Modified Variational Iteration Method Research Articles

Investigation of the complex dynamical structure of bifurcation and dark soliton solutions to fractional generalized double [formula omitted]-Gordon equation

In the current research work, the classical wave equation is combined with a nonlinear sinh source term in the sinh-Gordon equation. It has been used in several scientific domains such as differential geometry theory, integrable quantum field theory, kink dynamics, and statistical mechanics. It makes more comprehensible the dynamics of strings and multi-strings in the constant curvature space. The current study has three main objectives. Examine the governing model to get novel solutions by employing the modified Kudryashov technique. Then compare it with the numerical technique of modified variational iteration method (MVIM) to calculate the error terms. Furthermore, employing bifurcation theory to produce a dynamical system. Additionally, use the dynamical system’s sensitivity analysis to investigate the model’s sensitivity. At last, for the validation of acquired results, the cryptography technique of novel image encryption and decryption is used. The research is greatly enhanced by the presentation of thorough 2D and 3D phase portraits. The field of mathematics and other sciences will benefit from these discoveries.

On the Semi-Analytical Solutions for the Kudryashov-Sinelshchikov Dynamical Equation Arising in Mixtures of Liquid and Gas Bubbles Without Neglecting of Heat Transfer and Viscosity

In this study, Banach contraction method (BCM), Daftardar-Jafari method (DJM) and modified variational iteration method (MVIM) are proposed for the semi-analytical solutions of the Kudryashov-Sinelshchikov (K-S) dynamical equation. It has been shown that the analytical and semi-analytical solutions for the K-S dynamical equation with initial value problems by using semi-analytical methods can be obtained. In addition, the effectiveness and usefulness of the semi-analytical methods used are supported by tables and 3D figures. As the number of iteration or terms increases, how the semi-analytical solutions behave over time and converge to the exact solution is shown in detail with 2D figures. Also, it is shown comparison of semi-analytical solutions with exact solutions and error analysis with the help of tables. It has been discussed the methods are compared with each other and whether they are suitable for the K-S dynamical equation.

Solving the coupled Schrödinger -Korteweg- de-Vries system by modified variational iteration method with genetic algorithm

A system of nonlinear partial differential equations was solved using a modified variational iteration method (MVIM) combined with a genetic algorithm. The modified method introduced an auxiliary parameter (p) in the correction functional to ensure convergence and improve the outcomes. Before applying the modification, the traditional variational iteration method (VIM) was used firstly. The method was applied to numerically solve the system of Schrödinger-KdV equations. By comparing the two methods in addition to some of the previous approaches, it turns out the new algorithm converges quickly, generates accurate solutions and shows improved accuracy. Additionally, the method can be easily applied to various linear and nonlinear differential equations.

Modified Variational Iteration Method Approach for Fuzzy Initial Value Problem

In this paper, an application of  modified variational iteration method (MVIM) for solving homogeneous and nonhomogeneous ordinary differential equations with fuzzy initial value conditions (FDEs) are examined. This modified form is about a procedure some changes in the general formula of the variational iteration method (VIM) . MVIM showed a qualitative and excellent improvement over the standard VIM. To illustrate how this modified form is works we picked some numerical examples and solved them. The efficacy and validity of MVIM have been confirmed through results obtained.  

A Computational Approach to a Model for HIV and the Immune System Interaction

This study deals with the numerical solution of the human immunodeficiency virus (HIV) infection model, which is a significant problem for global public health. Acquired immunodeficiency syndrome (AIDS) is a communicable disease, and HIV is the causative agent for AIDS, which damages the ability of the body to fight against disease and easily usual innocuous infections attack the body. On entering the body, HIV infects a large amount of CD4+ T-cells and disturbs the supply rate of these cells from the thymus. Herein, we consider the model with variable source terms in which the production of these cells is a monotonically decreasing function of viral load. Based on the reproduction number, we describe the stability of free equilibrium. The continuous Galerkin–Petrov method, in particular the cGP(2)-method, is implemented to determine the numerical solutions of the model. The influence of different parameters on the population dynamics of healthy/infected CD4+ T-cells and free HIV particles are examined, and the results are presented graphically. On the other hand, the model is solved using the fourth-order Runge–Kutta method, and briefly, the RK4-method, and the results of the proposed schemes are compared with those obtained from other classical schemes such as the Bessel collocation method (BCM), Laplace Adomian decomposition method (LADM), perturbation iteration algorithm (PIA), modified variational iteration method (MVIM), differential transform method (DTM), and exponential Galerkin method (EGM), numerically. Furthermore, absolute errors relative to the RK4 method are computed to describe the accuracy of the proposed scheme. It is presented that the cGP(2)-method gains accurate results at larger time step sizes in comparison with the results of the aforementioned methods. The numerical and graphical comparison reveals that the proposed scheme yields more accurate results relative to other traditional schemes from the literature.

