darboux transformation with dihedral reduction group we construct the darboux transformation with dihedral reduction group for the 2 dimensional generalisation of the periodic volterra lattice the resulting backlund transformation can be viewed as a nonevolutionary integrable differential difference equation we also find its generalised symmetry and the lax representation for this symmetry using formal diagonalisation of the darboux matrix we obtain local conservation laws of the system

he s variational iteration method for treating nonlinear singular boundary value problems this paper applies he s variational iteration method for solving nonlinear singular boundary value problems the solution process is illustrated and various physically relevant results are obtained comparison of the obtained results with exact solutions shows that the method used is an effective and highly promising method for treating various classes of both linear and nonlinear singular boundary value problems

discrete breathers in a two dimensional fermi pasta ulam lattice using asymptotic methods we investigate whether discrete breathers are supported by a two dimensional fermi pasta ulam lattice a scalar one component two dimensional fermi pasta ulam lattice is shown to model the charge stored within an electrical transmission lattice a third order multiple scale analysis in the semi discrete limit fails since at this order the lattice equations reduce to the 2 1 dimensional cubic nonlinear schrodinger nls equation which does not support stable soliton solutions for the breather envelope we therefore extend the analysis to higher order and find a generalised 2 1 dimensional nls equation which incorporates higher order dispersive and nonlinear terms as perturbations we find an ellipticity criterion for the wave numbers of the carrier wave numerical simulations suggest that both stationary and moving breathers are supported by the system calculations of the energy show the expected threshold behaviour whereby the energy of breathers does em not go to zero with the amplitude we find that the energy threshold is maximised by stationary breathers and becomes arbitrarily small as the boundary of the domain of ellipticity is approached

numerical method for non linear wave and diffusion equations by the variational iteration method this paper applies he s variational iteration to the wave equations in an infinite one dimensional medium and some non linear diffusion equations a suitable choice of an initial solution can lead to the needed exact solution by a few iterations copyright 2007 john wiley sons ltd

application of the variational iteration method for solving nth order integro differential equations in this paper we solve nth order integro differential equations by changing the problem to a system of ordinary integro differential equations and using the variational iteration method the variational iteration formula is derived and the lagrange multiplier can be effectively identified it is well known that one of the advantages of he s variational iteration method is the free choice of initial approximation therefore we use this advantage to construct an initial values without unknown parameters some examples are given and the results reveal that the method is very effective and simple compared with the homotopy perturbation method hpm

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

a simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint in this paper a new and simple approach is presented to exactly calculate the critical buckling loads of beams with arbitrarily axial inhomogeneity for various end boundary conditions we transform the governing equation with varying coefficients to linear algebraic equations then a characteristic equation in critical buckling loads will be obtained several examples of estimating buckling loads under typical end supports are discussed by comparing our numerical results with the exact and existing results for homogeneous and nonhomogeneous beams it can be found that our method has fast convergence and the obtained numerical results have high accuracy moreover the buckling behavior of a functionally graded beam composed of aluminum and zirconia as two constituent phases is investigated for axially varying material properties the effects of gradient parameters on the critical buckling loads are elucidated finally we give an example to illustrate the enhancement of the load carrying capacity of tapered beams for admissible shape profiles with constant volume or weight the proposed method is of benefit to optimum design of beams against buckling in engineering applications

on the convergence of he s variational iteration method in this work we will consider he s variational iteration method for solving second order initial value problems we will discuss the use of this approach for solving several important partial differential equations this method is based on the use of lagrange multipliers for identification of optimal value of a parameter in a functional this procedure is a powerful tool for solving the large amount of problems using the variational iteration method it is possible to find the exact solution or an approximate solution of the problem this technique provides a sequence of functions which converges to the exact solution of the problem our emphasis will be on the convergence of the variational iteration method in the current paper this scheme will be investigated in details and efficiency of the approach will be shown by applying the procedure on several interesting and important models

improvement of he s variational iteration method for solving systems of differential equations in recent years a lot of attention from researchers has been attracted to the various aspects of the well known he s variational iteration method this method is a very powerful method for solving a large amount of problems it provides a sequence which converges to the solution of the problem without discretization of the variables in this work an idea is proposed that accelerates the convergence of the sequences which result from the variational iteration method for solving systems of differential equations illustrative examples are presented to show the validity of the new method

