the modified extended tanh function method for solving burgers type equations abstract by means of the modified extended tanh function metf method the multiple travelling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system solutions for the nonlinear equations such as one dimensional burgers kdv burgers coupled burgers and two dimensional burgers equations are obtained precisely and so the efficiency of the method can be demonstrated

discrete exact solutions to some nonlinear differential difference equations via the g g expansion method we extended the g g expansion method to two well known nonlinear differential difference equations the discrete nonlinear schrodinger equation and the toda lattice equation for constructing traveling wave solutions discrete soliton and periodic wave solutions with more arbitrary parameters as well as discrete rational wave solutions are revealed it seems that the utilized method can provide highly accurate discrete exact solutions to nddes arising in applied mathematical and physical sciences

the extended tanh method for new solitons solutions for many forms of the fifth order kdv equations abstract the extended tanh method is used to derive new solitons solutions for several forms of the fifth order nonlinear kdv equation the forms include the lax sawada kotera sk sawada kotera parker dye skpd kaup kupershmidt kk kaup kupershmidt parker dye kkpd and the ito equations the criteria established in a m wazwaz abundant solitons solutions for several forms of the fifth order kdv equation by using the tanh method appl math comput in press doi 10 1016 j amc 2006 02 047 to build up reliable relations between the parameters of the equation are confirmed by using this new approach entirely new bell shaped solitons are determined

new solitary wave solutions to the modified forms of degasperis procesi and camassa holm equations abstract in this work we establish new solitary wave solutions to the modified forms of degasperis procesi and camassa holm equations unlike the standard degasperis procesi and camassa holm equations where multi peakon solutions arise the modified forms cause a change in the characteristic of these solutions and change it to bell shaped solitons periodic and complex solutions the extended tanh method the rational hyperbolic functions method and the rational exponential functions method are employed to reveal these new solutions

modified extended tanh function method and its applications to nonlinear equations new exact travelling wave solutions for the generalized shallow water wave equation the improved boussinesq equation and the coupled system for the approximate equations for water waves are found using a modified extended tanh function method the obtained results include rational periodic singular and solitary wave solutions

new solitary wave solutions to the kuramoto sivashinsky and the kawahara equations abstract in this work the kuramoto sivashinsky equation and the kawahara equation are studied the tanh method and the extended tanh method are used for analytic treatment for these two equations by means of these methods new solitary wave solutions are determined for each equation the two approaches are reliable and manageable

effects of corrugation shape on frequency band gaps for longitudinal wave motion in a periodic elastic layer the paper concerns determining frequency band gaps for longitudinal wave motion in a periodic waveguide the waveguide may be considered either as an elastic layer with variable thickness or as a rod with variable cross section as a result widths and locations of all frequency band gaps are determined by means of the method of varying amplitudes for the general symmetric corrugation shape the width of each odd band gap is controlled only by one harmonic in the corrugation series with its number being equal to the number of the band gap widths of even band gaps however are influenced by all the harmonics involved in the corrugation series so that the lower frequency band gaps can emerge these are band gaps located below the frequency corresponding to the lowest harmonic in the corrugation series for the general non symmetric corrugation shape the mth band gap is controlled only by one the mth harmonic in the corrugation series the revealed insights into the mechanism of band gap formation can be used to predict locations and widths of all frequency band gaps featured by any corrugation shape these insights are general and can be valid also for other types of wave motion in periodic structures e g transverse or torsional vibration

a note on travelling wave solutions to lax s seventh order kdv equation ganji and abdollahzadeh d d ganji m abdollahzadeh appl math comput 206 2008 438 444 derived three supposedly new travelling wave solutions to lax s seventh order kdv equation each solution was obtained by a different method it is shown that any two of the solutions may be obtained trivially from the remaining solution furthermore it is noted that one of the solutions has been known for many years