Modified Variational Iteration Method with Chebyshev Polynomials for Solving 12th order Boundary Value problems

We consider in this paper an illustration of the modified variational iteration method (MVIM) as an effective and accurate solver of 12th order boundary value problem (BVP). For this reason, the Chebyshev polynomials of the principal kind was utilized as a premise capabilities in the guess of the logical capability of the given issue. The strategy is applied in an immediate manner without utilizing linearization or irritation. The subsequent mathematical confirmations recommend that the strategy is without a doubt successful and exact as applied to a few direct and nonlinear issues as mathematical trial and error. Maple 18 was used for all computational simulations carried out in this research.©2022 JNSMR UIN Walisongo. All rights reserved.

Modified Variational Iteration Method for Solving Nonlinear Partial Differential Equation Using Adomian Polynomials

The aim of this paper is to solve numerically the Cauchy problems of nonlinear partial differential equation (PDE) in a modified variational iteration approach. The standard variational iteration method (VIM) is first studied before modifying it using the standard Adomian polynomials in decomposing the nonlinear terms of the PDE to attain the new iterative scheme modified variational iteration method (MVIM). The VIM was used to iteratively determine the nonlinear parabolic partial differential equation to obtain some results. Also, the modified VIM was used to solve the nonlinear PDEs with the aid of Maple 18 software. The results show that the new scheme MVIM encourages rapid convergence for the problem under consideration. From the results, it is observed that for the values the MVIM converges faster to exact result than the VIM though both of them attained a maximum error of order 10^-9. The resulting numerical evidences were competing with the standard VIM as to the convergence, accuracy and effectiveness. The results obtained show that the modified VIM is a better approximant of the above nonlinear equation than the traditional VIM. On the basis of the analysis and computation we strongly advocate that the modified with finite Adomian polynomials as decomposer of nonlinear terms in partial differential equations and any other mathematical equation be encouraged as a numerical method.

A hybrid parametrization approach for a class of nonlinear optimal control problems

In this paper, a suitable hybrid iterative scheme for solving a class of non-linear optimal control problems (NOCPs) is proposed. The technique is based upon homotopy analysis and parametrization methods. Actually an appropriate parametrization of control is applied and state variables are computed using homotopy analysis method (HAM). Then performance index is transformed by replacing new control and state variables. The results obtained from the given method are compared with the results which are obtained using the spectral homotopy analysis method (SHAM), homotopy perturbation method (HPM), optimal homotopy perturbation method (OHPM), modified variational iteration method (MVIM) and differential transformations. The existence and uniqueness of the solution are presented. The comparison and ability of the given approach is illustrated via two examples.

Some New Existence, Uniqueness and Convergence Results for Fractional Volterra-Fredholm Integro-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.

Modified variational iteration method for straight fins with temperature dependent thermal conductivity

The modified variational iteration method (MVIM) has been used to calculate the efficiency of straight fins with temperature dependent thermal conductivity. The obtained results are compared with homotopy analysis method (HAM), homotopy perturbation method (HPM), and Adomian decomposition method (ADM). It is used w ? 0 auxiliary parameter to keep under control convergence region of solution series in MVIM. As a result, although MVIM and HAM give results close to each other; HPM and ADM give divergent results from analytical solution.

Modified Variational Iteration Method for the Numerical Solutions of some Non-Linear Fredholm Integro-Differential Equations of the Second Kind

This paper provides approximate solutions to some nonlinear Fredholm-Integro differential equations of the second kind by using a Modified Variational Iteration Method. Comparison of the approximate solutions of this method with other known methods shows that the Modified Variational Iteration scheme is more accurate, reliable and readily implemented.

A study of nonlinear age-structured population models

This paper presents a novel analysis for the solution of nonlinear age-structured problem which is of extreme importance in biological sciences. The presented model is very useful but quite complicated. Modified variational iteration method (MVIM) coupled with auxiliary parameter is used to cope with the complexity of the model which subsequently shows better results as compared to some existing results available in literature. Furthermore, an appropriate way is used for the identification of auxiliary parameter by means of residual function. Numerical examples are presented for the analysis of the proposed algorithm. Graphical results along with the discussions re-confirm the efficiency of proposed algorithm. The work proposes a new algorithm where He’s polynomials and an auxiliary parameter are merged with correction functional. The suggested scheme is implemented on nonlinear age-structured population models. Graphs are plotted for the residual function that reflects the accuracy and convergence of the presented algorithm.

