Variational Iteration Method for Solving Time Fractional Burgers Equation Using Maple

The Time Fractional Burger equation was solved in this study using the Mabel software and the Variational Iteration approach. where a number of instances of the Time Fractional Burger Equation were handled using this technique. Tables and images were used to present the collected numerical results. The difference between the exact and numerical solutions demonstrates the effectiveness of the Mabel program’s solution, as well as the accuracy and closeness of the results this method produced. It also demonstrates the Mabel program’s ability to quickly and effectively produce the numerical solution.

Variational iteration solving method for El Nio phenomenon atmospheric physics of nonlinear model

A class of E1 Niйo atmospheric physics oscillation model is considered. The E1 Niйo atmospheric physics oscillation is an abnormal phenomenon involved in the tropical Pacific ocean-atmosphere interactions. The conceptual oscillator model should consider the variations of both the eastern and western Pacific anomaly patterns. An E1 Niйo atmospheric physics model is proposed using a method for the variational iteration theory. Using the variational iteration method, the approximate expansions of the solution of corresponding problem are constructed. That is, firstly, introducing a set of functional and accounting their variationals, the Lagrange multiplicators are counted, and then the variational iteration is defined, finally, the approximate solution is obtained. From approximate expansions of the solution, the zonal sea surface temperature anomaly in the equatorial eastern Pacific and the thermocline depth anomaly of the sea-air oscillation for E1 Niйo atmospheric physics model can be analyzed. E1 Niйo is a very complicated natural phenomenon. Hence basic models need to be reduced for the sea-air oscillator and are solved. The variational iteration is a simple and valid approximate method.

Variational iteration method for solving the mechanism of the Equatorial Eastern Pacific El Nino-Southern Oscillation

A class of coupled system for the E1 Nifio-Southern Oscillation （ENSO） mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO model are obtained and the asymptotic behaviour of solution for corresponding problem is considered.

Doubly Periodic Wave Solutions of Jaulent-Miodek Equations Using Variational Iteration Method Combined with Jacobian-function Method

One of the advantages of the variational iteration method is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some unknown parameters. The Jaulent-Miodek equations are used to illustrate effectiveness and convenience of this method, some new explicit exact travelling wave solutions have been obtained, which include bell-type soliton solution, kink-type soliton solutions, solitary wave solutions, and doubly periodic wave solutions.

Variational iteration method for solving compressible Euler equations

This paper applies the variational iteration method to obtain approximate analytic solutions of compressible Euler equations in gas dynamics. This method is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Using this method, a rapid convergent sequence is produced which converges to the exact solutions of the problem. Numerical results and comparison with other two numerical solutions verify that this method is very convenient and efficient.

Variational iteration method for solving time-fractional diffusion equations in porous the medium

The variational iteration method is successfully extended to the case of solving fractional differential equations, and the Lagrange multiplier of the method is identified in a more accurate way. Some diffusion models with fractional derivatives are investigated analytically, and the results show the efficiency of the new Lagrange multiplier for fractional differential equations of arbitrary order.

Modified variational iteration method for an El Nio Southern Oscillation delayed oscillator

This paper studies a delayed air-sea coupled oscillator describing the physical mechanism of El Nino Southern Oscillation. The approximate expansions of the delayed differential equation＇s solution are obtained successfully by the modified variational iteration method. The numerical results illustrate the effectiveness and correctness of the method by comparing with the exact solution of the reduced model.

Variational Iteration Method for Solving Boussinesq Equations Using Maple

In this study, we applied the variational iteration method to solve the Boussinesq time equation. Bossiness’s article from 1872 introduced the equations that are now known as the Boussinesq equations. Numerical methods are commonly utilized to solve nonlinear equation systems. Several research papers have documented the values of the variational iteration method and its applications for various categories of differential equations. A comparison of the exact and numerical solutions was obtained using the variational iteration method. The variational iteration method shows that the proposed method is very effective and convenient. The results are shown for different specific cases of the problem. The variational iteration method is useful in numerical simulations and approximate analytical solutions, and it is used to resolve nonlinear differential equations in various situations using Maple. For example, the linear Boussinesq equation was resolved using the variational iteration method. By comparing the numerical results, we found that the variable repetition method produced accurate results and was close to the exact solution, allowing it to be widely applied to the Boussinesq equation. This proves the effectiveness of the method and the capability to quickly and effectively obtain the numerical number solution related to the exact solution using the Maple 18 program. Additionally, the outcomes are extremely precise.

