method of lines solution of the korteweg de vries equation abstract the korteweg de vries equation kdve is a classical nonlinear partial differential equation pde originally formulated to model shallow water flow in addition to the applications in hydrodynamics the kdve has been studied to elucidate interesting mathematical properties in particular the kdve balances front sharpening and dispersion to produce solitons i e traveling waves that do not change shape or speed in this paper we compute a solution of the kdve by the method of lines mol and compare this numerical solution with the analytical solution of the kdve in a second numerical solution we demonstrate how solitons of the kdve traveling at different velocities can merge and emerge the numerical procedure described in the paper demonstrates the ease with which the mol can be applied to the solution of pdes using established numerical approximations implemented in library routines

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

analytical and numerical studies of weakly nonlocal solitary waves of the rotation modified korteweg de vries equation abstract a century ago the korteweg de vries kdv equation was derived as a model for weakly nonlinear long waves propagating down a channel when cross channel and depth variations are sufficiently weak in this article we study the steadily translating coherent structures of a generalization of this equation the rotation modified kdv equation which applies when coriolis forces are significant u t uu x u xxx x 2 u 0 where x is the down channel coordinate this is also called the ostrovosky equation the rmkdv solitary waves are weakly nonlocal due to the radiation of long waves where the radiation is proportional to the small parameter because of its simplicity the rmkdv coherent structures are the prototype of nonlocal solitary waves where the amplitude is a power of the small parameter micropterons we extend the matched asymptotic expansions of hunter to third order we also compute direct numerical solutions and bifurcations of the nonlinear eigenvalue problem new modes with multiple kdv like peaks flanked by small quasi sinusoidal oscillations are shown to be well described by matched asymptotics too

pattern storage bifurcations and groupwise correlation structure of an exactly solvable asymmetric neural network model despite their biological plausibility neural network models with asymmetric weights are rarely solved analytically and closed form solutions are available only in some limiting cases or in some m

discontinuous fractional sturm liouville problems with transmission conditions abstract the main purpose of this study is the investigation of discontinuous sturm liouville problem with fractional derivatives we give an operator theoretic framework of the problem under consideration namely we define an operator a in the hilbert space l 2 1 1 the eigenvalues and corresponding eigenfunctions of which coincide with the eigenvalues and corresponding eigenfunctions of the boundary value problem respectively then we establish the characteristic function and prove that the eigenvalues of the considered problem coincide with the roots of this characteristic function

compactons and solitary patterns structures for variants of the kdv and the kp equations in this paper we discuss mathematical variants in higher dimensions of the kdv and the kp equations it is shown that the focusing branches of these variants exhibit compactons solitons with finite wavelength whereas the defocusing branches support solitary patterns solutions with infinite slopes or cusps the study presents a fairly complete understanding of the compact and noncompact dispersive structures

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

a repository of equations with cosine sine compactons nonlinear evolution equations with cosine sine compacton solutions are reviewed including the rosenau hyman equation and generalizations of korteweg de vries camassa holm boussinesq benjamin bona mahony klein gordon and other equations each equation is generalized to three dimensions and the conditions for its cosine solitary waves to be either a compacton or a soliton are determined several equations claimed in the literature to be different among them are found to be equivalent

the tanh method for generalized forms of nonlinear heat conduction and burgers fisher equations generalized forms of the kdv the nonlinear heat conduction and burgers fisher equations are investigated the analysis rests mainly on the standard tanh method the focus is on the case where the parameter m is noninteger a variety of exact travelling wave solutions of distinct physical structures are formally derived

analytic solution of nonlinear boundary value problems in several dimensions by decomposition abstract nonlinear partial differential equations are systematically solved by the decomposition method of g adomian for general boundary conditions described by boundary operator equations linear partial differential equations and linear or nonlinear ordinary differential equations become special cases all dealt with by a single procedure

compactons solitons and periodic solutions for variants of the kdv and the kp equations studying solitons and solitons with compact support is of important significance in nonlinear phenomena in this paper we study nonlinear variants of the kdv and the kp equations with positive and negative exponents we employ the sine cosine algorithm to back up our analysis exact solutions with compactons solitons solitary patterns and periodic structures are obtained for these variants

