Abstract
We propose a three-step block method of Adam’s type to solve nonlinear second-order two-point boundary value problems of Dirichlet type and Neumann type directly. We also extend this method to solve the system of second-order boundary value problems which have the same or different two boundary conditions. The method will be implemented in predictor corrector mode and obtain the approximate solutions at three points simultaneously using variable step size strategy. The proposed block method will be adapted with multiple shooting techniques via the three-step iterative method. The boundary value problem will be solved without reducing to first-order equations. The numerical results are presented to demonstrate the effectiveness of the proposed method.
Highlights
Boundary value problems (BVPs) arise in many areas of applied mathematics, for example, application to chemical reactor theory [1] and Bratu-type problem [2]
In Problem 3, 3SAM solved the system of boundary value problem subject to two boundary conditions and compared our result to SCM and LRBFM. 3SAM, 2P1BVS, and bvp4c are implemented by variable step size strategy and controlled by the tolerances while MLAM and COLHW used constant step size
In Problem 4, 3SAM solved the system of boundary value problem subject to the Neumann-Dirichlet type boundary conditions which is a free convective boundary-layer flow in a porous medium above a heated horizontal impermeable surface or below a cooled horizontal impermeable surface where wall temperature is a power function of distance from the origin
Summary
Boundary value problems (BVPs) arise in many areas of applied mathematics, for example, application to chemical reactor theory [1] and Bratu-type problem [2]. The homotopy perturbation method that has been introduced by Saadatmandi et al [5] to solve second-order BVPs has less iteration compared to Adomian decomposition method. Asadi et al [6] has extended the modified homotopy perturbation method to solve the nonlinear system of second-order BVPs. The quintic B-spline collocation method has been modified by Lang and Xu [7] to solve second-order BVPs. Srivastava et al [8] developed a numerical algorithm based on the nonpolynomial quintic spline functions for the solution of second-order BVPs with engineering applications, while Ibraheem and Khalaf [9] have proposed a shooting neural networks algorithm for solving. BVPs. Rahman et al [10] solved numerically second-order BVPs by the Galerkin method. The aim of this research is to propose a three-step block method to solve the BVPs directly using multiple shooting techniques with variable step size strategy.
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