Time Discretized Variational Iteration Method for the Stochastic Volatility Process with Jumps

A model for both stochastic jumps and volatility for equity returns in the area of option pricing is the stochastic volatility process with jumps (SVPJ). A major advantage of this model lies in the area of mean reversion and volatility clustering between returns and volatility with uphill movements in price asserts. Thus, in this article, we propose to solve the SVPJ model numerically through a discretized variational iteration method (DVIM) to obtain sample paths for the state variable and variance process at various timesteps and replications in order to estimate the expected jump times at various iterates resulting from executing the DVIM as n increases. These jumps help in estimating the degree of randomness in the financial market. It was observed that the average computed expected jump times for the state variable and variance process is moderated by the parameters (variance process through mean reversion), Θ (long-run mean of the variance process), σ (volatility variance process) and λ (constant intensity of the Poisson process) at each iterate. For instance, when = 0.0, Θ = 0.0, σ = 0.0 and λ = 1.0, the state variable cluttered maximally compared to the variance process with less volatility cluttering with an average computed expected jump times of 52.40607869 as n increases in the DVIM scheme. Similarly, when = 3.99, Θ = 0.014, σ = 0.27 and λ = 0.11, the stochastic jumps for the state variable are less cluttered compared to the variance process with maximum volatility cluttering as n increases in the DVIM scheme. In terms of option pricing, the value 52.40607869 suggest a better bargain compared to the value 20.40344029 due to the fact that it yields less volatility rate. MAPLE 18 software was used for all computations in this research.

Reduced Differential Transform Method for Solving Linear and Nonlinear Goursat Problem

In this paper a new method for solving Goursat problem is introduced using Reduced Differential Transform Method (RDTM). The approximate analytical solution of the problem is calculated in the form of series with easily computable components. The comparison of the methodology presented in this paper with some other well known techniques demonstrates the effectiveness and power of the newly proposed methodology.

Modified integral equation combined with the decomposition method for time fractional differential equations with variable coefficients

In this paper,the modified integral equation,namely,Elzaki transformation coupled with the Adomian decomposition method called Elzaki Adomian decomposition method(EADM)is used to investigate the solution of time-fractional fourth-order parabolic partial differential equations(PDEs)with variable coefficients.The introduced method is used to solve two models of the proposed problem,the analytical and approximate solutions of the models are obtained.The outcomes illustrate that the proposed technique is a highly accurate,and facilitates the process of solving differential equations by comparing it,with the exact solution and those obtained by the variation iteration method(VIM)and Laplace homotopy perturbation method(LHPM).

Numerical Solutions of Three-Dimensional Coupled Burgers’ Equations by Using Some Numerical Methods

In this paper, we found the numerical solution of three-dimensional coupled Burgers’ Equations by using more efficient methods: Laplace Adomian decomposition method, Laplace transform homotopy perturbation method, variational iteration method, variational iteration decomposition method and variational iteration homotopy perturbation method. Example is examined to validate the efficiency and accuracy of these methods and they reduce the size of computation without the restrictive assumption to handle nonlinear terms and it gives the solutions rapidly.

Exact Solutions of the Harry-Dym Equation

The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition, He＇s variational iteration, direct integration, and power series. We show that the two later methods are more successful than the two former to obtain more solutions of the equation.

Numerical solution of the imprecisely defined inverse heat conduction problem

This paper investigates the numerical solution of the uncertain inverse heat conduction problem. Uncertainties present in the system parameters are modelled through triangular convex normalized fuzzy sets. In the solution process, double parametric forms of fuzzy numbers are used with the variational iteration method （VIM）. This problem first computes the uncertain temperature distribution in the domain. Next, when the uncertain temperature measurements in the domain are known, the functions describing the uncertain temperature and heat flux on the boundary are reconstructed. Related example problems are solved using the present procedure. We have also compared the present results with those in [Inf. Sci. （2008） 178 1917] along with homotopy perturbation method （HPM） and [Int. Commun. Heat Mass Transfer （2012） 39 30] in the special cases to demonstrate the validity and applicability.