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

travelling wave solutions of generalized forms of burgers burgers kdv and burgers huxley equations in this work exact travelling wave solutions of generalized forms of burgers burgers kdv and burgers huxley equations are obtained the analysis rests mainly on the standard tanh method the work emphasizes the need for a transformation formula for the case where the parameter m is non integer the approach can be used in a variety of many types of nonlinearity

analytical approximations and pade approximants for volterra s population model in this paper an analytic approximation for volterra s model for population growth of a species in a closed system is presented the nonlinear integro differential model includes an integral term that characterizes accumulated toxicity on the species in addition to the terms of the logistic equation the series solution method and the decomposition method are implemented independently to the model and to a related ode the pade approximants that often show superior performance over series approximations are effectively used in the analysis to capture the essential behavior of the population u t of identical individuals

new solitons and kinks solutions to the sharma tasso olver equation abstract in this work the sharma tasso olver sto equation is investigated the tanh method the extended tanh method and other ansatze involving hyperbolic and exponential functions are efficiently used for analytic study of this equation new solitons and kinks solutions are formally derived the proposed schemes are reliable and manageable

the strong maximum principle revisited in this paper we rst present the classical maximum principle due to e hopf together with an extended commentary and discussion of hopf s paper we emphasize the comparison technique invented by hopf to prove this principle which has since become a main mathematical tool for the study of second order elliptic partial dierential equations and has generated an enormous number of important applications while hopf s principle is generally understood to apply to linear equations it is in fact also crucial in nonlinear theories such as those under consideration here in particular we shall treat and discuss recent generalizations of the strong maximum principle and also the compact support principle for the case of singular quasilinear ellip tic dierential inequalities under generally weak assumptions on the quasilinear operators and the nonlinearities involved our principal interest is in necessary and sucient con ditions for the validity of both principles in exposing and simplifying earlier proofs of corresponding results and in extending the conclusions to wider classes of singular oper ators than previously considered the results have unexpected rami cations for other problems as will develop from the exposition e g i two point boundary value problems for singular quasilinear ordinary dierential equations sections 3 4 ii the exterior dirichlet boundary value problem section 5 iii the existence of dead cores and compact support solutions i e dead cores at in nity section 7 iv euler lagrange inequalities on a riemannian manifold section 9 v comparison and uniqueness theorems for solutions of singular quasilinear dierential inequalities section 10 the case of p regular elliptic inequalities is briesy considered in section 11

generalized boussinesq type of equations with compactons solitons and periodic solutions generalized boussinesq type of equations with positive and negative exponents are examined the analysis depends mainly on the sine cosine ansatz it is formally shown that these nonlinear models give rise to compactons solitary patterns solitons and periodic solutions depending on the exponents and the coefficients of the derivatives of u x t the presented scheme reveals quite a number of remarkable features that will be helpful for identical problems

multiple soliton solutions for the calogero bogoyavlenskii schiff jimbo miwa and ytsf equations abstract in this work we employ the hirota s bilinear method to derive multiple front solutions for nonlinear evolution equations three models namely the 2 1 dimensional calogero bogoyavlenskii schiff equation the 3 1 dimensional jimbo miwa equation and the 3 1 dimensional potential ytsf equation are used as vehicles to conduct the analysis the work shows the power of the bilinear method and the variety of the obtained front solutions

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

the tanh and the sine cosine methods for the complex modified k dv and the generalized k dv equations the complex modified k dv cmk dv equation and the generalized k dv equation are investigated by using the tanh method and the sine cosine method a variety of exact travelling wave solutions with compact and noncompact structures are formally obtained for each equation the study reveals the power of the two schemes where each method complements the other

a comparison of adomian s decomposition method and wavelet galerkin method for solving integro differential equations this paper aims to introduce a comparison of adomian decomposition method and wavelet galerkin method for the solution of integro differential equations from the computational viewpoint the comparison shows that the adomian decomposition method is efficient and easy to use

on the analytic solutions of the nonhomogeneous blasius problem in this article a totally analytic solution of the nonhomogeneous blasius problem is obtained using the homotopy analysis method ham this solution converges for 0 l c an analytical value of the critical value of l c was also obtained for the first time