APPLICATION OF MVIM TO ANALYSIS AN ILL-POSED PROBLEM

In this paper, we consider the inverse heat conduction problem (IHCP) with a single sensor location inside the heat conduction body and solve it by applying modified variational iteration method (MVIM). Although the IHCP is ill-posed problem, we can find the numerical-analytic solution of the problem using the MVIM. This method has an excellent estima- tion of the solution compared with finite difference method even x vanishing. The approximations of the temperature and the heat flux at x = 0 are con- sidered. Moreover, MVIM does not require any discretization, linearization or small perturbation, thus it can be considered as an efficient method to solve this problem. To show the strength of the method, some examples are given.

Influence of Accelerating Force on the Electromechanical Instability of Paddle-Type and Double-Sided Sensors Made of Nanowires

The double-sided and paddle-type nanostructures are promising for developing miniature accelerometers. In the present work, the impact of the accelerating force on the pull-in performance of the double-sided and paddle-type sensors fabricated from nanowire are investigated. The proximity force approximation (PFA) is employed to consider the effect of Casimir attraction in the theoretical model. Using the modified couple stress theory, the constitutive equations of the sensors are derived. The governing equations are solved by two different approaches, i.e. modified variational iteration method (MVIM), and finite difference method (FDM). The influences of the Casimir and accelerating forces, geometrical parameters and the size phenomenon on the pull-in parameters are demonstrated.

A Computational Method for the Solution of Nonlinear Burgers’ Equation Arising in Longitudinal Dispersion Phenomena in Fluid Flow through Porous Media

This paper discusses the Modified Variational Iteration Method (MVIM) for the solution of nonlinear Burgers’ equation arising in longitudinal dispersion phenomena in fluid flow through porous media. The method is an elegant combination of Taylor’s series and the variational iteration method (VIM). Using Maple 18 for implementation, it is observed the procedure provides rapidly convergent approximation with less computational efforts. The result shows that the concentration C(x,t) of the contaminated water decreases as distance x increases for the given time t.

Modified variational iteration method for variant Boussinesq equation

In this paper, we solve the variant Boussinesq equation by the modified variational iteration method. The approximate solutions to the initial value problems of the variant Boussinesq equation are provided, and compared with the exact solutions. Numerical experiments show that the modified variational iteration method is efficient for solving the variant Boussinesq equation.

THE MODIFIED VARIATIONAL ITERATION METHOD FOR SOLVING THE IMPENETRABLE AGAR MODEL PROBLEM

In this paper, we expand the scope of application of the modified variational iteration method (MVIM) to solve the Impenetrable Agar Model Problem. This model represents the propagation of a bacterial exo-enzyme within an impenetrable host material which simulates the manifestations bac- teria do on tissue wounds in an infection. The numerical results show that the MVIM is convenient, efficient, and easy to use for the model problem.

Effects of size-dependent elasticity on stability of nanotweezers

It is well-recognized that the electromechanical response of a nanostructure is affected by its element size. In the present article, the size dependent stability behavior and nanotweezers fabricated from nanowires are investigated by modified couple stress elasticity (MCSE). The governing equation of the nanotweezers is obtained by taking into account the presence of Coulomb and intermolecular attractions. To solve the equation, four techniques, i.e., the modified variational iteration method (MVIM), the monotonic iteration method (MIM), the MAPLE numerical solver, and a lumped model, are used. The variations of the arm displacement of the tweezers versus direct current (DC) voltage are obtained. The instability parameters, i.e., pull-in voltage and deflection of the system, are computed. The results show that size-dependency will affect the stability of the nanotweezers significantly if the diameter of the nanowire is of the order of the length scale. The impact of intermolecular attraction on the size-dependent stability of the system is discussed.

A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials

Abstract In this paper, we introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. Also the fractional Riccati differential equation is solved by variational iteration method with considering Adomians polynomials for nonlinear terms. The main advantage of the MVIM is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval. The numerical results show that the method is simple and effective.

The Semianalytical Solutions for Stiff Systems of Ordinary Differential Equations by Using Variational Iteration Method and Modified Variational Iteration Method with Comparison to Exact Solutions

The Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM) are used to find solutions of systems of stiff ordinary differential equations for both linear and nonlinear problems. Some examples are given to illustrate the accuracy and effectiveness of these methods. We compare our results with exact results. In some studies related to stiff ordinary differential equations, problems were solved by Adomian Decomposition Method and VIM and Homotopy Perturbation Method. Comparisons with exact solutions reveal that the Variational Iteration Method (VIM) and the Modified Variational Iteration Method (MVIM) are easier to implement. In fact, these methods are promising methods for various systems of linear and nonlinear stiff ordinary differential equations. Furthermore, VIM, or in some cases MVIM, is giving exact solutions in linear cases and very satisfactory solutions when compared to exact solutions for nonlinear cases depending on the stiffness ratio of the stiff system to be solved.