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Keywords = dual-Lagrange multiplier variational iteration method

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16 pages, 351 KB  
Article
Iterative Integro-Differential Techniques Based on Green’s Function for Two-Point Boundary-Value Problems of Ordinary Differential Equations
by Juan I. Ramos
Axioms 2026, 15(1), 65; https://doi.org/10.3390/axioms15010065 (registering DOI) - 17 Jan 2026
Abstract
Several iterative integro-differential formulations for two-point, second- and third-order, nonlinear, boundary-value problems of ordinary differential equations based on Green’s functions and the method of variation of parameters are presented. It is shown that the generalized or dual Lagrange multiplier method (GVIM) previously developed [...] Read more.
Several iterative integro-differential formulations for two-point, second- and third-order, nonlinear, boundary-value problems of ordinary differential equations based on Green’s functions and the method of variation of parameters are presented. It is shown that the generalized or dual Lagrange multiplier method (GVIM) previously developed for the iterative solution of nonlinear, boundary-value problems of ordinary differential equations that makes use of modified functionals and two Lagrange multipliers, is nothing but an iterative Green’s function formulation that does not require Lagrange multipliers at all. It is also shown that the two Lagrange multipliers of GVIM are associated with the left and right Green’s functions. The convergence of iterative methods based on both the Green function and the method of variation of parameters is proven for nonlinear functions that depend on the dependent variable and is illustrated by means of two examples. Several new iterative integro-differential formulations based on Green’s functions that use a multiplicative function for convergence acceleration are also presented. Full article
(This article belongs to the Section Mathematical Analysis)
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