Variational Iteration Method Research Articles

Solutions of fractional differential equations by using a blend of variational iteration method with Sumudu transform and application to price adjustment equations

The presence of delays in a mathematical model can improve its vitality and suitability in describing several phenomena. However, in the presence of delays, nonlinear fractional differential equations are more difficult to study. This paper presents the use of a blend of variational iteration method with Sumudu transform for solving delay differential equations with Caputo derivatives of fractional variable order. Moreover, the paper introduces delays into the price adjustment equations to propose new price adjustment models with more potential for vitality and suitability. The paper assigns suitable real values to the parameters for the graphical display and comparison of the obtained solutions. The paper presents the interactions that exist among the price, demand, supply and dependence of supply and demand on the price, which can be applied to estimate the equilibrium price.

An application to formable transform: Novel numerical approach to study the nonlinear oscillator

Numerical methods in the area of nonlinear systems are extensively implemented for computing their approximate solutions because these systems are very difficult to tackle analytically. There are various numerical techniques available in the literature to find the solutions of nonlinear oscillators. Variational iteration method (VIM) is one of these approaches which is convenient to implement for these kinds of problems. In this work, our study aims to identify the numerical solution of nonlinear oscillator by making use of variational iteration method associated with Formable transformation. For the smooth utilization of this approach, we have to formulate the Lagrange multiplier through variational theory. Furthermore, we develop a new unified iterative scheme for the correction functional of VIM, considering the Formable transformation. Several new schemes of correction functional can be deduced from the newly proposed method considering the duality relation of Formable transform. In support of our primary finding, we discuss numerical example as application. A number of Physical applications of nonlinear oscillators are available in the field of vibrations and oscillations but in recent times nonlinear oscillators are used to describe complicated systems or to address mechanical, electrical, and other engineering phenomenon.

Analytical solution of local fractal continuum traffic flow model

This study developed a continuum traffic flow model with local fractional derivatives. This model overcomes the non-differentiable traffic parameters arising in vehicular traffic flow. The proposed model is solved using the local fractional Laplace variational iteration method (LFLVIM) and is well suited for analyzing the dynamical evolution of non-differentiable traffic density and speed function. Furthermore, the stability of the solution for the continuum model has also been discussed. Illustrative examples are also discussed to show the effectiveness of employing LFLVIM in the suggested model. Additionally, numerical simulations for each instance have been displayed. This research indicates that the utilized iterative approach is efficient and may be used to derive the non-differentiable solution to the proposed continuum traffic model.

Application of the Variational Iteration Method for the time-fractional Kaup-Kupershmidt Equation and the Boussinesq-Burger equation

The variational iteration method is used to deal with linear and nonlinear differential equations. The main characteristics of the method lie in its flexibility and ability to accurately and easily solve nonlinear equations. In this work, a general framework is presented for a variational iteration method for the analytical treatment of partial differential equations in fluid mechanics. The Caputo sense is used to describe fractional derivatives. The time-fractional Kaup-Kupershmidt (KK) equation is investigated, as it is the solution of the system of partial differential equations via the Boussinesq-Burger equation. By comparing the results that are obtained by the variational iteration method with those obtained by the two-dimensional Legendre multiwavelet, the optimal homotopy asymptotic method (OHAM), the q-homotopy analysis transform method, the Laplace Adomian Decomposition Method, and the homotopy perturbation method, the first method proved to be very effective and convenient. The main methodology in this work is anticipated to be applied to various fractional calculus, linear, and nonlinear problems.

Variational iteration method for the nanobeams-based N/MEMS system

The nano/microelectromechanical system (N/MEMS) has triggered worldwide concern, and its applications have revolutionized technologies in various advanced fields from wearable sensors, 5 G communication technology, to energy harvesting, to aerospace. However, when the applied force is sufficiently large, the pull-in instability arises, and reliable operation is forbidden. Therefore, it is extremely important to insight fast and accurately into the periodic motion of the system to prevent the system from its pull-in motion. The basic aim of this study is to demonstrate the applicability of the well-known variational iteration method (VIM) for predicting the dynamic behavior of N/MEMS. For this, a nanobeam-based microstructure with van der Waals force for actuation is used as an example to reveal its periodic properties. The governing equation for the oscillation of the microsystem is obtained from the Euler-Bernoulli beam principle, considering the midplane stretching effect. We then employ the Galerkin technique to transform the governing partial differential equation into an ordinary differential equation, which is highly nonlinear, making it extremely difficult to solve by some traditional analytical methods, however, the VIM shows its ability to elucidate accurately the basic properties of the N/MEMS by simple calculation. This paper offers a new road for fast and accurate prediction of the microsystem's properties, and the result can be used for optimizing the N/MEMS.•A nanobeam-based N/MEMS system with van der Waals force is considered.•A strongly governing equation without a linear term is obtained.•The variational iteration method is applied to figure out the basic properties of the system.