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

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

a computational approach to soliton solutions of the kadomtsev petviashvili equation in this paper we present a computational approach to develop soliton solutions of the nonlinear kadomtsev petviashvili equation our approach rests mainly on the adomian decomposition method to include few components of the decomposition series the proposed framework is presented in a general way so that it can be used in nonlinear evolution equations of the same type numerical examples are tested to illustrate the proposed scheme

a new modification of the adomian decomposition method for linear and nonlinear operators in this paper we present an efficient modification of the adomian decomposition method that will facilitate the calculations we then conduct a comparative study between the new modification and the modified decomposition method the study is conducted through illustrative examples the new modification introduces a promising tool for many linear and nonlinear models

general solutions with solitary patterns for the defocusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this work the defocusing branch of the genuinely nonlinear dispersive k n n equation that exhibits solutions with solitary patterns is investigated the equation with constant coefficients to the derivatives is examined in one two and three dimensional spaces general formulas for solutions with solitary patterns are developed for all positive integers n n 1 two sets of entirely new formulas one set produces solutions for even integers n whereas the second set produces solutions in opposite directions for odd integers n n 1 are established

numerical solution of singularly perturbed fifth order two point boundary value problem we will consider adomain decomposition method and the homotopy method to solve a fifth order singularly perturbed bvp arising in viscoelastic flows the success and pitfalls of the methods will be investigated numerical testing will be provided to show the efficiency of the methods proposed comparison with the work of others will also be done

adomian decomposition method for approximating the solution of the korteweg devries equation in this paper the adomian decomposition method for solving the nonlinear korteweg devries equation is implemented with appropriate initial conditions we discuss the case when the problem has either one or more solitons some numerical examples are presented

new solitary wave and periodic wave solutions to the 2 1 dimensional nizhnik novikov veselov system in this work the 2 1 dimensional nizhnik novikov veselov nnv system is studied the tanh method the extended tanh method and the cosh sinh method are used to investigate this system by means of these methods new solitary wave and periodic wave solutions are determined

a comparison between adomian s decomposition methods and perturbation techniques for nonlinear random differential equations abstract comparison of the adomian decomposition method and regular perturbation techniques applied to the solution of nonlinear vector random differential equations shows that the decomposition method is superior and generally applicable even for the cases where perturbation is applicable the comparison shows that the decomposition method is easier to compute and supplies quantitatively reliable results

a reliable modification of adomian decomposition method in this paper we propose a powerful modification of adomian decomposition method that will accelerate the rapid convergence of the series solution the validity of the modified technique is verified through illustrative examples in all cases of differential and integral equations we obtained excellent performance from the modified algorithm the modification could lead to a promising approach for many applications in applied sciences

double phase problems with variable growth abstract we consider a class of double phase variational integrals driven by nonhomogeneous potentials we study the associated euler equation and we highlight the existence of two different rayleigh quotients one of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues the notion of eigenvalue is understood in the sense of pairs of nonlinear operators as introduced by fucik necas soucek and soucek the analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the p x laplace operator the generalized mean curvature operator or the capillarity differential operator with variable exponent the results contained in this paper complement the pioneering contributions of marcellini mingione et al in the field of variational integrals with unbalanced growth

an efficient algorithm for the multivariable adomian polynomials abstract in this article the sum of the series of multivariable adomian polynomials is demonstrated to be identical to a rearrangement of the multivariable taylor expansion of an analytic function of the decomposition series of solutions u 1 u 2 u m about the initial solution components u 1 0 u 2 0 u m 0 of course the multivariable adomian polynomials were developed and are eminently practical for the solution of coupled nonlinear differential equations the index matrices and their simplified forms of the multivariable adomian polynomials are introduced we obtain the recurrence relations for the simplified index matrices which provide a convenient algorithm for rapid generation of the multivariable adomian polynomials another alternative algorithm for term recurrence is established in these algorithms recurrence processes do not require complicated operations such as parametrization expanding and regrouping derivatives etc as practiced in prior art the mathematica program generating the adomian polynomials based on the algorithm in this article is designed

distinct variants of the kdv equation with compact and noncompact structures in this paper we examine three distinct variants of the generalized kdv equation both the compact and the noncompact structures are treated for these variants we also emphasize the different physical structures of the two classes the focusing branch and the defocusing branch

measuring and testing dependence by correlation of distances distance correlation is a new measure of dependence between random vectors distance covariance and distance correlation are analogous to product moment covariance and correlation but unlike the classical definition of correlation distance correlation is zero only if the random vectors are independent the empirical distance dependence measures are based on certain euclidean distances between sample elements rather than sample moments yet have a compact representation analogous to the classical covariance and correlation asymptotic properties and applications in testing independence are discussed implementation of the test and monte carlo results are also presented

an extended simplest equation method and its application to several forms of the fifth order kdv equation abstract in this paper an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations as applications many new exact travelling wave solutions for several forms of the fifth order kdv equation are obtained by using our method the forms include the lax sawada kotera sawada kotera parker dye caudrey dodd gibbon kaup kupershmidt kaup kupershmidt parker dye and the ito forms

