yielding a simple algorithm for approximating the sequence of numerical solutions. We also perform a convergence analysis proving that our algorithm converges quadratically to the exact solution of the problem. Finally, we present numerical simulation results showing the capability of our algorithm to accurately compute the two-branches of the solution for the two-dimensional Bratu's problem.