dromions in 2 1 dimensional ferromagnetic spin chain with bilinear and biquadratic interactions we study the dynamics of 2 1 dimensional ferromagnetic spin chain with bilinear and biquadratic interactions by using holstein primakoff hp representation and the coherent state ansatz the dynamical equations of motion are obtained by using long wavelength approximation the sine cosine function method is used to study the complete nonlinear excitation and the effect of the different types of interactions is investigated graphically

lenard scheme for two dimensional periodic volterra chain we prove that for compatible weakly nonlocal hamiltonian and symplectic operators hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating nijenhuis operators are weakly nonlocal or not we construct a recursion operator of the two dimensional periodic volterra chain from its lax representation and prove that it is a nijenhuis operator furthermore we show that this system is a generalized bi hamiltonian system rather surprisingly the product of its weakly nonlocal hamiltonian and symplectic operators gives rise to the square of the recursion operator

the variational iteration method an efficient scheme for handling fractional partial differential equations in fluid mechanics variational iteration method has been used to handle linear and nonlinear differential equations the main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently in this work a general framework of the variational iteration method is presented for analytical treatment of fractional partial differential equations in fluid mechanics the fractional derivatives are described in the caputo sense numerical illustrations that include the fractional wave equation fractional burgers equation fractional kdv equation fractional klein gordon equation and fractional boussinesq like equation are investigated to show the pertinent features of the technique comparison of the results obtained by the variational iteration method with those obtained by adomian decomposition method reveals that the first method is very effective and convenient the basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus

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

nonlinear constructive interference in electrical lattices we analyze the constructive interference of input signals in nonlinear discrete two dimensional media the media consist of inductor capacitor lattices with saturating voltage dependent capacitors we find that nonlinearity significantly boosts the ability of such media to generate large amplitude output signals from small amplitude inputs to understand this boosting we develop a general perturbative method suitable for finding the steady state solution of a damped n ifmmode times else texttimes fi n nonlinear lattice that is driven at a single frequency we verify our theory using extensive numerical simulations

the variational iteration method for rational solutions for kdv k 2 2 burgers and cubic boussinesq equations in this work the reliable variational iteration method is used to determine rational solutions for the kdv the k 2 2 the burgers and the cubic boussinesq equations the study highlights the efficiency of the method and its dependence on the lagrange multiplier rational solutions are obtained directly in a straightforward manner

solving the breaking soliton equation by he s variational method he s variational method is applied to obtain the soliton solutions of the breaking soliton equation where the semi inverse method is applied to establish a needed variational functional

a residual method for solving nonlinear operator equations and their application to nonlinear integral equations using symbolic computation a derivative free residual method for solving nonlinear operator equations in real hilbert spaces is discussed this method uses in a systematic way the residual as search direction but it does not use first order information furthermore a convergence analysis and numerical results of the new method applied to nonlinear integral equations using symbolic computation are presented

on the solutions of nonlinear fractional klein gordon equation with modified riemann liouville derivative abstract in this paper fractional variational iteration method fvim is applied for finding approximate analytical solutions of nonlinear fractional klein gordon equation time fractional derivative is described in the riemann liouville sense a new application of fractional variational iteration method fvim is extended to derive analytical solutions in the form of a series for this equation the behavior of the solutions and the effects of different values of fractional order are indicated graphically it is shown that the solutions obtained by the fvim are reliable convenient and effective for strongly nonlinear partial equations with modified riemann liouville derivative

the variational iteration method a highly promising method for solving the system of integro differential equations this paper applies he s variational iteration method for solving two systems of volterra integro differential equations the solution process is illustrated and various physically relevant results are obtained comparison of the obtained results with exact solutions shows that the used method is an effective and highly promising method for various classes of both linear and nonlinear integro differential equations