finite element prediction of wave motion in structural waveguides a method is presented by which the wavenumbers for a one dimensional waveguide can be predicted from a finite element fe model the method involves postprocessing a conventional but low order fe model the mass and stiffness matrices of which are typically found using a conventional fe package this is in contrast to the most popular previous waveguide fe approach sometimes termed the spectral finite element approach which requires new spectral element matrices to be developed in the approach described here a section of the waveguide is modeled using conventional fe software and the dynamic stiffness matrix formed a periodicity condition is applied the wavenumbers following from the eigensolution of the resulting transfer matrix the method is described estimation of wavenumbers energy and group velocity discussed and numerical examples presented these concern wave propagation in a beam and a simply supported plate strip for which analytical solutions exist and the more complex case of a viscoelastic laminate which involves postprocessing an ansys fe model the method is seen to yield accurate results for the wavenumbers and group velocities of both propagating and evanescent waves

some applications of the g g expansion method to non linear partial differential equations in the present paper we construct the traveling wave solutions involving parameters of the 2 1 dimensional higher order broer kaup equations the 2 1 dimensional breaking soliton equations the 2 1 dimensional asymmetric nizhnik novikov vesselov equations and the 2 1 dimensional bkp equations in terms of the hyperbolic functions trigonometric functions and the rational functions by using a new approach namely the g g expansion method where g g x satisfies a second order linear ordinary differential equation when the parameters are taken special values the solitary waves are derived from the traveling waves the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions

exact solutions to the two dimensional korteweg de vries burgers equation two methods are described for obtaining exact travelling monotone shock wave solutions to the two dimensional korteweg de vries burgers equation they are compared with two methods that have been reported recently by other authors

application of the g g expansion to travelling wave solutions of the broer kaup and the approximate long water wave equations by using the g g expansion proposed recently the travelling wave solutions involving parameters of the broer kaup equations and the approximate long water wave equations are found out the travelling wave solutions are expressed by three types of functions which are the hyperbolic functions the trigonometric functions and the rational functions when the parameters are taken as special values the solitary wave solutions are obtained c 2008 elsevier inc all rights reserved

the extended tanh method for abundant solitary wave solutions of nonlinear wave equations the extended tanh method is used to establish abundant solitary wave solutions of nonlinear wave equations the obtained solutions include solitons kinks and plane periodic solutions the study is an extension to the remarkable development by malfliet 1 the extended tanh method presents a wider applicability for handling nonlinear wave equations

vibration suppression for strings with distributed loading using spatial cross section modulation abstract a problem of vibration suppression in any preassigned region of a bounded structure subjected to action of an external time periodic load which is distributed over its domain is considered a passive control is applied in which continuous spatially periodic modulations of structural parameters are used as a means for vibration suppression as an example stationary vibrations of a string under action of a distributed time periodic load are studied this system in a simplified form models such processes as interaction between membranes and colloids oscillations of transmission lines under action of rain and wind and dynamics of suspension bridges and stay cables suppression of vibration in predefined regions of the string is performed by continuous spatial modulation of its cross section for analyzing the problem considered a novel approach named the method of varying amplitudes is employed this approach is applicable for solving differential equations without a small parameter and may be considered as a natural continuation of the classical methods of harmonic balance and averaging as a result optimal parameters for the string cross sectional area modulation are determined for the cases of harmonically uniformly and arbitrarily distributed load which allows for completely suppressing or considerably reducing vibration in the prescribed part of the string compared to the case without modulation

wave propagation in continuous periodic structures research contributions from southampton 1964 1995 abstract after brief reference to some early studies by other investigators this paper focuses mainly on methods developed at the university of southampton since 1964 to analyze and predict the free and forced wave motion in continuous periodic engineering structures beginning with receptance methods which have been applied to periodic beams and rib skin structures it continues with a method of direct solution of the wave equation this uses floquet s principle and has been applied to beams and quasi one dimensional periodic plates and cylindrical shells sample curves of the propagation and attenuation constants pertaining to these structures are presented a limited discussion of the transfer matrix then follows after which the method of space harmonics is introduced as the method best suited to the prediction of sound radiated from a vibrating periodic structure reviewed next are some theorems and variational principles relating to periodic structures which have been developed at southampton and which form a basis for finding natural frequencies of finite structures or for computing free and forced wave notion by energy methods this has led to the finite element method in its standard and hierarchical forms being used to study wave motion in genuine two dimensional and three dimensional structures examples of this work are shown the method of phased array receptance functions is then introduced as possibly the easiest way of setting up exact equations for the propagation constants of uniform quasi one dimensional periodic structures a summary is finally presented of the limited and early work performed at southampton on simple disordered periodic structures

