two dimensional q differential transformation and its application the one dimensional q differential transformation was introduced in 8 for solving the ordinary q differential equations here we present the definition and operation of the two dimensional q differential transform a distinctive feature of the q differential transform is its ability to solve linear and nonlinear partial q differential equations

initial value problem of the linearized benjamin ono equation and its applications we consider the initial value problem of the benjamin ono bo equation linearized about the n soliton solution by establishing the completeness relation for the eigenfunctions of the linearized bo equation we construct the explicit solution to this problem as an application of the above result we investigate the linear stability of the n soliton solution we show that the wave under consideration is stable against infinitesimal perturbations thus we have a direct multisoliton perturbation theory for the bo equation without recourse to the inverse scattering transform in particular we can handle the first order solution beyond the adiabatic approximation the completeness relation established here enables us to give a general scheme for evaluating the first order correction to the leading order n soliton solution we also demonstrate that the first order solution satisfies an infinite set of conservation laws modified by the perturbations finally in the one soliton case we perform explicit calcu

new algorithms for taylor coefficients of indefinite integrals and their applications new recurrence algorithms for calculating the taylor coefficients of indefinite integral are introduced and proved in this paper these algorithms are simple in form only involve the analytic operations addition and multiplication and can be directly implemented in any symbolic language also no specific restriction on the kernel is required several types of volterra equations such as integral integro differential and system of integro differential equations with nondegenerate kernel are then solved to demonstrate the accuracy and efficiency of the proposed scheme

sobolev interpolation inequalities with weights we study weighted local sobolev interpolation inequalities of the form 1 r lu x i ph ui 2 x dx s 1 r lu x ip ii x dx h 1 1172 b j bl b j b x ibi p n r ly u x ipw l x dx 1 r lu x ipu x dx wl b jb v b jb where i i b is a ball in r n and 11 111 1 and 1112 are weight functions the case p 2 is of special importance in deriv ing regularity results for solutions of degenerate parabolic equations we also study the analogous inequality without the second summand on the right in the case u has compact support in b and we derive global landau inequalities iiy uiilq s cllull ally 2ull p 0 a i i p s q 00 when u has ii compact support

periodic solutions of the intermediate long wave equation a nonlinear superposition principle periodic stationary wave solutions of the intermediate long wave ilw equation are derived using the bilinear transformation method and a new expression for the dispersion relation is obtained the class of physically important real valued solutions is identified these solutions may be represented as an infinite superposition of solitary wave profiles a property shared by the related korteweg de vries kdv and benjamin ono bo equation this nonlinear superposition principle which has been the subject of various interpretations in the literature is discussed the ilw periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively for intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave in the shallow water kdv limit the ilw periodic solution leads to the familiar cnoidal wave whereas the deep water bo limit yields benjamin s periodic wave a previously unknown expression for the cnoidal wave dispersion relation in terms of theta functions is obtained

nonlinear evolution equations associated with a riemann hilbert scattering problem in an earlier paper nonlinear evolution equations associated with a riemann hilbert scattering problem which reduces in an appropriate limit to the zakharov shabat akns scattering problem were considered here we discuss certain necessary constraints associated with the scattering problem and their impact upon the associated evolution equations moreover the direct linearization of the nonlinear evolution equations and an algorithm to construct ann soliton solution are given

on the initial value problem for the modified benjamin ono equation the initial value problem for the modified benjamin ono equation is solved by exploiting the fokas ablowitz inverse scattering transform for a certain complex extension of the benjamin ono equation the asymptotics of the jost functions for small positive values of the spectral parameter are sharpened and lead to a previously overlooked nongeneric case

on the two dimensional differential transform method in this paper two dimensional differential transform method of solution of the initial value problem for partial differential equations pdes have been studied new theorems have been added and some linear and nonlinear pdes solved by using this method the method can be easily applied to linear or nonlinear problems and is capable of reducing the size of computational work in this work additionally analytical form solutions of two diffusion problems have been obtained and the solutions are compared very well with those obtained by decomposition method

periodic ilw equation with discrete laplacian we study an integro differential equation which generalizes the periodic intermediate long wave ilw equation the kernel of the singular integral involved is an elliptic function written as a second order difference of the weierstrass function using sato s formulation we show the integrability and construct some special solutions an elliptic solution is also obtained we present a conjecture based on a poisson structure that gives an alternative description of this integrable hierarchy we note that this poisson algebra in turn is related to a quantum algebra related to the family of macdonald difference operators

exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method the authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations the physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system the emphasis is however on the mathematical aspects i e the formal procedure necessary to find single solitary wave solutions by means of examples it is shown how various cases of pulse type and kink type solutions are to be obtained by this method an exhaustive list of equations so treated is presented in tabular form together with the particular intermediate relations necessary for deriving their solutions the extension of the technique to construct n soliton solutions and indicate connections with other existing methods is outlined

linear stability analysis for the multikink solution of the sine hilbert equation the linear stability analysis is performed for the multikink solution of the sine hilbert sh equation we first solve the eigenvalue problem of the sh equation linearized about the multikink solution and then establish the completeness relation among the eigenfunctions this relation enables us to solve the initial value problem of the linearized sh equation proving the stability of the solution against infinitesimal perturbations as an application of the completeness relation we develop a direct multikink perturbation theory while employing the method of multiple time scales we derive the evolution equations that describe the slow change of the kink parameters as well as a nonsecular expression for the first order correction to the multikink solution a novelty of the present analysis is that the proof of the completeness relation is done by purely algebraic means with the help of the classical theory of algebraic equations

approximate analytical solutions of fractional gas dynamic equations in this article differential transform method dtm has been successfully applied to obtain the approximate analytical solutions of the nonlinear homogeneous and non homogeneous gas dynamic equations shock wave equation and shallow water equations with fractional order time derivatives the true beauty of the article is manifested in its emphatic application of caputo fractional order time derivative on the classical equations with the achievement of the highly accurate solutions by the known series solutions and even for more complicated nonlinear fractional partial differential equations pdes the method is really capable of reducing the size of the computational work besides being effective and convenient for solving fractional nonlinear equations numerical results for different particular cases of the equations are depicted through graphs

final state problem for class of nonlinear nonlocal dispersive equation abstract we study the large time asymptotics of the solutions to nonlinear dispersive equations of the form 0 1 t u p i x x u x u 3 t 0 x r where p i x is a pseudo differential operator defined by the symbol p in this paper we consider the case that p is nonlocal and nonhomogeneous and show the existence of the solution to 0 1 around a given modified free solution

the interacton equation given the lie algebra lfp of vector fields of a free particle then in a joint field of interacting particles an interacton is defined as a direct summand isomorphic to lfp for several soliton equations it is shown how interactons can be obtained from various methods of symmetry analysis and how their dynamics can be expressed in terms of self interaction alone this procedure leads to new integrable systems furthermore in case of nonlinear schrodinger equations it is shown how the dynamics of interactons can be formulated in terms of self consistent potentials with antlinear parts

solutions of the system of differential equations by differential transform method in this study three dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time moreover as an application of two and three dimensional differential transform exact solutions of linear and non linear systems of partial differential equations have been investigated the results of the present method are compared very well with those obtained by decomposition method differential transform method can easily be applied to linear or non linear problems and reduces the size of computational work with this method exact solutions may be obtained without any need of cumbersome work and it is an useful tool for analytical and numerical solutions

status of the differential transformation method abstract further to a recent controversy on whether the differential transformation method dtm for solving a differential equation is purely and solely the traditional taylor series method it is emphasized that the dtm is currently used often only as a technique for analytically calculating the power series of the solution in terms of the initial value parameters sometimes a piecewise analytic continuation process is implemented either in a numerical routine e g within a shooting method or in a semi analytical procedure e g to solve a boundary value problem emphasized also is the fact that at the time of its invention the currently used basic ingredients of the dtm that transform a differential equation into a difference equation of same order that is iteratively solvable were already known for a long time by the traditional taylor method users notably in the elaboration of software packages numerical routines for automatically solving ordinary differential equations at now the defenders of the dtm still ignore the though much better developed studies of the traditional taylor method users who in turn seem to ignore similarly the existence of the dtm the dtm has been given an apparent strong formalization set on the same footing as the fourier laplace or mellin transformations though often used trivially it is easily attainable and easily adaptable to different kinds of differentiation procedures that has made it very attractive hence applications to various problems of the taylor method and more generally of the power series method including noninteger powers has been sketched it seems that its potential has not been exploited as it could be after a discussion on the reasons of the misunderstandings which have caused the controversy the preceding topics are concretely illustrated it is concluded that for the sake of clarity when the dtm is applied to odes it should be mentioned that the dtm exactly coincides with the traditional taylor method contrary to what is currently written