Approximate analytic solutions for a generalized Hirota—Satsuma coupled KdV equation and a coupled mKdV equation

This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.

Numerical Simulation for Nonlinear Water Waves Propagating along the Free Surface

The main aim of this work is to introduce the analytical approximate solutions of the water wave problem for a fluid layer of finite depth in the presence of gravity. To achieve this aim, we begun with the derivation of the Korteweg-de Vries equations for solitons by using the method of multiple scale expansion. The proposed problem describes the behavior of the system for free surface between air and water in a nonlinear approach. To solve this problem, we use the well-known analytical method, namely, variational iteration method (VIM). The proposed method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. The proposed method provides a sequence of functions which may converge to the exact solution of the proposed problem. Finally, we observe that the elevation of the water waves is in form of traveling solitary waves.

A Comparison Results of Some Analytical Solutions of Model in Double Phase Flow through Porous Media

In this paper, we consider Variational Iteration Method （VIM） and q-Homotopy Analysis Method （q-HAM） to sotve me partial differential equation resulted from Fingero Imbibition phenomena in double phase flow through porous media. We further compare the results obtained here with the solution obtained in [ 12] using Adomian Decomposition Method. Numerical results are obtained, using Mathematica 9, to show the effectiveness of these methods to our choice of problem especially for suitable values ofh and n.

Ma’s Variation of Parameters Method for Fisher’s Equations

In this paper,we apply Ma’s variation of parameters method(VPM)for solving Fisher’s equations.The suggested algorithm proved to be very efficient and finds the solution without any discretization,linearization,perturbation or restrictive assumptions.Numerical results reveal the complete reliability of the proposed VPM.

Two efficient methods for solving Schlömilch’s integral equation

Purpose–In this paper,the exact solutions of the Schlömilch’s integral equation and its linear and non-linear generalized formulas with application are solved by using two efficient iterative methods.The Schlömilch’s integral equations have many applications in atmospheric,terrestrial physics and ionospheric problems.They describe the density profile of electrons from the ionospheric for awry occurrence of the quasi-transverse approximations.The paper aims to discuss these issues.Design/methodology/approach–First,the authors apply a regularization method combined with the standard homotopy analysis method to find the exact solutions for all forms of the Schlömilch’s integral equation.Second,the authors implement the regularization method with the variational iteration method for the same purpose.The effectiveness of the regularization-Homotopy method and the regularizationvariational method is shown by using them for several illustrative examples,which have been solved by other authors using the so-called regularization-Adomian method.Findings–The implementation of the two methods demonstrates the usefulness in finding exact solutions.Practical implications–The authors have applied the developed methodology to the solution of the Rayleigh equation,which is an important equation in fluid dynamics and has a variety of applications in different fields of science and engineering.These include the analysis of batch distillation in chemistry,scattering of electromagnetic waves in physics,isotopic data in contaminant hydrogeology and others.Originality/value–In this paper,two reliable methods have been implemented to solve several examples,where those examples represent the main types of the Schlömilch’s integral models.Each method has been accompanied with the use of the regularization method.This process constructs an efficient dealing to get the exact solutions of the linear and non-linear Schlömilch’s integral equation which is easy to implement.In addition to that,the accompanied regularization method with each of the two used methods proved its efficiency in handling many problems especially ill-posed problems,such as the Fredholm integral equation of the first kind.

Steady state heat transfer analysis in a rectangular moving porous fin

In this article,the variation of temperature distribution and fin efficiency in a porous moving fin of rectangular profile is studied.This study is performed using Darcy's model to formulate the governing heat transfer differential equation.The approximate analytical solution is generated using the variational iteration method(VIM).The power series solution is validated by benchmarking it against the numerical solution obtained by applying the Runge-Kutta fourth order method.A good agreement between the analytical and numerical results is observed.The effects of porosity parameter,Peclet number and other thermo-physical parameters,such as the power index of heat transfer coefficient,convective-conductive parameter,radiative-conductive parameter,thermal conductivity gradient and non-dimensional ambient temperature on non-dimensional temperature are also studied and explained.The results indicate that the fin rapidly dissipates heat to the ambient temperature with an increase in the Peclet number,convection-radiation parameters and the porosity parameter.