Solving nonlinear partial differential equations using the modified variational iteration Padé technique

In this paper, the modified variational iteration method (MVIM) is reintroduced with the enhancement of Padé approximants to lengthen the interval of convergence of VIM or MVIM when used alone in solving nonlinear problems. KdV, mKdV, Burger's and Lax's equations are used as examples to illustrate the effectiveness and convenience of the proposed technique.     

Numerical solutions of nonlinear evolution equations using variational iteration method

The variational iteration method is used to solve three kinds of nonlinear partial differential equations, coupled nonlinear reaction diffusion equations, Hirota–Satsuma coupled KdV system and Drinefel’d–Sokolov–Wilson equations. Numerical solutions obtained by the variational iteration method are compared with the exact solutions, revealing that the obtained solutions are of high accuracy. He's variational iteration method is introduced to overcome the difficulty arising in calculating Adomian polynomial in Adomian method. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

Approximate analytical solution for the Zakharov–Kuznetsov equations with fully nonlinear dispersion

In this paper, variational iteration method (VIM) is used to obtain numerical and analytical solutions for the Zakharov–Kuznetsov equations with fully nonlinear dispersion. Comparisons with exact solution show that the VIM is a powerful method for the solution of nonlinear equations.     

The generalized quasilinearization method for second-order three-point boundary value problems

In this paper, we obtain the existence and uniqueness results for general second-order three-point boundary value problems by applying the method of upper and lower solutions together with Leray–Schauder degree, as well as obtaining monotone iteration schemes which converge quadratically to the unique solution of some specific second-order three-point boundary value problem by using the method of quasilinearization.     

Estimates for eigenvalues of quasilinear elliptic systems

In this paper we introduce the generalized eigenvalues of a quasilinear elliptic system of resonant type. We prove the existence of infinitely many continuous eigencurves, which are obtained by variational methods. For the one-dimensional problem, we obtain an hyperbolic type function defining a region which contains all the generalized eigenvalues (variational or not), and the proof is based on a suitable generalization of Lyapunov's inequality for systems of ordinary differential equations. We also obtain a family of curves bounding by above the kth variational eigencurve.     

Solvability of strongly nonlinear boundary value problems for second order differential inclusions

In this paper we study a problem for a second order differential inclusion with Dirichlet, Neumann and mixed boundary conditions. The equation is driven by a nonlinear, not necessarily homogeneous, differential operator satisfying certain conditions and containing, as a particular case, the p$p$-Laplacian operator. We prove the existence of solutions both for the case in which the multivalued nonlinearity has convex values and for the case in which it has not convex values. The presence of a maximal monotone operator in the equation make the results applicable to gradient systems with non-smooth, time invariant, convex potential and differential variational inequalities.     

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.     

Existence of doubly periodic solutions for a class of telegraph system with indefinite weight

Using the method of monotone iteration and Schauder fixed point theorem, we establish the existence of doubly periodic solutions for a coupled nonlinear telegraph system with indefinite weight.     

Wave propagation in RTD-based cellular neural networks

This work investigates the existence of monotonic traveling wave and standing wave solutions of RTD-based cellular neural networks in the one-dimensional integer lattice . For nonzero wave speed c, applying the monotone iteration method with the aid of real roots of the corresponding characteristic function of the profile equation, we can partition the parameter space (γ,δ)-plane into four regions such that all the admissible monotonic traveling wave solutions connecting two neighboring equilibria can be classified completely. For the case of c=0, a discrete version of the monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Furthermore, if γ or δ is zero then the profile equation for the standing waves can be viewed as an one-dimensional iteration map and we then prove the multiplicity results of monotonic standing waves by using the techniques of dynamical systems for maps. Some numerical results of the monotone iteration scheme for traveling wave solutions are also presented.     

THE KACANOV METHOD FOR A NONLINEAR VARIATIONAL INEQUALITY OF THE SECOND KIND ARISING IN ELASTOPLASTICITY

The authors first prove a convergence result on the Ka(?)anov method for solving generalnonlinear variational inequalities of the second kind and then apply the Kacanov method tosolve a nonlinear variational inequality of the second kind arising in elastoplasticity. In additionto the convergence result, an a posteriori error estimate is shown for the Kacanov iterates. Ineach step of the Ka(?)anov iteration, one has a (linear) variational inequality of the secondkind, which can be solved by using a regularization technique. The Ka(?)anov iteration andthe regularization technique together provide approximations which can be readily computednumerically. An a posteriori error estimate is derived for the combined effect of the Ka(?)anoviteration and the regularization.     

