Abstract
Abstract In this study, a new iterative method for third-order boundary value problems based on embedding Green’s function is introduced. The existence and uniqueness theorems are established, and necessary conditions are derived for convergence. The accuracy, efficiency and applicability of the results are demonstrated by comparing with the exact results and existing methods. The results of this paper extend and generalize the corresponding results in the literature.
Highlights
The iterative methods are used to solve initial and boundary value problems (BVPs) in image and restoration problems, variational inequality problems and etc
Several notable researchers introduced many fixed point iterative methods to approximate the solution of a given problem for better approximation with a minimum error
Finding the solution of nonlinear initial and BVPs, second- or third-order differential equations, has become a very interesting problem. [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] and references therein are some of these studies
Summary
The iterative methods are used to solve initial and boundary value problems (BVPs) in image and restoration problems, variational inequality problems and etc. Several notable researchers introduced many fixed point iterative methods to approximate the solution of a given problem for better approximation with a minimum error Khan-Green’s fixed point iterative method is generalized and extended for the approximate solution of third-order BVPs. The existence and uniqueness theorems for generalized method are established, and necessary conditions are derived for convergence. The new method is implemented on several numerical examples including linear and nonlinear third-order BVPs. Effectiveness is established with better approximation with minimum error when compared to exact solutions and Picard-Green’s solutions
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