Some modifications of the quasilinearization method with higher‐order convergence for solving nonlinear BVPs

In this paper, modifications of the quasilinearization method with higher‐order convergence for solving nonlinear differential equations are constructed. A general technique for systematically obtaining iteration schemes of order m ( > 2) for finding solutions of highly nonlinear differential equations is developed. The proposed iterative schemes have convergence rates of cubic, quartic and quintic orders. These schemes were further applied to bifurcation problems and to obtain critical parameter values for the existence and uniqueness of solutions. The accuracy and validity of the new schemes is tested by finding accurate solutions of the one‐dimensional Bratu and Frank‐Kamenetzkii equations.