automorphic lie algebras with dihedral symmetry the concept of automorphic lie algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems automorphic lie algebras are obtained by imposing a discrete group symmetry on a current algebra of krichever novikov type past work shows remarkable uniformity between algebras associated to different reduction groups for example if the base lie algebra is sl2 c and the poles of the automorphic lie algebra are restricted to an exceptional orbit of the symmetry group changing the reduction group does not affect the lie algebra structure in this research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base lie algebra we find a uniform description of automorphic lie algebras with dihedral symmetry valid for poles at exceptional and generic orbits

a new modification of the variational iteration method this paper presents a new variational iteration algorithm which provides great freedom in choosing linear operators for various nonlinear equations so that the lagrange multiplier can be effectively identified

steady state perturbative theory for nonlinear circuits we study the steady state behavior of a damped driven nonlinear lrc oscillator where the nonlinearity arises due to voltage dependent capacitance the driving or input signal is assumed to be a pure tone using an iterative perturbative solution technique combined with an energy conservation argument we show that the oscillator transfers energy from the fundamental to higher harmonics we determine a series expansion of the two norm of the steady state output signal and show that in a large region of parameter space the two norm depends superlinearly on the input amplitude we also use the two norm calculation to devise a performance goal that the infinity norm of the steady state output signal should satisfy in order for the nonlinear system to have a genuine boost over the corresponding linear system taken together these results are a step toward the automatic design of nonlinear systems that have an optimal boost over corresponding linear systems

a piecewise variational iteration method for riccati differential equations in this paper we introduce a piecewise variational iteration method for riccati differential equations which is a modified variational iteration method mvim the solutions of riccati differential equations obtained using the traditional variational iteration method vim give good approximations only in the neighborhood of the initial position however the solutions obtained using the mvim give good approximations for a larger interval rather than a local vicinity of the initial position some numerical examples are selected to illustrate the effectiveness and simplicity of the method numerical results show that the method does not share the drawback of the conventional vim and is a satisfactory method for riccati differential equations

numerical treatment of stochastic models used in statistical systems and financial markets in this paper by means of the variational iteration method numerical solutions are computed for some stochastic models without any linearization or weak assumptions two stochastic models the fokker planck equation for non equilibrium statistical systems and the black scholes model for pricing stock options are solved numerically in this approach the solution is found in the form of a convergent series with easily computed components the behavior of the approximate solutions is shown graphically

variational iteration method new development and applications variational iteration method has been favourably applied to various kinds of nonlinear problems the main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper recent trends and developments in the use of the method are reviewed major applications to nonlinear wave equation nonlinear fractional differential equations nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed the confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods variational iteration method is uniquely qualified to address this challenge the flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions this paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations a new iteration formulation is suggested to overcome the shortcoming a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested examples are given to illustrate the solution procedure

variational iteration method for solving burger s and coupled burger s equations by means of variational iteration method the solutions of burger s equation and coupled burger s equations are exactly obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

classes of solutions in the problem of stability analysis in bars with varying cross section and axial distributed loading we construct classes of analytical solutions for the linear ordinary differential equation of variable coefficients governing the stability analysis of bars with varying cross section and axial distributed loading the analytical solutions obtained by li et al 1995 int j solids structures for two kinds of variable stiffness and axial loading in such types of bars result as special cases

a variational approach to analyzing catalytic reactions in short monoliths in this paper a variational principle is established for analyzing catalytic reactions in short monoliths by he s semi inverse method on the basis of the established variational formulation the solutions can be easily obtained by the ritz method

solution of a quadratic nonlinear oscillator by the method of harmonic balance abstract this paper deals with the quadratic nonlinear oscillator qno x x e x 2 0 this oscillator is compared with the nonlinear oscillators x x e x x 0 which are solved by the method of harmonic balance therefore we obtain the first approximate solution to the qno this solution is more accurate than the second approximate solution obtained by the lindstedt poincare method for qnos it has been pointed out that the angular frequency for x 0 should be different from the angular frequency for x 0 if solutions of higher degree of accuracy are required

exact solutions for cluster growth kinetics with evolving size and shape profiles in this paper we construct a model for the simultaneous compaction by which clusters are restructured and growth of clusters by pairwise coagulation the model has the form of a multi component aggregation problem in which the components are cluster mass and cluster diameter following suitable approximations exact explicit solutions are derived which may be useful for the verification of simulations of such systems numerical simulations are presented to illustrate typical behaviour and to show the accuracy of approximations made in deriving the model the solutions are then simplified using asymptotic techniques to show the relevant timescales of the kinetic processes and elucidate the shape of the cluster distribution functions at large times