COMPARISON OF LAPLACE-ADOMIAN’S DECOMPOSITION METHOD AND LAPLACE VARIATIONAL ITERATION METHOD IN NONLINEAR BOUSSINESQ EQUATION

In this paper, we apply two techniques Laplace Decomposition Method (LADM) and Laplace Variational Iteration Method (LVIM) to determine the analytical solution of Boussinesq Equation.

A Review of Analytical Algorithms for Approximate Solutions of Ordinary Differential Equation

This review article aims to provide an in-depth analysis of a wide range of analytical methods used for solving ordinary differential equations (ODEs). ODEs are fundamental in numerous scientific and engineering disciplines, making the development, and understanding of effective solution techniques crucial. We explore various approaches, including the Adomian decomposition method, homotopy perturbation method, homotopy analysis method, variational iteration method, Daftardar-Jafari method, successive approximation method, power series method, and modified Adomian decomposition method. Each method is discussed in terms of its principles, applications, advantages, limitations, and computational considerations. This comprehensive overview will serve as a valuable resource for researchers, practitioners, and students interested in solving ODEs.

Analytical assessment of heat transfer due to Williamson hybrid nanofluid (MoS2 + ZnO) with engine oil base material due to stretched sheet

The efficiency of fuel consumptions and improved performances is real challenge in the vehicle engines. Recent advances in thermal engineering suggested various tools to improve the capacitance of engine oil for which the interaction of nanoparticles is more dynamical. The aim of current research is to enhance the thermal aspects of engine oil with suspension of Williamson hybrid nanofluid. The hybrid nanofluid is the suspension of molybdenum disulfide MoS2 and zinc oxide ZnO. In order to increase the heating capability of system, external heat source and viscous dissipation features are contributed. The flow analysis is based on moving surface contains porous medium. After modelling the problem, the analytical outcomes of attributed with variational iteration method (VIM). The computational simulations are performed with help of Lagrange Multiplier technique. The impact of numerous physical parameters on skin fraction, Nusselt numbers, temperature distribution and velocity profile is exhibited.

Analysis of the Axial Vibration of Non-Uniform and Functionally Graded Rods via an Analytical-Based Numerical Approach

In this study, an analytical-based numerical approach was proposed for the analysis of the free axial vibration of homogeneous and functionally graded rods with varying cross-sectional areas. The proposed approach is based on analytical approximation techniques, such as the Adomian decomposition method, variational iteration method, and homotopy perturbation method. However, the governing equations of the problems solved in this study were variable coefficient differential equations. These equations provide analytical solutions for strictly limited cases. Analytical approximation methods easily handle problems with uniform material properties and constant cross-sections, whereas with varying cross-sectional areas, the analytical integration process becomes a difficult task for the software. If the rod’s material is functionally graded with varying cross-sectional areas, the analytical integration process becomes a cumbersome task. The proposed approach eliminates all difficulties and requires computation within several seconds. The application of this method is straightforward, and the results obtained in this study are in excellent agreement with the solutions provided in the literature.

NUMERICAL ACCURACY OF FREDHOLM LINEAR INTEGRO-DIFFERENTIAL EQUATIONS BY USING ADOMIAN DECOMPOSITION METHOD, MODIFIED ADOMIAN DECOMPOSITION METHOD AND VARIATIONAL ITERATION METHOD

In this article, we present as a comparative result of Adomian Decomposition Method (ADM), Modified Adomian Decomposition Method (MADM) and Variational Iteration Method (VIM). These methods used for developed to find the analytical approximate solution of linear Fredholm integro-differential equations. The main purpose of this paper was to show a better method for Numerical equations which does not give easily analytical solution. So, in this paper, we find approximate solutions of linear Fredholm integro-differential equations. We explain the convergence of ADM, MADM and VIM by using examples of a deterministic model by graphs and tables. All the calculations performed by the help of MATLAB (2018) Version 9.4.