stationary kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity this paper deals with the existence and the asymptotic behavior of non negative solutions for a class of stationary kirchhoff problems driven by a fractional integro differential operator lk and involving a critical nonlinearity in particular we consider the problem m u 2 lku f x u u 2s 2uin u 0in rn where rn is a bounded domain 2s is the critical exponent of the fractional sobolev space hs rn the function f is a subcritical term and is a positive parameter the main feature as well as the main difficulty of the analysis is the fact that the kirchhoff function m could be zero at zero that is the problem is degenerate the adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature

compacton solutions in a class of generalized fifth order korteweg de vries equations solitons play a fundamental role in the evolution of general initial data for quasilinear dispersive partial differential equations such as the korteweg de vries kdv nonlinear schrodinger and the kadomtsev petviashvili equations these integrable equations have linear dispersion and the solitons have infinite support we have derived and investigate a new kdv like hamiltonian partial differential equation from a four parameter lagrangian where the nonlinear dispersion gives rise to solitons with compact support compactons the new equation does not seem to be integrable and only mass momentum and energy seem to be conserved yet the solitons display almost the same modal decompositions and structural stability observed in integrable partial differential equations the compactons formed from arbitrary initial data are nonlinearly self stabilizing and maintain their coherence after multiple collisions the robustness of these compactons and the inapplicability of the inverse scattering tools that worked so well for the kdv equation make it clear that there is a fundamental mechanism underlying the processes beyond integrability we have found explicit formulas for multiple classes of compact traveling wave solutions when there are more than one compacton solution for a particular set of parameters the wider compacton is the minimum of a reduced hamiltonian and is the only one that is stable

application of the extended simplest equation method to the coupled schrodinger boussinesq equation in this paper we successfully construct the new exact traveling wave solutions of the coupled schrodinger boussinesq equation by using the extended simplest equation method the exact traveling wave solutions with double arbitrary parameters are expressed by the hyperbolic functions the trigonometric functions and the rational functions respectively when the parameters are taken as special values some solitary wave solutions are derived from the hyperbolic function solutions

generalized solitary and periodic wave solutions to a 2 1 dimensional zakharov kuznetsov equation abstract in this paper the exp function method is employed to the zakharov kuznetsov equation as a 2 1 dimensional model for nonlinear rossby waves the observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient the obtained results may be useful for better understanding the properties of two dimensional coherent structures such as atmospheric blocking events

eigenvalue problems with unbalanced growth nonlinear patterns and standing wave solutions abstract we consider two classes of nonlinear eigenvalue problems with double phase energy and lack of compactness we establish existence and non existence results and related properties of solutions our analysis combines variational methods with the generalized pucci serrin maximum principle

higher integrability for constrained minimizers of integral functionals with p q growth in low dimension abstract we prove higher summability for the gradient of minimizers of strongly convex integral functionals of the calculus of variations f x d u x d x u r n s n 1 with growth conditions of p q type p f x c q 1 p q in low dimension our procedure is set in the framework of fractional sobolev spaces and renders the desired regularity as the result of an approximation technique relying on estimates obtained through a careful use of difference quotients

a simple derivation of the n soliton solutions to the korteweg devries equation the n soliton solutions to the korteweg devries kdv equation are derived from wentzel kramers brilbuin wkb theory without using inverse scattering theory only elementary manipulations are needed to construct the solutions

a comparison between adomian decomposition method and taylor series method in the series solutions in this work we will compare the performance of the adomian decomposition method and the taylor series method applied to the solution of linear and nonlinear ordinary differential equations the comparison shows that the decomposition method is reliable efficient and easy to use from a computational viewpoint although the two methods provide the solution in an infinite series the decomposition method provides a fast convergent series of easily computable components and eliminates cumbersome computational work needed by taylor series method

absolute stability conditions for discrete time recurrent neural networks an analysis of the absolute stability for a general class of discrete time recurrent neural networks rnn s is presented a discrete time model of rnn s is represented by a set of nonlinear difference equations some sufficient conditions for the absolute stability are derived using ostrowski s theorem and the similarity transformation approach for a given rnn model these conditions are determined by the synaptic weight matrix of the network the results reported in this paper need fewer constraints on the weight matrix and the model than in previously published studies