darboux transformations finite reduction groups and related yang baxter maps in this paper we construct yang baxter yb maps using darboux matrices which are invariant under the action of finite reduction groups we present six dimensional yb maps corresponding to darboux transformations for the nonlinear schrodinger nls equation and the derivative nonlinear schrodinger dnls equation these yb maps can be restricted to four dimensional yb maps on invariant leaves the former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries fordy a and kassotakis a 2013 j phys a math theor 46 205201 we give a six dimensional yb map corresponding to the darboux transformation for a deformation of the dnls equation we also consider vector generalizations of the yb maps corresponding to the nls and dnls equations

buckling and post buckling analysis of extensible beam columns by using the differential quadrature method buckling and post buckling analysis of extensible beam columns is performed numerically in this paper it was experienced earlier that in some cases the numerical integration would not produce the convergent post buckling solutions especially under high loads therefore a new differential quadrature dq based iterative numerical integration method is proposed to solve post buckling differential equations of extensible beam columns six cases including five classical euler buckling cases are analyzed critical loads and convergent post buckling solutions under different applied loads are obtained the results are compared with the existing multiple scales solutions it is found that under high applied loads the small rotation assumption in obtaining multiple scales solutions is no longer valid the proposed iterative dq based numerical integration method can yield reliable and accurate post buckling solutions even at high applied loads for the cases investigated

some numerical methods for solving nonlinear equations by using decomposition technique in this paper we use the system of coupled equations involving an auxiliary function together with decomposition technique to suggest and analyze some new classes of iterative methods for solving nonlinear equations these new methods include the halley method and its variant forms as special cases various numerical examples are given to illustrate the efficiency and performance of the new methods these new iterative methods may be viewed as an addition and generalization of the existing methods for solving nonlinear equations

variational iteration method for solving a generalized pantograph equation the variational iteration method is applied to solve the generalized pantograph equation this technique provides a sequence of functions which converges to the exact solution of the problem and is based on the use of lagrange multipliers for identification of optimal value of a parameter in a functional employing this technique it is possible to find the exact solution or an approximate solution of the problem some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results

the variational iteration method for cauchy problems in this paper the variational iteration method is used to solve the cauchy problem some examples are given to elucidate the solution procedure and reliability of the obtained results the variational iteration algorithm leads to exact solutions in the present study

elastic stability of euler columns with a continuous elastic restraint using variational iteration method in this paper analysis of elastic stability for continuously restrained euler columns is conducted by means of variational iteration method vim a uniform homogeneous column is assumed to be restrained along its length the restraint considered in this study is an elastic foundation model in engineering practice and it is of great interest to foundation engineers hence the variation of the critical buckling loads with the stiffness of elastic restraint is investigated using vim for the restrained columns with different end conditions obtaining analytical solutions for these types of problems is not a simple procedure since the equations of stability criteria are highly nonlinear this study presents the application of vim for obtaining exact solutions for continuously restrained euler columns the study proves that vim is a very efficient and promising approach in the elastic stability analysis of specified problems