Approximate Solutions for Optimal Control of Fixed Boundary Value Problems Using Variational and Minimum Approaches

The optimal control is the process of finding a control strategy that extreme some performance index for a dynamic system (partial differential equation) over the class of admissibility. The present work deals with a problem of fixed boundary with a control manipulated in the structure of the partial differential equation. An attractive computational method for determining the optimal control of unconstrained linear dynamic system with a quadratic performance index is presented. In the proposed method the difference between every state variable and its initial condition is represented by a finite - term polynomial series, this representation leads to a system of linear algebraic equations which represents the necessary condition of optimality. The linear algebraic system is solved by using two approaches namely the variational iteration method and the minimization approach for unconstrained optimization problem with estimation of gradient and Hessian matrix. These approaches are illustrated by two application examples.

The Analytical Methods Of Volterra Integral Equations of The Second Kind

This paper discussed the analytic methods to solve Second order Volterra integral equations form using different methods. The domain decomposition method is the first technique. Depending on the hypothesis, the solution is by sequence. The second method is the successive approximation technique, which is used Picard iteration method. The third method used Laplace transformation. The modified decomposition technique is used as the fourth method. Depending on the Taylor series, the fifth method is called the series method. The last method solves the VIE using the functional correction technique called the variational iteration method. We introduce some examples to illustrate these methods.

Numerical inspection of (3 + 1)- perturbed Zakharov–Kuznetsov equation via fractional variational iteration method with Caputo fractional derivative

In this paper, the implementation of the Fractional Variational iteration method (FVIM) is used to approximate the generalized fractional perturbed Zakharove-Kuznetsov equation. The obtained results show that the suggested approach is very efficient for dealing with various forms of differential equations of fractional order. The numerical comparison for the attained resolution is made through the q-homotopy analysis transmute manner (q-HATM) which confirms the convergence of the devised method. The applicability of the method is observed by applying the proposed technique to two numerical examples. Furthermore, the solutions obtained by the FVIM are completely well-matched with the solutions presented in the literature.

Variational Iteration Approach for Solving Fractional Integro-Differential Equations with Conformable Differintegrals

Fractional integro-differential equations involving conformable differintegrals have promising applications but lack solution methods. This work develops a variational iteration approach for such equations. Conformable fractional derivatives and integrals that generalize integer-order equivalents are defined. Existence and uniqueness results are established for a class of nonlinear fractional integro-differential equations. These exist under less restrictive smoothness assumptions than integer-order cases. The variational iteration method (VIM) is then applied, providing an analytical approximate technique for integro-differential problems of fractional order. Convergence analysis demonstrates the efficiency and accuracy of the VIM solutions. Several test cases validate the VIM, matching analytical solutions available for simpler fractional differential sub-cases. The proposed technique advances available methods for this emerging class of fractional integro-differential equations. Significantly, it enables application of such models by allowing accurate solution of associated mathematical system representations. Extensions to include more singular equation classes, comparisons with other methods, and real-world applications are suggested as future work.
 The abstract summarizes the key points: significance of problem, fractional models used, existence/uniqueness proofs, VIM approach and analysis, test case validations, implications for applications

Advanced numerical scheme and its convergence analysis for a class of two-point singular boundary value problems

In the past decades, many applications related to applied physics, physiology and astrophysics have been modelled using a class of two-point singular boundary value problems (SBVPs). In this article, a novel approach based on the shooting projection method and the Legendre wavelet operational matrix formulation for approximating a class of two-point SBVPs with Dirichlet and Neumann–Robin boundary conditions is proposed. For the new approach, an initial guess is postulated in contrast to the boundary conditions in the first step. The second step deals with the usage of the Legendre wavelet operational matrix method to solve the initial value problem (IVP). Further, the resulting solution of the IVP is utilized at the second endpoint of the domain of a differential equation in a shooting projection method to improve the initial condition. These two steps are repeated until the desired accuracy of the solution is achieved. To support the mathematical formulation, a detailed convergence analysis of the new approach is conducted. The new approach is tested against some existing methods such as various types of the variational iteration method, considering several numerical examples to which it provides high-quality solutions.