exact and explicit solutions to some nonlinear evolution equations by utilizing the g g expansion method in this paper we demonstrate the effectiveness of the so called g g expansion method by examining some nonlinear evolution equations with physical interest our work is motivated by the fact that the g g expansion method provides not only more general forms of solutions but also periodic and solitary waves if we set the parameters in the obtained wider set of solutions as special values then some previously known solutions can be recovered the method appears to be easier and faster by means of a symbolic computation system

observations on the tanh coth expansion method for finding solutions to nonlinear evolution equations abstract the tanh coth expansion method for finding solitary travelling wave solutions to nonlinear evolution equations has been used extensively in the literature it is a natural extension to the basic tanh function expansion method which was developed in the 1990s it usually delivers three types of solution namely a tanh function expansion a coth function expansion and a tanh coth expansion it is known that for every tanh function expansion solution there is a corresponding coth function expansion solution it is shown that there is a tanh coth expansion solution that is merely a disguised version of the coth solution in many papers such tanh coth solutions are erroneously claimed to be new however other tanh coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh function method similar remarks apply to tan cot and tan cot expansion solutions

the tanh coth method for solitons and kink solutions for nonlinear parabolic equations abstract the tanh coth method is used to derive solitons and kink solutions for some of the well known nonlinear parabolic partial differential equations the equations include the fisher equation newell whithead equation allen cahn equation fitzhugh nagumo equation fisher s equation and the burgers fisher equation the new tanh coth approach provides abundant solitons and kink solutions in addition to the existing ones the power of this manageable method is confirmed

the potential for efficiency gains in estimation from the use of additional moment restrictions it is frequently assumed that in generalised method of moments estimation the explicit utilization of valid additional moment restrictions will lead to asymptotic efficiency gains in this paper i develop a general framework for examining the possibility of such efficiency gains using this framework i then consider in more detail the potential for efficiency gains from utilizing a constant conditional disturbance variance restriction when estimating a single equation from a nonlinear simultaneous equations model sem by nonlinear two stage least squares 2sls in general such efficiency gains will arise however for a linear sem with constant disturbance covariances and zero third moments no efficiency gains arise for fairly general instrument sets this generalises in certain respects the usual result that 2sls is asymptotically efficient for estimating a single equation from a linear sem with independently identically distributed normal disturbances

space time topology optimization for one dimensional wave propagation abstract a space time extension of the topology optimization method is presented the formulation with design variables in both the spatial and temporal domains is used to create structures with an optimized distribution of material properties that can vary in time the method is outlined for one dimensional transient wave propagation in an elastic rod with time dependent young s modulus by two simulation examples it is demonstrated how dynamic structures can display rich dynamic behavior such as wavenumber frequency shifts and lack of energy conservation the optimization method s potential for creating structures with novel dynamic behavior is illustrated by a simple example it is shown that an elastic rod in which the optimized stiffness distribution is allowed to vary in time can be much more efficient in prohibiting wave propagation compared to a static bandgap structure optimized designs in form of spatio temporal laminates and checkerboards are generated and discussed the example lays the foundation for creating designs with more advanced functionalities in future work

application of g g expansion method to two nonlinear evolution equations abstract two nonlinear evolution equations are separately reduced to nonlinear ordinary differential equations ode by using a simple transformation when the parameters are taken as special values the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions the proposed method can be used for many other nonlinear evolution equations

constructing of new exact solutions to the gkdv mkdv equation with any order nonlinear terms by g g expansion method abstract the g g expansion method is introduced to construct more general exact travelling wave solutions to the generalized kdv mkdv equation with any order nonlinear terms as a result hyperbolic function solution trigonometric function solution and rational solution with parameters are obtained when the parameters take up special values the solitary waves are also derived from the travelling wave solutions the g g expansion method presents a wider applicability for handling nonlinear wave equations