necessary conditions for the appearance of noise terms in decomposition solutions series in this paper the phenomena of the noise terms that appear in the components of the series solution introduced by adomian and rach 1 will be examined extensively the phenomena is useful in demonstrating a fast convergence of the exact solution however the condition set by adomian and rach 1 in that these noise terms appear for every nonhomogeneous equation is not always true and the nonhomogeneity condition is not sufficient another necessary condition will be established in this study to ensure the appearance of noise terms in specific cases several differential and integral equations will be tested to confirm the newly developed condition

adomian decomposition method for solution of nonlinear differential algebraic equations in m m hosseini adomian decomposition method with chebyshev polynomials appl math comput in press an efficient modification of the adomian decomposition method was presented by using chebyshev polynomials also in m m hosseini adomian decomposition method for solution of differential algebraic equations j comput appl math in press solution of linear differential algebraic equations was considered by adomian decomposition method in this paper standard and modified adomian decomposition methods are applied to non linear differential algebraic equations the schemes are tested for some examples and the results demonstrate reliability and efficiency of the proposed methods

symmetry reductions and new exact non traveling wave solutions of potential kadomtsev petviashvili equation with p power with the aid of maple symbolic computation and lie group method pkpp equation is reduced to some 1 1 dimensional partial differential equations in which there are linear pde with constant coefficients nonlinear pde with constant coefficients and nonlinear pde with variable coefficients using the separation of variables homoclinic test technique and auxiliary equation methods we obtain new abundant exact non traveling solutions with arbitrary functions for the pkpp

homoclinic breather wave solutions and doubly periodic wave solutions for coupled kdv equations abstract in this paper we first find out a proper variable transformation by the ideas of painleve expansion then we apply the extended homoclinic test approach to obtain two cycle breathing places wave solutions of eq 1 which describe interactions between two physical waves and these special solutions can be applied to explain the structure of certain physical phenomena thus this method can be applied to the study of other similar nonlinear coupled system

auxiliary equation method for the mkdv equation with variable coefficients abstract by using solutions of an ordinary differential equation an auxiliary equation method is described to seek exact solutions of nonlinear evolution equations with variable coefficients being concise and straightforward this method is applied to the mkdv equation with variable coefficients as a result new explicit solutions including solitary wave solutions and trigonometric function solutions are obtained with the aid of symbolic computation

solitons and periodic solutions for the rosenau kdv and rosenau kawahara equations in this work we use the sine cosine and the tanh methods for solving the rosenau kdv and rosenau kawahara equations the two methods reveal solitons and periodic solutions the study confirms the power of the two schemes

new solutions of distinct physical structures to high dimensional nonlinear evolution equations abstract in this work we employ the tanh coth method for a reliable treatment on nonlinear evolution equations three models namely the 3 1 dimensional jimbo miwa equation the 2 1 dimensional calogero bogoyavlenskii schiff equation and the 3 1 dimensional potential ytsf equation are use as vehicles to conduct the analysis new travelling wave solutions are formally derived the work shows the power of the proposed scheme and the variety of the obtained solutions

a modification of the adomian decomposition method for a class of nonlinear singular boundary value problems in this paper the adomian decomposition method is modified to solve a class of nonlinear singular boundary value problems which arise as nonlinear normal modal equations in nonlinear conservative vibratory systems the effectiveness of the modified method is verified by three examples

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

application of the g g expansion method to 3 1 dimensional nonlinear evolution equations in this paper the g g expansion method is employed to solve the 3 1 dimensional kadomtsev petviashvili kp equation the 3 1 dimensional potential ytsf equation and the 3 1 dimensional jimbo miwa jm equation exact traveling wave solutions are obtained the traveling wave solutions are expressed in terms of hyperbolic functions the trigonometric functions and the rational functions

a study on a boundary layer equation arising in an incompressible fluid in this paper a differential equation of mixing layer that arises in viscous incompressible fluid will be investigated the series solution method will be effectively implemented to a related integro differential equation the condition at infinity will be applied to a related pade approximants of the obtained series solution

solitary wave solutions and periodic solutions for higher order nonlinear evolution equations the sine cosine and the tanh methods combined with the homogeneous balance method are used for analytic study for higher order nonlinear evolution equations exact solitary wave solutions and periodic solutions are developed a comparison between the two used methods is presented

compact structures for variants of the generalized kdv and the generalized kp equations in this paper we discuss mathematical variants in higher dimensions of the generalized kdv and the generalized kp equations it is shown that the focusing branches of these variants exhibit compactons solitons with finite wavelength whereas the defocusing branches support solitary patterns solutions with infinite slopes or cusps the compact and the noncompact dispersive structures are examined in a general way