Positive Periodic and Homoclinic Solutions for Nonlinear Differential Equations with Nonsmooth Potential

We study the existence of positive solutions and of positive homoclinic (to zero) solutions for a class of periodic problems driven by the scalar ordinary p-Laplacian and having a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory and our results extend the recent works of Korman–Lazer (Electronic JDE (1994)) and of Grossinho–Minhos–Tersian (J. Math. Anal. Appl. 240 (1999)).     

Existence and multiplicity results for nonlinear nonautonomous second-order systems

In this paper we consider nonlinear periodic systems driven by the ordinary p$p$-Laplacian and having a nonsmooth potential function. Under minimal and natural hypotheses on the nonsmooth potential and using variational methods based on the nonsmooth critical point theory we prove four existence theorems and a multiplicity theorem. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (i.e. it is unbounded both above and below).     

Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations

In this paper, we consider the existence of at least three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. The associated Green’s function for the boundary value problems is first given. The proofs of our main results are based upon the Leggett–Williams fixed point theorem. Finally, we give an example to demonstrate our result.     

Singularly perturbed differential inclusions: An averaging approach

We consider nonlinear, singularly perturbed differential inclusions and apply the averaging method in order to construct a limit differential inclusion for slow motion. The main approximation result states that the existence and regularity of the limit differential inclusion suffice to describe the limit behavior of the slow motion. We give explicit approximation rates for the uniform convergence on compact time intervals. The approach works under controllability or stability properties of fast motion.     

On Legendre polynomial approximation with the VIM or HAM for numerical treatment of nonlinear fractional differential equations

The variational iteration method and the homotopy analysis method, as alternative methods, have been widely used to handle linear and nonlinear models. The main property of the methods is their flexibility and ability to solve nonlinear equations accurately and conveniently. This paper deals with the numerical solutions of nonlinear fractional differential equations, where the fractional derivatives are considered in Caputo sense. The main aim is to introduce efficient algorithms of variational iteration and homotopy analysis methods that can be simply used to deal with nonlinear fractional differential equations. In these algorithms, Legendre polynomials are effectively implemented to achieve better approximation for the nonhomogeneous and nonlinear terms that leads to facilitate the computational work. The proposed algorithms are capable of reducing the size of calculations, improving the accuracy and easily overcome the difficulty arising in calculating complicated integrals. Numerical examples are examined to show the efficiency of the algorithms.     

An effective modification of He’s variational iteration method

It is well known that one of the advantages of He’s variational iteration method is the free choice of initial approximation. Therefore, in this paper, we use this advantage to propose a reliable modification of He’s variational iteration method. Indeed, this constructs an initial trial-function without unknown parameters, which is called the modified variational iteration method. Some of the nonlinear and linear equations are examined by the modified method to illustrate the effectiveness and convenience of this method, and in all cases, the modified technique performed excellently. The results reveal that the proposed method is very effective and simple and gives exact solutions. The modification could lead to a promising approach for many applications in applied sciences.     

The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method

In this paper, a scheme is developed to study numerical solution of the space- and time-fractional Burgers equations with initial conditions by the variational iteration method (VIM). The exact and numerical solutions obtained by the variational iteration method are compared with that obtained by Adomian decomposition method (ADM). The results show that the variational iteration method is much easier, more convenient, and more stable and efficient than Adomian decomposition method. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed.     

An approximation solution of a nonlinear equation with Riemann–Liouville's fractional derivatives by He's variational iteration method

In this article, an application of He's variational iteration method is proposed to approximate the solution of a nonlinear fractional differential equation with Riemann–Liouville's fractional derivatives. Also, the results are compared with those obtained by Adomian's decomposition method and truncated series method. The results reveal that the method is very effective and simple.     

Analysis for an N-stepped Rayleigh bar with sections of complex geometry

In this paper we analyse the vibrations of an N-stepped Rayleigh bar with sections of complex geometry, supported by end lumped masses and springs. Equations of motion and boundary conditions are derived from the Hamilton’s variational principal. The solutions for tapered and exponential sections are given. Two types of orthogonality for the eigenfunctions are obtained. The analytic solution to the N-stepped Rayleigh model is constructed in terms of Green function.     

Theorems on Schur complement of block diagonally dominant matrices and their application in reducing the order for the solution of large scale linear systems

We firstly consider the block dominant degree for I-(II-)block strictly diagonally dominant matrix and their Schur complements, showing that the block dominant degree for the Schur complement of an I-(II-)block strictly diagonally dominant matrix is greater than that of the original grand block matrix. Then, as application, we present some disc theorems and some bounds for the eigenvalues of the Schur complement by the elements of the original matrix. Further, by means of matrix partition and the Schur complement of block matrix, based on the derived disc theorems, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and the numerical example illustrates that the iteration can compute out the results faster.