New Solution of the Sine-Gordon Equation by the Daftardar-Gejji and Jafari Method

In this article, the Daftardar-Gejji and Jafari method (DJM) is used to obtain an approximate analytical solution of the sine-Gordon equation. Some examples are solved to demonstrate the accuracy of DJM. A comparison study between DJM, the variational iteration method (VIM), and the exact solution are presented. The comparison of the present symmetrical results with the existing literature is satisfactory.

Modelling of natural growth with memory effect in economics: An application of adomian decomposition and variational iteration methods

The power-law memory effect is taken into consideration in a generalisation of the economic model of natural growth. The memory effect refers to a process's reliance on its current state and its history of previous changes. However, the study that focuses on natural growth in economics considering the memory effect with fractional order-linear differential equation model is still limited. The current investigation seeks to solve the natural growth with memory effect in the economics model and decide the best model using fractional differential equation (FDE), namely Adomian Decomposition and Variational Iteration Methods. Also, this study assumes the level of consumer loss memory during a certain time interval denoted by a parameter (α). This study showed the model of loss memory effect with 0 < α ≤ 1 given a slowdown in output growth compared to a model without memory effect. Besides that, this study also found that output Y(t) is growing faster with the Variational Iteration method compared to the Adomian decomposition method. Also, using graphical simulation, this study found the output Y(t) is closer to the exact solution with α=0.4 and α=0.9. In conclusion, this study successfully solved natural growth with memory effect in economics and decided the best model between FDE, namely Adomian decomposition and Variational iterative methods using numerical analysis.

Approximate Solution of Linear Fuzzy Initial Value Problems Using Modified Variaional Iteration Method

The main objective of this paper is to solve fuzzy initial value problems, in which the fuzziness occurs in the initial conditions. The proposed approach, namely the modified variational iteration method, will be used to find the solution of fuzzy initial value problem approximately and to increase the rate of convergence of the variational iteration method. From the obtained results, as it is expected, the approximate results of the proposed method are more accurate than those results obtained without using the modified variational iteration method.

Semianalytical Solutions of Some Nonlinear-Time Fractional Models Using Variational Iteration Laplace Transform Method

In this work, we combined two techniques, the variational iteration technique and the Laplace transform method, in order to solve some nonlinear-time fractional partial differential equations. Although the exact solutions may exist, we introduced the technique VITM that approximates the solutions that are difficult to find. Even a single iteration best approximates the exact solutions. The fractional derivatives being used are in the Caputo-Fabrizio sense. The reliability and efficiency of this newly introduced method is discussed in details from its numerical results and their graphical approximations. Moreover, possible consequences of these results as an application of fixed-point theorem are placed before the experts as an open problem.