Analytical solution for a suspension bridge by applying HPM and VIM

In this paper, homotopy perturbation method (HPM) and variational iteration method (VIM) are used to solve the large amplitude torsional oscillations equations in a nonlinearly suspension bridge. This paper compares the HPM and VIM in order to solve the equations of nonlinearly suspension bridge. A comparative study between the HPM and VIM is presented in this work. The achieved results reveal that the HPM and VIM are very effective, convenient and quite accurate to nonlinear partial differential equations. These methods can be easily extended to other strongly nonlinear oscillations and can be found widely applicable in engineering and science. The Laplace transform method is applied to obtaining the Lagrange multiplier in the VIM solution.     

Second-order two-point boundary value problems using the variational iteration algorithm-II

In this paper, we introduce an improved variational iteration method (VIM) for nonlinear second-order boundary value problems. The main advantage of this modification is that it can avoid additional computation in determining the unknown parameters in initial approximation when solving boundary value problems using the conventional VIM. Also, iterative sequences obtained using the improved VIM do satisfy the boundary conditions while iterative sequences obtained using conventional VIM may not, in general, satisfy the boundary conditions. Numerical results reveal that the improved method is accurate and efficient.     

An iterative method for suboptimal control of a class of nonlinear time-delayed systems

This work presents an approximate solution method for the infinite-horizon nonlinear time-delay optimal control problem. A variational iteration method (VIM) is applied to design feedforward and feedback optimal controllers. By using the VIM, the original optimal control is transformed into a sequence of nonhomogeneous linear two-point boundary value problems (TPBVPs). The existence and uniqueness of the optimal control law are proved. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. The feedback term is determined by solving Riccati matrix differential equation. By using the finite-step iteration of a nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation results demonstrate the validity and applicability of the VIM.     

Numerical simulations of the Boussinesq equation by He's variational iteration method

In this paper, we present a reliable algorithm to study the well-known model of nonlinear dispersive waves proposed by Boussinesq. We solve the Cauchy problem of Boussinesq equation using variational iteration method (VIM). The numerical results of this method are compared with the exact solution of an artificial model to show the efficiency of the method. The approximate solutions show that VIM is a powerful mathematical tool for solving nonlinear problems.     

Approximate analytical solutions of systems of PDEs by homotopy analysis method

In this paper, the homotopy analysis method (HAM) is applied to obtain series solutions to linear and nonlinear systems of first- and second-order partial differential equations (PDEs). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown in particular that the solutions obtained by the variational iteration method (VIM) are only special cases of the HAM solutions.     

The variational iteration method for solving systems of equations of Emden–Fowler type

In this paper, the variational iteration method (VIM) is used to study systems of linear and nonlinear equations of Emden–Fowler type arising in astrophysics. The VIM overcomes the singularity at the origin and the nonlinearity phenomenon. The Lagrange multipliers for all cases of the parameter α,α>0, are determined. The work is supported by examining specific systems of two or three Emden–Fowler equations where the convergence of the results is emphasized.     

Variational iteration method for singular perturbation initial value problems

In this paper, the variational iteration method (VIM) is applied to solve singular perturbation initial value problems (SPIVPs). The obtained sequence of iterates is based on the use of Lagrange multipliers. Some convergence results of VIM for solving SPIVPs are given. Moreover, the illustrative examples show the efficiency of the method.     

Application of He's homotopy perturbation and He's variational iteration methods for solution of Benney–Lin equation

In this paper, we study the Benney–Lin equation using He's homotopy perturbation method (HPM) and He's variational iteration method (VIM). We compare HPM and VIM methods and show that the results of the HPM method are in excellent agreement with the results of the VIM method and the obtained solutions are shown graphically. Several cases are considered to apply HPM and VIM, which demonstrate the reliability and efficiency of these two methods in solving such a complicated equation with various initial conditions.     

On the numerical solution of the model for HIV infection of CD4 T cells

In this article, a variational iteration method (VIM) is performed to give approximate and analytical solutions of nonlinear ordinary differential equation systems such as a model for HIV infection of CD4⁺ T cells. A modified VIM (MVIM), based on the use of Padé approximants is proposed. Some plots are presented to show the reliability and simplicity of the methods.     

Two-dimensional differential transform method,Adomian's decomposition method,and variational iteration method for partial differential equations

The implementation of the two-dimensional differential transform method (DTM), Adomian's decomposition method (ADM), and the variational iteration method (VIM) in the mathematical applications of partial differential equations is examined in this paper. The VIM has been found to be particularly valuable as a tool for the solution of differential equations in engineering, science, and applied mathematics. The three methods are compared and it is shown that the VIM is more efficient and effective than the ADM and the DTM, and also converges to its exact solution more rapidly. Numerical solutions of two examples are calculated and the results are presented in tables and figures.     