new application of the g g expansion method to higher order nonlinear equations in this paper we extend the g g expansion method to higher order nonlinear equations as a result a series of new travelling wave solutions of the modified kawahara equation and the fifth order kdv equation have been obtained the travelling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions it is shown that the g g expansion method presents a wider applicability for handling nonlinear evolution equations

effects of quadratic and cubic nonlinearities on a perfectly tuned parametric amplifier abstract we consider the performance of a parametric amplifier with perfect tuning two to one ratio between the parametric and direct excitation frequencies and quadratic and cubic nonlinearities a forced duffing mathieu equation with appended quadratic nonlinearity is considered as the model system and approximate analytical steady state solutions and corresponding stabilities are obtained by the method of varying amplitudes some general effects of pure quadratic and mixed quadratic and cubic nonlinearities on parametric amplification are shown in particular the effects of mixed quadratic and cubic nonlinearities may generate additional amplitude frequency solutions in this case an increased response and a more phase sensitive amplitude phase between excitation frequencies is obtained as compared to the case with either pure quadratic or cubic nonlinearity furthermore jumps and bi stability in the amplitude phase characteristics are predicted supporting previously reported experimental observations

the tanh coth method for new compactons and solitons solutions for the k n n and the k n 1 n 1 equations abstract the nonlinear k n n and the k n 1 n 1 equations are analytically studied the tanh coth method the rational hyperbolic functions method and the rational exponential function method are applied for this analytic study we show that these equations exhibit new compactons solitons and periodic solutions the change of the physical structure of the solutions is caused by variation of the exponents and the coefficients of the derivatives

the g g expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics i the present paper we construct the traveling wave solutions involving parameters of the combined korteweg de vries modified korteweg de vries equation the reaction diffusion equation the compound kdv burgers equation and the generalized shallow water wave equation by using a new approach namely the g g expansion method where g g satisfies a second order linear ordinary differential equation when the parameters take special values the solitary waves are derived from the traveling waves the traveling wave solutions are expressed by the hyperbolic functions the trigonometric functions and the rational functions

a new approach to the analysis of oscillations of one dimensional spatially periodic structures abstract a new approach to the analysis of oscillations of spatially periodic structures which is based on the method of direct separation of motions mdsm is proposed in the paper as an example of the approach application the study of oscillations of a string with variable cross section is conducted as the result analytical expressions for the eigenmodes and the eigenfrequencies of the system are obtained it is shown that modulation of the string cross section leads to a change of the eigenfrequencies as compared with their non modulated values and to the emergence of a spectrum of additional high eigenfrequencies which correspond to large wave lengths it is noted that application of the multiple scales method and other classical asymptotic methods for solving the considered equations is rather cumbersome and leads to erroneous results because an unknown parameter is present in them the fundamental difference between the mdsm and these methods is revealed in particular it is shown that the mdsm may be employed for solving equations without small parameter thereby the applicability range of this method turns out to be broader than the one of the classical asymptotic methods to illustrate the validity of this statement a classical problem about the stability of a pendulum with vibrating suspension axis is considered

exact traveling solutions of some nonlinear evolution equation by g g expansion method in this paper we establish exact solutions for some nonlinear evolution equations the g g expansion method was used to construct traveling wave solutions of ramani s equations in this method we take the advantages of the general solutions of the second order lode to solve effectively many nonlinear evolution equations the g g expansion method presents a wider applicability for handling nonlinear wave equations

the tanh coth and the sine cosine methods for kinks solitons and periodic solutions for the pochhammer chree equations abstract in this paper the tanh coth method and the sine cosine method are used for a reliable treatment of the pochhammer chree equations which include a variety of models travelling wave solutions of distinct structures namely bell shaped solitons kink shaped solitons and periodic solutions are formally derived the two methods demonstrate power reliability and efficiency

analytic study on two nonlinear evolution equations by using the g g expansion method abstract the validity and reliability of the so called g g expansion method is tested by applying it to two nonlinear evolutionary equations solutions in more general forms are obtained when the parameters are taken as special values it is observed that the previously known solutions can be recovered new rational function solutions are also presented being concise and less restrictive the method can be applied to many nonlinear partial differential equations