Bratu’s problem: A novel approach using fixed-point iterations and Green’s functions

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In this article, the one-dimensional non-linear Bratu’s boundary value problem is solved via a novel approach that combines Green’s function and fixed point iterative schemes, such as Picard’s and Krasnoselskii–Mann’s. The convergence of the introduced iterative algorithm is proved using the contraction principle. The method is supported by considering a number of numerical examples that correspond to different cases of eigenvalues. The procedure underlying the strategy reduces calculations and provides highly accurate results in comparison with the exact solution and/or numerical solutions provided in the literature. The current method overcomes the difficulty of treating the problem for eigenvalues near and at the critical value, such as λ=3 and λ=3.51, and handles them reliably and very efficiently.