Variational Iteration Approach for Solving Two-Points Fuzzy Boundary Value Problems

The main objective of this paper is to introduce interval two-point fuzzy boundary value problems, in which the fuzziness course when the coefficients of the governing ordinary differential equation and/or the boundary conditions include fuzzy numbers of either triangular or trapezoidal types. Such equations will be solved by introducing the concept of α – level sets, α  [0,1] to treat the fuzzy ordinary differential equation into two nonfuzzy ordinary differential equations, which correspond to the lower and upper solutions of the interval fuzzy solutions. The well-known variational iteration method has been used to solve two-point fuzzy boundary value problems and linear equations have been examined.

Wave propagation and evolution in a (1+1)-dimensional spatial-temporal domain: A comprehensive study

In this investigation, we utilize two recent analytical schemes to unveil novel solitary wave solutions for the [Formula: see text]-dimensional Mikhailov–Novikov–Wang integrable equation. The said equation serves as a mathematical model that captures specific physical phenomena, albeit lacking a direct physical interpretation. Nevertheless, it finds relevance in various systems within the realm of nonlinear waves in physics. Through the application of the aforementioned analytical schemes, we derive fresh solutions and evaluate their accuracy by employing the variational iteration method. The congruence observed between the analytical and numerical solutions of the investigated model serves as validation for the constructed solutions. Furthermore, we delve into exploring the implications of obtaining precise and ground breaking solitary wave solutions on the practical applications associated with the studied model.

Using Variational Iteration Method for Solving Linear Fuzzy Random Ordinary Differential Equations

Using Variational Iteration Method for Solving Linear Fuzzy Random Ordinary Differential Equations

Waves propagation of optical waves through nonlinear media; modified Kawahara equation

The focus of this study is to investigate novel and precise solitary wave solutions of the modified Kawahara (MK) equation using recent and accurate computational techniques. The MK equation, a nonlinear partial differential equation, has significant applications in various fields, including fluid dynamics and plasma physics The comprehension of nonlinear wave phenomena relies heavily on solitary wave solutions, making it necessary to develop accurate and efficient computational techniques for their identification. In this study, the Khater II method is employed as a computational technique, while the variational iteration method is used as a numerical scheme.The Khater II method is a powerful computational technique that has exhibited promising outcomes in solving nonlinear wave equations. It uses an analytical framework to transform the partial differential equation into a set of ordinary differential equations that are amenable to straightforward solutions, thereby enabling the construction of exact solutions. Conversely, the variational iteration method is a numerical scheme that enhances solution accuracy through iterative approximations.By using the Khater II method as a computational technique and the variational iteration method as a numerical scheme, this study identifies novel and precise solitary wave solutions of the MK equation. These solutions provide valuable insights into the dynamics and behavior of nonlinear waves across various applications. The effectiveness of the employed techniques is demonstrated by comparing them with other commonly utilized computational methods for solving the MK equation. The results indicate that the combined application of the Khater II method and the variational iteration method yields more accurate and efficient solutions. This heightened accuracy and efficiency are critical for comprehending and predicting the behavior of the studied model in real-world applications. Thus, this study underscores the significance of employing recent and accurate computational techniques, such as the Khater II method and the variational iteration method, to identify novel and precise solitary wave solutions of the MK equation. The obtained solutions contribute to an improved understanding of the model’s behavior and hold practical implications across diverse scientific and engineering domains.

A generalized local fractional LWR model of vehicular traffic flow and its solution

In this study, a generalized nonlinear local fractional Lighthill‐Whitham‐Richards (LFLWR) model has been developed. The local fractional variational iteration method (LFVIM) solves and analyzes the proposed model. Numerous works have been described in the past to address linear LWR and linear LFLWR models. This research highlighted on generalized nonlinear LFLWR model, and LFVIM is employed to derive non‐differentiable solutions of the suggested model. The existence and uniqueness of the solution of LFLWR model have also been established. Furthermore, several exemplary instances are discussed to demonstrate the success of implementing LFVIM to the proposed model. The numerical simulations for each of the cases have also been shown. Additionally, the obtained solutions of the suggested model have been compared with the solutions of the classical LWR model with non‐differentiable conditions in few examples. The study demonstrates that the employed iterative scheme is quite efficient and can be utilized to obtain the non‐differentiable solution to the proposed generalized nonlinear LFLWR traffic flow model.