变分迭代算法在非线性微分方程中的应用

应用何吉欢的变分迭代算法,求解了一类强非线性振动方程．其中一阶近似解已有非常高的精度,并且得到的近似解在全域内一致有效．     

Numerical comparison of methods for solving fractional differential–algebraic equations (FDAEs)

In this paper, the variational iteration method (VIM) and the Adomian decomposition method (ADM) are implemented to give approximate solutions for fractional differential–algebraic equations (FDAEs). Both methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. This paper presents a numerical comparison between these two methods and the homotopy analysis method (HAM) for solving FDAEs. Numerical results reveal that the VIM and the ADM are quite accurate and applicable.     

Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations

In this study it is shown that the numerical solutions of linear Fredholm integro-differential equations obtained by using Legendre polynomials can also be found by using the variational iteration method. Furthermore the numerical solutions of the given problems which are solved by the variational iteration method obviously converge rapidly to exact solutions better than the Legendre polynomial technique. Additionally, although the powerful effect of the applied processes in Legendre polynomial approach arises in the situations where the initial approximation value is unknown, it is shown by the examples that the variational iteration method produces more certain solutions where the first initial function approximation value is estimated. In this paper, the Legendre polynomial approximation (LPA) and the variational iteration method (VIM) are implemented to obtain the solutions of the linear Fredholm integro-differential equations and the numerical solutions with respect to these methods are compared.     

Modified variational iteration method (nonlinear homogeneous initial value problem)

The modified variational iteration method is applied for analytical treatment of nonlinear homogeneous initial value problem. The modified variational iteration method accelerates the convergence of the power series solution and reduces the size of work. A comparison between modified variational iteration method (MVIM) and variational iteration method (VIM) was made. The comparison enhances the use of the modified variational iteration method if we wish to obtain an approximate power series solution that converges faster to the closed form solution. The method is very simple and easy.     

Numerical solutions of Duffing equations involving both integral and non-integral forcing terms

In this paper, an improved variational iteration method is presented for solving Duffing equations involving both integral and non-integral forcing terms. The main advantage of this modification over the standard variational iteration method (VIM) is that it can avoid unnecessary repeated computation in determining the unknown parameters in the initial solution. Numerical results reveal that the improved method is simple and efficient.     

An analytical approach to the Fornberg–Whitham type equations by using the variational iteration method

In this paper, He’s variational iteration method (VIM) is applied to solve the Fornberg–Whitham type equations. The VIM provides fast converged approximate solutions to nonlinear equations without linearization, discretization, perturbation, or the Adomian polynomials. This makes the method attractive and reliable for solving the Fornberg–Whitham type equations. Numerical examples related to two initial value problems are presented to show the efficiency of the VIM.     

Multi-point boundary value problem for optimal bridge design

Based upon He's homotopy perturbation and variational iteration methods, we present a method for approximate solutions of nonlinear second-order multi-point boundary value problems (BVPs) in bridge design. Two numerical experiments are carried out to demonstrate the efficiency of the present method. The results reveal that the proposed method is very effective for second-order multi-point BVPs in bridge design.     

The variational iteration method for solving linear and nonlinear Volterra integral and integro-differential equations

The variational iteration method is used for solving the linear and nonlinear Volterra integral and integro-differential equations. The method is reliable in handling Volterra equations of the first kind and second kind in a direct manner without any need for restrictive assumptions. The method significantly reduces the size of calculations.     

Solution of a nonlinear time-delay model in biology via semi-analytical approaches

The delay logistic equations have been extensively used as models in biology and other sciences, with particular emphasis on population dynamics. In this work, the variational iteration and Adomian decomposition methods are applied to solve the delay logistic equation. The variational iteration method is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. On the other hand, the Adomian decomposition method approximates the solution as an infinite series and usually converges to the accurate solution. Moreover, these techniques reduce the volume of calculations because they have no need of discretization of the variables, linearization or small perturbations. Illustrative examples are included to demonstrate the validity and applicability of the presented methods.     

Reliable algorithms for solving integro-differential equations with applications

This paper suggests four different methods to solve nonlinear integro-differential equations, namely, He's variational iteration method, Adomian decomposition method, He's homotopy perturbation method and differential transform method. To assess the accuracy of each method, a test example with known exact solution is used. The study outlines significant features of these methods as well as sheds some light on advantages of one method over the other. The results show that these methods are very efficient, convenient and can be adapted to fit a larger class of problems. The comparison reveals that, although the numerical results of these methods are similar, He's homotopy perturbation method is the easiest, the most efficient and convenient. Moreover, we applied modified forms of He's variational iteration method and differential transform method to solve a mathematical model, which describes the accumulated effect of toxins on populations living in a closed system.