eigenfunctions and eigenvalues for a scalar riemann hilbert problem associated to inverse scattering a complete set of eigenfunctions is introduced within the riemann hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations in particular we consider the time independent and time dependent schrodinger problems which are related to the kdv and kpi equations possessing solitons and lumps respectively non standard scalar products orthogonality and completeness relations are derived for these problems the complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations we classify two different types of bifurcations of new eigenvalues and analyze their characteristic features one type corresponds to thresholdless generation of solitons in the kdv equation while the other predicts a threshold for generation of lumps in the kpi equation

the non classical solution of the inhomogeneous non linear diffusion equation a complete classification of the symmetry reductions of the inhomogeneous nonlinear diffusion inld equation using the non classical method leads to one family of symmetry generators the resulting ordinary differential equation is of second order and non linear new solutions are obtained for some physical processes corresponding to particular values of n and p m

singularity confinement analysis of integrodifferential equations of benjamin ono type we analyse the integrability of various types of intregrodifferential equations belonging to the benjamin ono family the method used is a combination of two well known integrability detectors the painleve method for continuous systems and the singularity confinement for discrete systems we confirm the results of hietarinta based on the study of multisoliton solutions in particular we show that the third order extension to benjamin ono that he proposed does pass the test and is thus an excellent candidate for integrability

dynamics of solitons in a damped sine hilbert equation a damped sine hilbert sh equation is proposed it can be linearized by a dependent variable transformation which enables one to solve an initial value problem of the equation the n soliton solution is obtained explicitly and its properties are investigated in comparison with those of the n soliton solution of the sh equation in particular the interaction of the two solitons is explored in detail with the aid of the pole representation it is found that the interaction process is classified into the two types according to the initial amplitudes and positions of both solitons in the general n soliton case the long time behavior of the solution is shown to be characterized by the positive n zeros of the hermite polynomial of degree 2n finally a linearized version of the damped sh equation is briefly discussed

weighted poincare and sobolev inequalities and estimates for weighted peano maximal functions 1 introduction in this paper we derive local poincare and sobolev inequalities involving two weight functions the estimates we obtain include those proved by fabes kenig and serapioni in 8 for the one weight function case we also consider the notion of weighted peano derivatives and show that the peano derivative formed with a given weight function can be estimated in terms of a maximal function involving a second weight function in the special case of ordinary lebesgue measure results of this kind lead to the theorems of a p calderon and a zygmund about local differentiability of functions in sobolev spaces and of functions which are of bounded variation in the sense of tonelli see 3 and 4 all our results are based on variants of the basic estimate in 5 but we do not presuppose any facts from 5 let v x be a weight function on r i e let v x be nonnegative and locally integrable with respect to lebesgue measure we shall use the notation v e se v x dx ordinary lebesgue measure will be denoted ie l if bh x is the open ball with center x and radius h we say v satisfies the doubling condition with respect to lebesgue measure if

on the initial value problem for a class of nonlinear integral evolution equations including the sine hilbert equation a method for solving a class of nonlinear singular integral evolution equations for decaying initial values on the line is presented the underlying scattering problem is a matrix riemann hilbert problem scattering analysis shows that the spectrum is purely discrete an application is to the so called sine hilbert equation h t c sin where c is a constant and h denotes the hilbert transform

multidimensional nonlinear evolution equations and inverse scattering abstract in this paper we will review some recent work done in the field of integrable nonlinear evolution equations and inverse scattering we will concentrate on the basic underlying areas and refer interested readers to suitable references for complete details specifically background material can be found in various texts on this subject e g 1 by ablowitz and segur more recent references will be given as necessary the outline of the paper is as follows 1 1 introductory remarks 2 2 a discussion of two separate but related issues namely a solving certain nonlinear evolutions equations in infinite space and b inverse scattering these are important problems having many physical applications moreover they are related to each other by what we refer to as the inverse scattering transform ist 3 3 at the end of the paper we will make some remarks on the possibility of solving nonlinear evolution equations in high dimensions i e equations with more than two spatial and one time variable by using the ist method as we now understand it