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

preface the variational iteration method reliable efficient and promising in the last few decades considerable work has been invested in developing new methods for analytical and numerical solutions of differential and integral equations linear or nonlinear there has been a great deal of research work done to address the issues of nonlinearity and singularity phenomena that arise in many scientific and engineering problems moreover there has been a great need for effective algorithms to avoid the onerous work required by traditional techniques as a result of this need many analytical and numerical tools such as grid points techniques perturbation techniques spline solutions the adomian method and others have been developed however each of these methods suffers from one or more limitations the grid points techniques define the solution at grid points only the spline solution requires restrictions on boundary points the perturbation method suffers from a high computational workload especially when the degree of nonlinearity increases in addition the adomian method suffers from the complication of the algorithms needed to calculate adomian polynomials that are necessary for nonlinear problems this special issue of computers and mathematics with applications conveys a strong reliable efficient and promising development ji huan he developed the variational iteration method vim in 1999 in this method the equations are initially approximated with possible unknowns a correction functional is established by a general lagrange multiplier which can be identified optimally via the variational theory the method gives rapidly convergent successive approximations of the exact solution the vim has no restrictions or unrealistic assumptions such as linearization or small parameters that are used for nonlinear operators the vim handles linear and nonlinear problems in a like manner the convergence concept has been proved a substantial amount of research work has been directed to the study of the variational iteration method by many well known researchers several studies have been conducted to compare the vim method with existing techniques and it was shown by all that this method gives exact solutions faster than other methods the method has been applied in a wide variety of scientific and engineering applications these works confirmed the fact that this methodology is reliable efficient and promising as well as having a wider applicability included herein is a collection of original refereed research papers by well established researchers in the field of applied mathematics we hope that this issue will prove to be a timely and valuable reference for researchers in this area special thanks go to the referees for their valuable work we here thank dr ervin y rodin for providing us with the opportunity to produce this special issue on this exciting method

determination of critical buckling load for elastic columns of constant and variable cross sections using variational iteration method in this paper variational iteration method vim is applied to the problem of determination of critical buckling loads for euler columns with constant and variable cross sections vim is a powerful method for the solution of nonlinear ordinary and partial differential equations and integral equations hence it is a suitable approach for the analysis of engineering problems where an exact solution is difficult to obtain this study presents the application of vim to various buckling cases and results are produced for columns with different support conditions and with different variation of cross sections the results obtained are accurate which show that variational iteration method is a very efficient technique in the analysis of elastic stability problems

variational iteration method for solving integral equations in this paper several integral equations are solved by he s variational iteration method comparison with exact solution shows that the method is very effective and convenient for solving integral equations

a new dq based method and its application to the stability of columns a numerical picard like method which combines successive approximations with differential quadrature is discussed for application to the stability of columns unlike other methods the present method satisfies all the boundary conditions without any additional computation for each iteration and it does not require any subdivision of the problem domain despite of its recursive nature the approximate solution can be expressed by operational matrices and vectors of known quantities such as the solution obtained by advanced polynomial expansions several examples show the satisfactorily behaviour of the proposed method in comparison to the existing methods both in the linear and nonlinear range

on the classification of automorphic lie algebras the problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants this provides a powerful tool of analysis and it opens the road to new applications of automorphic lie algebras beyond the con text of integrable systems in this paper it is shown that sl2 c based automorphic lie algebras associated to the icosahedral group i the octahedral group o the tetrahedral group t and the dihedral group dn are isomorphic the proof is based on techniques from classical invariant theory and makes use of clebsch gordan decomposition and transvectants molienfunctionsandthetrace form thisresultprovidesacompleteclas sification of sl2 c based automorphic lie algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of automorphic lie algebras

numerical approach to differential equations of fractional order in this paper the variational iteration method and the adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order the two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations in these schemes the solution takes the form of a convergent series with easily computable components this paper presents a numerical comparison between the two methods for solving systems of fractional differential equations numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order

two methods for solving integral equations the objective of this work is to introduce two efficient strategies that produce the accurate solutions to linear and nonlinear volterra and fredholm integral equations of the second kind for illustration several linear nonlinear and integro differential examples have been tested and the results showed improvements over existing techniques

stability analysis of bars with varying cross section abstract the exact solutions for stability analysis of bars with varying cross sections subjected to simple or complicated loads including concentrated and variably distributed axial loads are presented in this paper the distribution of flexural stiffness of the bar and that of axial loads acting on the bar are expressed as power functions or exponential functions also the extracted exact solutions are expressed in terms of bessel functions and super geometric series

module structure of the center of the universal central extension of a genus zero krichever novikov algebra abstract we describe how the center of the universal central extension of the genus zero krichever novikov current algebra decomposes as a direct sum of irreducible modules for automorphism group of the coordinate ring of this algebra

application of g g expansion method to regularized long wave rlw equation in this paper the g g expansion method with the aid of maple is used to obtain a generalized soliton solution for the generalized regularized long wave rlw equation each of the obtained solutions namely hyperbolic function solutions and trigonometric function solutions contain an explicit linear function of the variables in the considered equation it is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems

