Multistage Variational Iteration Method Research Articles

Approximate-Analytic Solution of Hyperchaotic Finance System by Multistage Approach

This paper devotes to constructing an approximate analytic solution for the hyperchaotic finance model. The model describes the time variation of the interest rate, the investment demand, the price exponent, and the average profit margin. The multistage homotopy analysis method (MHAM) and multistage variational iteration method (MVIM) are utilized to generate the analytical solutions. The solutions are presented in terms of continuous piecewise functions without interpolation. These procedures prove their applicability for this kind of model due to rapidly convergent series solutions with easily computable terms, iterates, and efficiently obtained by applying it over multiple time intervals. We also provide the convergences theorem of the MHAM. Numerical comparisons are displayed with the results obtained by MHAM, MVIM, and the fourth-order Runge-Kutta method to demonstrate the validity and effectivity of this procedure.

Multistage variational iteration method for a SEIQR COVID-19 epidemic model with isolation class

The SEIQR COVID-19 epidemic model is in the form of the system of first-order nonlinear differential equations. In this paper we propose multistage versions of the variational iteration method (VIM) to solve this COVID-19 epidemic model. The idea of multistage version is to divide the entire time domain into a finite number of subintervals and then implementing the VIM piecewisely on each subinterval. There are two kinds of multistage methods discussed in this paper, where the difference between the two methods lies in the number of restricted variations used in the correction functional. The multistage methods generally give more accurate solutions on longer time intervals than the classical versions. The multistage VIM with less number of restricted variations has the best performance among all types of variational iteration methods discussed in this paper. The accuracy of multistage VIM solution can be increased by using smaller size of subinterval or by implementing more iterations in each subinterval.

A fast and accurate analytical-numerical method for solving the prey-predator model

We consider the prey-predator mathematical model of population dynamics. For this model, we propose a new analytical-numerical method. Our proposed analytical-numerical method is much faster in terms of computation and more accurate in terms of approximation than an available existing multistage variational iteration method. Our proposed method is promising to be extended for solving other mathematical models.

An Alternative Approach for Nonlinear Optimization Problem with Caputo - Fabrizio Derivative

In this study, a fractional mathematical model with steepest descent direction is proposed to find optimal solutions for a class of nonlinear programming problem. In this sense, Caputo-Fabrizio derivative is adapted to the mathematical model. To demonstrate the solution trajectory of the mathematical model, we use the multistage variational iteration method (MVIM). Numerical simulations and comparisons on some test problems show that the mathematical model generated using Caputo-Fabrizio fractional derivative is both feasible and efficient to find optimal solutions for a certain class of equality constrained optimization problems.

Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM

In this paper, a class of Nonlinear Programming problem is modeled with gradient based system of fractional order differential equations in Caputo's sense. To see the overlap between the equilibrium point of the fractional order dynamic system and theoptimal solution of the NLP problem in a longer timespan the Multistage Variational ?teration Method isapplied. The comparisons among the multistage variational iteration method, the variationaliteration method and the fourth order Runge-Kutta method in fractional and integer order showthat fractional order model and techniques can be seen as an effective and reliable tool for finding optimal solutions of Nonlinear Programming problems.

A new efficient method for solving quadratic Riccati differential equation

A new efficient method called the multistage variational iteration method (MVIM) is applied to the solution of quadratic Riccati differential equations. A comparison between MVIM solution with classical variational iteration method (VIM) and exact solution has been made and show that the MVIM is a powerful method to the solution of nonlinear differential equations.

A Multistage Variational Iteration Method for Solution of Delay Differential Equations

Abstract In this paper, the approximate solution of delay differential equations is obtained by a reliable algorithm based on an adaptation of the classical variational iteration method (VIM), which is called the multi-stage variational iteration method (MSVIM). Examples are carried out and numerical results with comparison to exact solutions, VIM and MATLAB dde23 solver are given. The comparison reveals that the MSVIM is a promising tool to solve delay differential equations.

