In this article, we implement Variational Iteration method (VIM) and Adomian Decomposition method (ADM) for finding exact solutions of Poisson equation with Dirichlet or Neumann boundary conditions. To illustrate the simplicity and reliability of the methods, three examples are provided. The results obtained by VIM are compared with those of ADM, and then the ability of each method is also discussed. 

A universal approach by Laplace transform to the Variational Iteration method for Fractional derivatives with the nonsingular kernel is presented; in particular, the Caputo-Fabrizio Fractional derivative and the Atangana-Baleanu Fractional derivative with the non-singular kernel is considered. The analysis elaborated for both non-singular kernel derivatives is shown the necessity of considering the Modified Caputo-Fabrizio Fractional derivative and the analogous modifications for the Atangana-Baleanu Fractional derivative with non-singular Mittag-Leffler kernel in order to satisfy the initial conditions for some Fractional differential equations.

This paper demonstrates a study on some significant latest innovations in the approximated techniques to find the approximate solutions of Caputo Fractional Volterra-Fredholm integro-differential equations. To this aim, the study uses the Modified Adomian Decomposition method (MADM) and the Modified Variational Iteration method (MVIM). A wider applicability of these techniques are based on their reliability and reduction in the size of the computational work. This study provides an analytical approximate to determine the behavior of the solution. It proves the existence and uniqueness results and convergence of the solution. In addition, it brings an example to examine the validity and applicability of the proposed techniques.

The Variational Iteration method (VIM) was extended to find approximate solutions of Fractional chemical engineering equations. The Lagrange multipliers of the VIM were not identified explicitly. In this paper we improve the VIM by using concept of fixed point Iteration method. Then this method was implemented for solving system of the time Fractional chemical engineering equations. The obtained approximate solutions are compared with the numerical results in the literature to show the applicability, efficiency and accuracy of the method.

The Haar wavelet collocation with Iteration technique is applied for solving a class of time-Fractional physical equations. The approximate solutions obtained by two dimensional Haar wavelet with Iteration technique are compared with those obtained by analytical methods such as Adomian Decomposition method (ADM) and Variational Iteration method (VIM). The results show that the present scheme is effective and appropriate for obtaining the numerical solution of the timeFractional Modified Camassa-Holm equation and Time Fractional Modified Degasperis-Procesi equation.

The Fractional Variational Iteration technique (FVIM), a dependable semi-analytic approach for solving multi-dimensional Navier-Stokes equations, is explained in this article. The accuracy, efficiency, and convergence of the provided approach are tested using a variety of demonstrative instances.

‎    This paper introduces a Modified version of the Variational Iteration Method, incorporating $\mathbb{P}$-transformation. We propose a novel semi-analytical technique named the Modified Variational Iteration method   for addressing Fractional differential equations featuring tempered Liouville-Caputo derivatives. The Modified Variational Iteration method emerges as a highly efficient and powerful mathematical tool, offering exact or approximate solutions for a diverse range of real-world problems in engineering and the natural sciences, specifically those expressed through differential equations. To assess its effectiveness and accuracy, we scrutinize the Modified Variational Iteration method by applying it to three problems related to the heat-like multidimensional diffusion equation with a Fractional time derivative in a tempered Liouville-Caputo form.