he s fractional derivative a powerful tool to fractional fornberg whitham equation abstract in this paper fractional fornberg whitham equation is studied in a fractal process based on he s fractional derivative the fractional complex transform is used to convert the studied fractional equation into an ordinary differential equation and gejji jafari iteration method is used to solve the equation

a reliable treatment for mixed volterra fredholm integral equations the main goal of this paper is to demonstrate the use of the modified decomposition method for mixed nonlinear volterra fredholm integral equations the modified method combined with the noise terms phenomena may provide the exact solution by using two iterations only two numerical illustrations are given to show the pertinent features of the technique

variational iteration method for twelfth order boundary value problems in this paper he s variational iteration method is applied to solve twelfth order boundary value problems the numerical results obtained with minimum amount of computation are compared with the exact solutions to show the efficiency of the method the results show that the variational iteration method is of high accuracy more convenient and efficient for solving high order boundary value problems

analytical treatment of differential equations with fractional coordinate derivatives this paper outlines a reliable strategy to use the homotopy perturbation method based on jumarie s derivative for solving fractional differential equations in this framework compact structures of fourth order fractional diffusion wave equations are considered as prototype examples moreover convergence of the proposed approach for these types of equations is investigated results show that the response expressions are mittag leffler stable

variational iteration method some recent results and new interpretations this paper is an elementary introduction to the concepts of variational iteration method first the main concepts in variational iteration method such as general lagrange multiplier restricted variation correction functional are explained heuristically subsequently the solution procedure is systematically addressed in particular for nonlinear oscillators particular attention is paid throughout the paper to give an intuitive grasp for the method the main motivation is to put things together in a convenient form for later reference and systematic use

the variational iteration method for solving a neutral functional differential equation with proportional delays in this paper the variational iteration method is applied to neutral functional differential equations with proportional delays illustrative examples are given to show the efficiency of the method we also compare the performance of the method with that of a particular runge kutta method and a one leg q method

exp function method for riccati equation and new exact solutions with two arbitrary functions of 2 1 dimensional konopelchenko dubrovsky equations in this paper new exact solutions with two arbitrary functions of the 2 1 dimensional konopelchenko dubrovsky equations are obtained by means of the riccati equation and its generalized solitary wave solutions constructed by the exp function method it is shown that the exp function method provides us with a straightforward and important mathematical tool for solving nonlinear evolution equations in mathematical physics

darboux transformations with tetrahedral reduction group and related integrable systems in this paper we derive new two component integrable differential difference and partial difference systems by applying a lax darboux scheme to an operator formed from an x1d530 x1d529 3 based automorphic lie algebra the integrability of the found systems is demonstrated via lax pairs and generalised symmetries

new applications of variational iteration method abstract the variational iteration method is used for solving three types of nonlinear partial differential equations such as coupled schrodinger kdv generalized kdv and shallow water equations the exact and numerical solutions obtained by variational iteration method are compared with that obtained using adomian decomposition method the comparison shows that the two solutions obtained are excellent agreement the method is made showing that the former is more effective than the latter in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

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

reduction groups and automorphic lie algebras we study a new class of infinite dimensional lie algebras which has important applications to the theory of integrable equations the construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic lie algebras for automorphic lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly these algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts

variational iteration method for nonlinear age structured population models this paper investigates nonlinear age structured population models the numerical solutions are obtained by using the variational iteration method and are of high accuracy two numerical examples are presented to illustrate the strength of the method

a modified spectral method for solving operator equations in this paper we introduce a modified spectral method for solving the linear operator equation l u f l d l h 1 h 2 where h 1 and h 2 are normed vector spaces with norms 1 and respectively and d l is the domain of l also for each h h 2 h 2 h h where is an inner product on h 2 in this method we make a new set n n 0 for h 1 using l and two sets in h 1 and h 2 then using the new set n n 0 we solve this linear operator equation we show that this method does not have some shortcomings of spectral method also we prove the stability and convergence of the new method after introducing the method we give some conditions that under them the nonlinear operator equation l u n u f can be solved some examples are considered to show the efficiency of method