an application of the inverse scattering transform to the modified intermediate long wave equation the modified intermediate long wave milw equation is a 1 1 dimensional nonlinear singular integro differential equation that possesses soliton solutions in an appropriate limit the milw equation reduces to the well known modified korteweg de vries equation in this paper we solve the initial value problem for the milw equation through a suitable implementation of the inverse scattering transform and use of the miura type transformation that maps solutions of the milw equation into solutions of a complexified version of the standard intermediate long wave ilw equation the initial value used for the milw equation is assumed to be real valued sufficiently smooth and decaying to zero as the absolute value of the spatial variable approaches large values an interesting feature of the procedure we develop is that soliton solutions for the ilw and milw equations can be derived by appropriate specializations of a master set of equations

the scattering transform for the benjamin ono equation in the small dispersion limit abstract using exact formulae for the scattering data of the benjamin ono equation valid for general rational potentials recently obtained in miller and wetzel 17 we rigorously analyze the scattering data in the small dispersion limit in particular we deduce precise asymptotic formulae for the reflection coefficient the location of the eigenvalues and their density and the asymptotic dependence of the phase constant associated with each eigenvalue on the eigenvalue itself our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering and they also provide new details not previously reported in the literature

a new zk ilw equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism abstract the research of algebraic gravity solitary waves is an advanced field which has important practical and theoretical value in physical oceanography aerology and etc by calculation condition and theoretical method limit previous researches mainly focused on the 1 1 dimensional models and 2 1 dimensional models were few considered in this paper from the non static equilibrium equation a new zk ilw equation is derived by using multi scale analysis and perturbation method in finite depth stratified atmosphere which is the first time obtained the model can reduce to zk bo model h and zk model h 0 and is the generalization of the above two models in order to further understand the nature of algebraic gravity solitary waves we get the analytical solution of zk ilw equation by using the trial function method and discuss the conservation laws furthermore the fission process of algebraic gravity solitary waves is studied and we can judge that one of the possible formation mechanism of squall lines is the fission of algebraic gravity solitary waves

a spectral transform for the intermediate nonlinear schrodinger equation a new spectral transform system is introduced to solve the initial value problem for the intermediate nonlinear schrodinger inls equation describing envelope waves in a deep stratified fluid the spectral system is a combination of the zakharov shabat linear system and a local riemann hilbert problem in a strip of the complex plane the inverse scattering transform technique is developed and the backlund darboux transformation soliton solutions and an infinite number of conservation laws are constructed it is shown that all these properties of the inls equation are closely related to those of the intermediate long wave equation

differential transform method for the system of two dimensional nonlinear volterra integro differential equations in this paper we develop the differential transform method dtm for solving a class of the system of two dimensional linear and nonlinear volterra integro differential equations of the second kind to this end we give some preliminary results of the differential transform and describe the method of this paper we also give some examples to demonstrate the accuracy of the presented method

direct and inverse scattering problems of the nonlinear intermediate long wave equation the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed the equation describes long internal gravity waves in a stratified fluid of finite depth and reduces to the korteweg de vries equation as shallow water limit and the benjamin ono equation as deep water limit both limits of the method and novel aspects of the theory are also discussed

uniqueness properties of solutions to the benjamin ono equation and related models abstract we prove that if u 1 u 2 are real solutions of the benjamin ono equation defined in x t r 0 t which agree in an open set r 0 t then u 1 u 2 we extend this uniqueness result to a general class of equations of benjamin ono type in both the initial value problem and the initial periodic boundary value problem this class of 1 dimensional non local models includes the intermediate long wave equation we relate our uniqueness results with those for a water wave problem finally we present a slightly stronger version of our uniqueness results for the benjamin ono equation

backlund transformation conservation laws and inverse scattering transform of a model integrodifferential equation for water waves the backlund transformation bt an infinite number of conservation laws and the inverse scattering transform ist of a model integrodifferential equation for water waves in fluids of finite depth y matsuno j math phys 29 49 1989 are constructed by employing the bilinear transformation method the model equation is also shown to pass the painleve test these facts prove the complete integrability of the equation both the deep and shallow water limits of various results thus obtained are then investigated in detail in addition a new method to evaluate conserved quantities for pure n soliton is developed by utilizing actively the time part of the bt it is found that the structure of conservation laws exhibits peculiar characteristics in comparison with those of usual water wave equations such as the benjamin ono and the korteweg de vries equations the most important problem left open in this paper is to solve various ist equations

