Refined variational iteration method through optimal linear operator and Lagrange multiplier: Local/global convergence

Our focus is on refining the Variational Iteration Method (VIM), which will henceforth be called R-VIM. We begin by introducing a general linear operator and consequently the Lagrange's multiplier; then, minimizing the residual error, we subsequently find the optimal linear operator as well as the best fitting Lagrange's multiplier for a given differential equation. By means of Banach Fixed Point Theorem, the necessary conditions for the convergence of the solutions of VIM, Optimal VIM, and R-VIM are established involving the Lagrange's multiplier; it has been demonstrated (and subsequently numerically certified) that solutions of VIM and Optimal VIM locally converge. While R-VIM solutions, with the aid of necessary and sufficient criteria, have been shown to converge globally for appropriate values of the unknown parameters involved in the general linear operator. We make sure that the convergence of the VIM solution depends critically on the optimal selection of the linear operator and, consequently, the best suited Lagrange's multiplier. Our bird's eye view would be to differentiate between the contributions of the homotopy equation in homotopy-based approaches and Lagrange's multiplier-based correction functional of the VIM for the fastest convergence. Furthermore, in order to achieve fast convergence, we additionally examine if the optimal VIM's regulating parameter offers any further benefit above proposed RVIM. We aim to apply R-VIM to problems with multiple solutions to ensure that all solutions are accurately identified. Our proposed method has been proven superior and more widely applicable through both theoretical and numerical illustrations. However, it also has its limits. Hence, we suggest the implementation of the multistep R-VIM, referred to as MR-VIM, as a means to solve chaotic dynamical systems.