Enhanced Multistage Differential Transform Method: Application to the Population Models

We present an efficient computational algorithm, namely, the enhanced multistage differential transform method (E‐MsDTM) for solving prey‐predator systems. Since the differential transform method (DTM) is based on the Taylor series, it is difficult to obtain accurate approximate solutions in large domain. To overcome this difficulty, the multistage differential transform method (MsDTM) has been introduced and succeeded to have reliable approximate solutions for many problems. In MsDTM, it is the key to update an initial condition in each subdomain. The standard MsDTM utilizes the approximate solution directly to assign the new initial value. Because of local convergence of the Taylor series, the error is accumulated in a large domain. In E‐MsDTM, we propose the new technique to update an initial condition by using integral operator. To demonstrate efficiency of the proposed method, several numerical tests are performed and compared with ones obtained by other numerical methods such as MsDTM, multistage variational iteration method (MVIM), and fourth‐order Runge‐Kutta method (RK4).

On solving the chaotic Chen system: a new time marching design for the variational iteration method using Adomian’s polynomial

This paper centres on the effectiveness of the variational iteration method and its modifications for numerically solving the chaotic Chen system, which is a three-dimensional system of ODEs with quadratic nonlinearities. This research implements the multistage variational iteration method with an emphasis on the new multistage hybrid of variational iteration method with Adomian polynomials. Numerical comparisons are made between the multistage variational iteration method, the multistage variational iteration method using the Adomian’s polynomials and the classic fourth-order Runge-Kutta method. Our work shows that the new multistage hybrid provides good accuracy and efficiency with a performance that surpasses that of the multistage variational iteration method.

Introducing variational iteration method to a biochemical reaction model

A basic enzyme kinetics is used to test the effectiveness of an analytical method, called the variational iteration method (VIM). This enzyme–substrate reaction is formed by a system of nonlinear ordinary differential equations. We shall compare the classical VIM against a modified version called the multistage VIM (MVIM). Additional comparison will be made against the conventional numerical method, Runge–Kutta (RK4)(fourth-order). Numerical results were obtained for these three methods and we found that MVIM and RK4 are in excellent conformance.

A new application of variational iteration method for the chaotic Rössler system

This paper emphasizes the strength of variational iteration method (VIM) in numerically solving the chaotic Rössler system. The Rössler system is a three-dimensional system of ODEs with quadratic nonlinearities. An adaptation of the VIM was implemented and is called the multistage VIM (MVIM). Numerical comparisons are made between MVIM and the classical fourth-order Runge–Kutta (RK4) with results showing extremely good performance by MVIM, yielding great accuracy and efficiency.

Prescribing a multistage analytical method to a prey–predator dynamical system

This article discusses the effectiveness of a fresh analytical method in solving a prey–predator problem, which is described as a system of two nonlinear ordinary differential equations. The method of interest is the multistage variational iteration method (MVIM), which provides a slight modification of the classical variational iteration method (VIM). We shall compare solutions of the classical VIM along with MVIM and match them against the conventional numerical method, Runge–Kutta (RK4) (fourth-order).

Efficacy of variational iteration method for chaotic Genesio system – Classical and multistage approach

This is a case study of solving the Genesio system by using the classical variational iteration method (VIM) and a newly modified version called the multistage VIM (MVIM). VIM is an analytical technique that grants us a continuous representation of the approximate solution, which allows better information of the solution over the time interval. Unlike its counterpart, numerical techniques, such as the Runge–Kutta method, provide solutions only at two ends of the time interval (with condition that the selected time interval is adequately small for convergence). Furthermore, it offers approximate solutions in a discretized form, making it complicated in achieving a continuous representation. The explicit solutions through VIM and MVIM are compared with the numerical analysis of the fourth-order Runge–Kutta method (RK4). VIM had been successfully applied to linear and nonlinear systems of non-chaotic in nature and this had been testified by numerous scientists lately. Our intention is to determine whether VIM is also a feasible method in solving a chaotic system like Genesio. At the same time, MVIM will be applied to gauge its accuracy compared to VIM and RK4. Since, for most situations, the validity domain of the solutions is often an issue, we will consider a reasonably large time frame in our work.

The multistage variational iteration method for a class of nonlinear system of ODEs

This paper implements the multistage variational iteration method (MVIM) to solve a class of nonlinear system of first-order ordinary differential equations (ODEs). The domain of validity of the solutions via the standard variational iteration method (VIM) is extended by the simple multistage strategy. Comparisons with the exact solution and the fourth-order Runge–Kutta (RK4) method show that the MVIM is a reliable method for nonlinear equations.