self improving properties of inequalities of poincare type on measure spaces and applications abstract we show that the self improving nature of poincare estimates persists for domains in rather general measure spaces we consider both weak type and strong type inequalities extending techniques of b franchi c perez and r wheeden as an application in spaces of homogeneous type we derive global poincare estimates for a class of domains with rough boundaries that we call john domains and we show that such domains have the requisite properties this class includes john or boman domains as well as s john domains further applications appear in a companion paper

numeric analytic solutions of mixed type systems of balance laws the aim of the present analysis is to apply two relatively recent methods reduced differential transform method rdtm and differential transform method dtm for the solution of balance law systems new generalized transformed formulas are derived the new approaches provided the solution in the form of a rapidly convergent series with easily computable components in the rdtm case and costly components for the dtm a comparison between the two methodologies shows that the rdtm is more effective efficient powerful and able to be applicable for large class of nonlinear partial differential equations than the dtm two test modeling problems are discussed to illustrate the effectiveness and performance of rdtm

on the limit from the intermediate long wave equation to the benjamin ono equation the intermediate long wave equation is a physically important singular integrodifferential equation containing a parameter referred to here as for it reduces to the benjamin ono equation it has been recently shown that the inverse scattering transform schemes of the above equations have certain significant differences here it is shown that for the inverse scattering transform scheme of the intermediate long wave equation reduces to that of the benjamin ono equation

a new homotopy perturbation method for solving systems of partial differential equations abstract in this paper a new homotopy perturbation method nhpm is introduced for obtaining solutions of systems of non linear partial differential equations theoretical considerations are discussed to illustrate the capability and reliability of the method three examples are provided comparison of the results of applying nhpm with those of applying hpm reveal the effectiveness and convenience of the new technique

a numerical solution of second order linear partial differential equations by differential transform two dimensional differential transform provides an effective tool to numerically solve partial differential equations in this paper we propose a general numerical solution of second order linear partial differential equations using two dimensional differential transform numerical examples justify the proposed scheme

multisoliton perturbation theory for the benjamin ono equation and its application to real physical systems a direct perturbation theory is developed to study the effects of small perturbations on the interaction process of algebraic solitons of the benjamin ono bo equation using the method of multiple scales the modulation equations for the amplitude and the phase of each soliton are derived in the lowest approximation as practical applications of the theory the interaction of two solitons is investigated for the two different types of perturbations that appear in real physical systems one is a dissipative perturbation bo burgers equation and the other is a dispersive perturbation higher order bo equation in both cases the changes of the soliton parameters due to small perturbation are calculated by numerical integrations and their characteristics are elucidated in detail among them the phase shift caused by the dispersive perturbation is a remarkable feature that has never been observed in the collision process of algebraic solitons

weighted inequalities on john domains we study two weighted inequalities on john domains we first introduce domains that generalize the c m domains defined by hajlasz and koskela j london math soc to appear and then show that our domains are actually just john domains we then extend an interesting and nice result of hajlasz and koskela by modifying their proof and then obtain some interesting consequences that include weighted sobolev interpolation inequalities and an almost necessary and sufficient condition for two weighted poincare inequality on cubes

kinks of the sine hilbert equation and their dynamical motions an exact method for constructing exact solutions of the sine hilbert equation is developed it is shown that the sh equation can be transformed into a system of non linear ordinary differential equations through a dependent variable transformation furthermore this system of equations is transformed into a system of decoupled linear ordinary differential equations exact solutions of the sh equation are then constructed by means of a simple integration the dynamical properties of the solutions are also studied in detail the solution is interpreted as a superposition of n pulses which may be termed kinks where n is a positive integer asymptotically for large time it is composed of a pulse with a constant velocity and n 1 static pulses this is a novel characteristic of the solution unlike the well known n soliton solution which behaves like n moving pulses for large time

escape to infinity under the action of a potential and a constant electromagnetic field escape to infinity is proved for a great variety of potentials including the potential created by an infinite number of sources relativistic escape is studied escape in the presence of a constant electromagnetic field and a potential is also considered

stability of nonlinear periodic internal waves in a deep stratified fluid stability of the nonlinear periodic solutions of the benjamin ono equation with respect to one dimensional perturbations and taking into account transverse modulations is analytically studied the full set of eigenfrequencies and of small perturbations is found in the one dimensional case the dispersion equation for long wave oblique perturbations is obtained these perturbations are shown to be unstable under either sign of the dispersion of the linear dispersion relation the instability limits and the maximum growth rates are determined in the extreme cases of strongly and weakly nonlinear waves

