An iterative technique for a class of Dirichlet nonlinear BVPs: Troesch’s problem

A coupled iterative approach based on quasilinearization and Krasnoselskii–Mann’s approximation for evaluating the solutions of nonlinear two point Dirichlet boundary value problems (BVPs) is illustrated. The nonlinear problems (including extremely nonlinear Troesch’s problem) are reduced to a sequence of linear equations by using the quasilinearization approach with Green’s function. Additionally, the Krasnoselskii–Mann’s approximation technique is applied to enhance the efficiency of the proposed approach. Some numerical problems, including Troesch’s problems, have been solved to exemplify the method’s ease of implementation. The comparison (in term of absolute errors) with existing techniques like Laplace, Sinc-Galerkin, homotopy perturbation method (HPM), Picard embedded Green’s function method (PGEM), Mann’s embedded Green’s function method (MGEM) and B-Spline method etc. demonstrate that the proposed method is extremely precise and converges quickly.