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Some modifications of the quasilinearization method with higher-order convergence for solving nonlinear BVPs. (English) Zbl 1271.65115
Summary: 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 are 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.

##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
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