comments on solving a class of two dimensional linear and nonlinear volterra integral equations by the differential transform method tari et al a tari m y rahimi s shahmorad f talati solving a class of two dimensional linear and nonlinear volterra integral equations by the differential transform method j comput appl math 228 2009 70 76 presented some fundamental properties of tdtm for the kernel functions in two dimensional volterra integral equations here we suggest simple proofs of those fundamental properties by using the basic properties of tdtm furthermore we present some fundamental properties of tdtm for the kernel functions of a quotient type in two dimensional volterra integral equations numerical illustrations are demonstrated to show the effectiveness of the tdtm for solving two dimensional volterra integral equations

a new method for a generalized hirota satsuma coupled kdv equation abstract in this paper differential transform method dtm which is one of the approximate methods is implemented for solving the nonlinear hirota satsuma coupled kdv partial differential equation a variety of initial value system is considered and the convergence of the method as applied to the hirota satsuma coupled kdv equation is illustrated numerically the obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient numerical examples are illustrated the pertinent features of the proposed algorithm

a numerical approach for solving generalized abel type nonlinear differential equations in this paper a numerical power series algorithm which is based on the improved taylor matrix method is introduced for the approximate solution of abel type differential equations and also riccati differential equations the technique is defined and illustrated with some numerical examples the obtained results reveal that the method is very effective simple and valid high accuracy the method can be easily extended to other nonlinear equations

solving partial differential equations by two dimensional differential transform method using two dimensional differential transform to solve partial differential equations pde is proposed in this study first the theory of two dimensional differential transform is introduced second taking two dimensional differential transform of a pde problem a set of difference equations is derived doing some simple mathematical operations on these equations we can get a closed form series solution or an approximate solution quickly finally three pde problems with constant and variable coefficients are solved by the present method the calculated results are compared very well with those obtained by other analytical or approximate methods

generalized intermediate long wave hierarchy in zero curvature representation with noncommutative spectral parameter the simplest generalization of the intermediate long wave hierarchy ilw is considered to show how to extend the zakharov shabat dressing method to nonlocal i e integro partial differential equations the purpose is to give a procedure of constructing the zero curvature representation of this class of equations this result obtains by combining the drinfeld sokolov formalism together with the introduction of an operator valued spectral parameter namely a spectral parameter that does not commute with the space variable x this extension provides a connection between the ilwk hierarchy and the saveliev vershik continuum graded lie algebras in the case of ilw2 the fairlie zachos sinh algebra was found

oscillatory finite time singularities in finance population and rupture abstract we present a simple two dimensional dynamical system where two nonlinear terms exerting respectively positive feedback and reversal compete to create a singularity in finite time decorated by accelerating oscillations the power law singularity results from the increasing growth rate the oscillations result from the restoring mechanism as a function of the order of the nonlinearity of the growth rate and of the restoring term a rich variety of behavior is documented analytically and numerically the dynamical behavior is traced back fundamentally to the self similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin the interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log periodic oscillations with remarkable scaling properties three domains of applications are discussed 1 the stock market with a competition between nonlinear trend followers and nonlinear value investors 2 the world human population with a competition between a population dependent growth rate and a nonlinear dependence on a finite carrying capacity 3 the failure of a material subjected to a time varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

exact solutions to nonlinear diffusion equations obtained by the decomposition method in this paper we develop a framework to obtain exact solutions to the nonlinear diffusion equations by employing the adomian decomposition method exact solutions are obtained for some important physical processes of power law diffusivities the method is capable of greatly reducing the size of computational domain and is presented in a general way so that it can be used in more diffusion classes

pulse formation in a dissipative nonlinear system a nonlocal nonlinear evolution equation is proposed that describes pulse formation in a dissipative system a novel feature of the equation is that it can be solved exactly through a linearization procedure the solutions are constructed under appropriate initial and boundary conditions and their properties are investigated in detail of particular interest is pulse formation which is caused by a balance between nonlinearity and dissipation the asymptotic behavior of the solution for large time is then represented by a train of moving pulses with equal amplitudes the corresponding position of each pulse is shown to be characterized by the zero of the hermite polynomial irrespective of initial conditions