discrete breathers in a two dimensional hexagonal fermi pasta ulam lattice we consider a two dimensional fermi pasta ulam fpu lattice with hexagonal symmetry using asymptotic methods based on small amplitude ansatz at third order we obtain a eduction to a cubic nonlinear schr o dinger equation nls for the breather envelope however this does not support stable soliton solutions so we pursue a higher order analysis yielding a generalised nls which includes known stabilising terms we present numerical results which suggest that long lived stationary and moving breathers are supported by the lattice we find breather solutions which move in an arbitrary direction an ellipticity criterion for the wavenumbers of the carrier wave symptotic estimates for the breather energy and a minimum threshold energy below which breathers cannot be found this energy threshold is maximised for stationary breathers and becomes vanishingly small near the boundary of the elliptic domain where breathers attain a maximum speed several of the results obtained are similar to those obtained for the square fpu lattice butt wattis em j phys a bf 39 4955 2006 though we find that the square and hexagonal lattices exhibit different properties in regard to the generation of harmonics and the isotropy of the generalised nls equation

variational iteration method for unsteady flow of gas through a porous medium using he s polynomials and pade approximants in this paper we apply the variational iteration method using he s polynomials for finding the analytical solution of unsteady flow of gas through a porous medium the proposed method is an elegant combination of he s variational iteration and the homotopy perturbation methods the suggested algorithm is quite efficient and is practically well suited for use in such problems the proposed iterative scheme finds the solution without any discretization linearization or restrictive assumptions the diagonal pade approximants are effectively used in the analysis to capture the essential behavior of the solution and to determine the initial slope a clear advantage of this technique over the decomposition method is that no calculation of adomian s polynomials is needed

variational iteration method for solving cubic nonlinear schrodinger equation the variational iteration method is applied to solve the cubic nonlinear schrodinger cnls equation in one and two space variables in both cases we will reduce the cnls equation to a coupled system of nonlinear equations numerical experiments are made to verify the efficiency of the method comparison with the theoretical solution shows that the variational iteration method is of high accuracy

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

an analytical approach to the dynamic analysis of a rotating electric machine this study presents an analytical approach to the dynamic problem encountered by a rotating electric machine supported by nonlinear bearings with damping properties subjected to a forcing excitation caused by an unbalanced force and a parametric excitation caused by an axial thrust the method employed in this analytical study is the optimal variational iteration method ovim which proves to be very effective and accurate in solving nonlinear problems the results are presented in the form of a time displacement response the results obtained from numerical simulations are in very good agreement with the approximate analytical results obtained through the proposed method

variable separation method for a nonlinear time fractional partial differential equation with forcing term abstract in this paper a variable coefficient nonlinear time fractional partial differential equation pde with initial and boundary conditions is solved by using the variable separation method as a result some new explicit and exact solutions of the time fractional pde are obtained including airy function solution hyperbolic function solution trigonometric function solution and rational solution it is shown that the variable separation method can provide a useful mathematical tool for solving some other nonlinear time fractional pdes in science and engineering

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

reductions of integrable equations dihedral group we discuss the algebraic and analytic structure of rational lax operators with algebraic reductions of lax equations we associate a reduction group a group of automorphisms of the corresponding infinite dimensional lie algebra we present a complete study of dihedral reductions for sl 2 c lax operators with simple poles and corresponding integrable equations in the last section we give three examples of dihedral reductions for sl n c lax operators

approximate solution of nonlinear differential equations with convolution product nonlinearities in this paper a new iteration method is proposed to solve nonlinear problems special attention is paid to nonlinear differential equations with convolution product nonlinearities the results reveal the approximations obtained by the proposed method are uniformly valid for both small and large parameters in nonlinear problems furthermore the first order of approximations are more accurate than perturbation solutions at high order of approximation

a reliable algorithm for obtaining positive solutions for nonlinear boundary value problems abstract in this paper we propose an algorithm for solving the nonlinear two point boundary value problem that has at least one positive solution 1 6 for in a compatible interval our method stems mainly from combining the decomposition series solution obtained by adomian decomposition method with pade approximates the validity of the approach is verified through illustrative numerical examples