An optimal analytical method for nonlinear boundary value problems based on method of variation of parameter

Abstract

<span style="font-family: 'Times New Roman','serif'; font-size: 14pt;">In this paper, the authors present a modified convergent analytic algorithm for the solution of nonlinear boundary value problems by means of a variable parameter method and briefly, the method is called optimal variable parameter method. This method, based on the embedding of a parameter and an auxiliary operator, provides a computational advantage for the convergence of the approximate solutions of nonlinear differential equations. The developed convergence has been shown and its details are discussed. Additionally, a convenient method is considered for selecting an optimal value of the auxiliary parameter, via minimizing the residual error over the domain of problem.</span><br /> <span style="font-family: 'Times New Roman','serif'; font-size: 14pt;">The effectiveness of the method and the accuracy of the proposed algorithm are illustrated by the implementation of physical problems such as Sturm-Liouville problem, Airy equation, and Quantum mechanical harmonic oscillator problem.</span><br /> <span style="font-family: 'Times New Roman','serif'; font-size: 14pt;">The numerical results and obtained demonstrate clearly reflect the accuracy of the method and its convergence.</span>