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

linearization of novel nonlinear diffusion equations with the hilbert kernel and their exact solutions two novel nonlinear diffusion equations with the hilbert kernel are proposed the equations can be linearized by introducing appropriate dependent variable transformations the initial value problems for the proposed equations are then solved exactly through the linearization and explicit nonperiodic and periodic solutions are constructed the properties of solutions are investigated in detail it is found that the blow up of solutions occurs at a finite time for both the nonperiodic and periodic cases due to the breakdown of certain analytic conditions imposed on the dependent variables

two dimensional differential transform for partial differential equations the differential transform is a numerical method for solving differential equations in this paper we present the definition and operation of the two dimensional differential transform a distinctive feature of the differential transform is its ability to solve linear and nonlinear differential equations partial differential equation of parabolic hyperbolic elliptic and nonlinear types can be solved by the differential transform we demonstrate that the differential transform is a feasible tool for obtaining the analytic form solutions of linear and nonlinear partial differential equation

bifurcations of new eigenvalues for the benjamin ono equation a criterion for the emergence of new eigenvalues is found for the linear scattering problem associated with the benjamin ono bo equation this bifurcation occurs due to perturbations of nongeneric potentials which include the soliton solutions of the bo equation the asymptotic approximation of an exponentially small new eigenvalue is derived the method is based on the expansion of a localized function through a complete set of unperturbed eigenfunctions explicit expressions are obtained for the soliton potentials

new integrable nonlinear integrodifferential equations and related solvable finite dimensional dynamical systems a new integrable nonlinear integrodifferential equation nide is proposed this equation may be interpreted as a model equation for deep water waves the n periodic and n soliton solutions for the equation are constructed by means of the bilinear transformation method these solutions have the same structure as that for the benjamin ono equation which describes internal waves in stratified fluids of great depth furthermore it is shown that the motion of the positions of the poles of solutions is related to certain solvable finite dimensional dynamical systems described by first order nonlinear ordinary differential equations the discussion is also made on a more general nide that may be interpreted as a model equation describing nonlinear waves in fluids of finite depth

new classes of similarity solutions of the inhomogeneous nonlinear diffusion equations the lie similarity method has been used to extend the similarity solutions of the one dimensional inhomogeneous nonlinear diffusion equations we determine the lie point symmetry vector fields and calculate the similarity ansatz then we discuss the resulting nonlinear ordinary differential equations exact solutions are found and their relations to some real physical situations are discussed

on genus two solutions for the ilw equation the existence of theta function solutions of genus two for the intermediate long wave equation is established a numerical example is also presented the method basically goes along with krichever s construction of theta function solutions for soliton equations such as the kronecker product equation this idea leads us to a question whether a riemann surface exists which allows a peculiar abelian integral of the third kind the answer is affirmative at least for genus two curves

a new algorithm for calculating two dimensional differential transform of nonlinear functions a new algorithm for calculating the two dimensional differential transform of nonlinear functions is developed in this paper this new technique is illustrated by studying suitable forms of nonlinearity three strongly nonlinear partial differential equations are then solved by differential transform method to demonstrate the validity and applicability of the proposed algorithm the present framework offers a computationally easier approach to compute the transformed function for all forms of nonlinearity this gives the technique much wider applicability

darboux transformations infinitesimal symmetries and conservation laws for the nonlocal two dimensional toda lattice the technique of darboux transformation is applied to the nonlocal partner of the two dimensional periodic an 1 toda lattice this system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear systems with a quantized spectral parameter we give the generalization of the darboux transformation technique on linear equations of this type we present the connections between the solutions of overdetermined linear systems and their expansions in series in the neighbourhood of singular points the solutions of the nonlocal toda lattice and infinite hierarchies of the infinitesimal symmetries and conservation laws are obtained

on the zero dispersion limit of the benjamin ono cauchy problem for positive initial data we study the cauchy initial value problem for the benjamin ono equation in the zero dispersion limit and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by lax and levermore to analyze the corresponding limit for the korteweg de vries equation 2010 wiley periodicals inc

two dimensional dynamical system associated with abel s nonlinear differential equation a two dimensional dynamical system is proposed that is described by a pair of nonlinear ordinary differential equations odes with a complex parameter it reduces to abel s nonlinear ode of the first kind by an appropriate transformation using this fact the properties of solutions are investigated in detail with the aid of numerical computations it is found that various types of bifurcation phenomena occur depending on the values of the parameter in particular the solution is shown to blow up in finite time under certain conditions in order to visualize the behaviors of dynamical motions the trajectories of solutions are depicted in the plane finally a discussion is made on some generalizations of the proposed system