the tanh and the sine cosine methods for compact and noncompact solutions of the nonlinear klein gordon equation the nonlinear klein gordon equation is used as a vehicle to employ the tanh method and the sine cosine method to formally derive a number of travelling wave solutions the study features a variety of solutions with distinct physical structures the work shows that one method complements the other and each method gives solutions of formal properties the obtained solutions include compactons solitons solitary patterns and periodic solutions

kinks and solitons solutions for the generalized kdv equation with two power nonlinearities in this work the generalized kdv equation with two power nonlinearities is studied the tanh method and two sets of ansatze involving hyperbolic functions are introduced for analytic study of this equation new kinds of solitary wave solutions are formally derived

on the adomian decomposition method and comparisons with picard s method abstract this paper shows that the adomian decomposition method stochastic systems academic press new york 1983 is not to be identified with any of the earlier methods since it solves a wide class of nonlinear and stochastic equations without the often unphysical assumptions which have become customary similarities between the picard method and the decomposition method are purely superficial

a novel fast modification of the adomian decomposition method to solve integral equations of the first kind with hypersingular kernels abstract in this paper using the modified adomian decomposition method a novel fast scheme has been presented being interactive to be used for solving a class of hypersingular integral equations of the first kind compared to the standard adomian decomposition method this method expedites the quick accumulation of a series of solutions different problems for exemplification have been solved in order to display the accuracy and validity of the aforementioned method according to the results it gets known that the novel fast modified adomian decomposition method is so efficient and simple which gives the exact solution and also shuns the complicated function of long theoretical calculations of collocation methods based on polynomial producing the exact solution

the tanh method for compact and noncompact solutions for variants of the kdv burger and the k n n burger equations abstract in this paper we examine variants of the kdv burger equation and the k n n burger equation both compact and noncompact structures are treated for these models the tanh method is employed as a vehicle to achieve this goal three different physical structures are emphasized compactons solitary patterns and blow up solutions

remark on modifications of adomian decomposition method a vital exception to the superiority of the chebyshev polynomial modification of the adomian decomposition method to other recent modifications is shown in this article also a new modification of the decomposition method is proposed and shown to be reliable

nonlinear variants of the improved boussinesq equation with compact and noncompact structures variants of the improved boussinesq equation with positive and negative exponents are investigated it is formally shown that these nonlinear variants give rise to compact and non compact physical structures where compactons solitons solitary patterns and periodic solutions are developed the presented sine cosine ansatz is reliable

a novel expansion iterative method for solving linear partial differential equations of fractional order in this manuscript we implement a relatively new analytic iterative technique for solving time space fractional linear partial differential equations subject to given constraints conditions based on the generalized taylor series formula the solution methodology is based on generating the multiple fractional power series expansion solution in the form of a rapidly convergent series with minimum size of calculations this method can be used as an alternative to obtain analytic solutions of different types of fractional linear partial differential equations applied in mathematics physics and engineering some numerical test applications were analyzed to illustrate the procedure and to confirm the performance of the proposed method in order to show its potentiality generality and accuracy for solving such equations with different constraints conditions numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the suggested algorithm

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

derivation of the adomian decomposition method using the homotopy analysis method abstract adomian decomposition method has been used intensively to solve nonlinear boundary and initial value problems it has been proved to be very efficient in generating series solutions of the problem under consideration under the assumption that such series solution exits however very little has been done to address the mathematical foundation of the method and its error analysis in this article the mathematical derivation of the method using the homotopy analysis method is presented in addition an error analysis is addressed as well as the convergence criteria

variants of the two dimensional boussinesq equation with compactons solitons and periodic solutions variants of the two dimensional boussinesq equation with positive and negative exponents are studied the sine cosine ansatz is fruitfully used to carry out the analysis exact solutions of different physical structures compactons solitary patterns solitons and periodic solutions are obtained the quantitative change in the physical structure of the solutions is shown to depend mainly on the exponent of the wave function u x t and on the ratio a b of the derivatives of u x t

equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations abstract we consider the solution of partial differential equations for initial boundary conditions using the decomposition method the partial solutions obtained from the seperate equations for the highest ordered linear operator terms are shown to be identical when the boundary conditions are general and asymptotically equal when the boundary conditions in one independent variable are independet of other variables

weighted residual method for obtaining positive solutions of two point nonlinear boundary value problems this paper applies the weighted residual method to solving two point nonlinear boundary value problems that have at least one positive solution numerical verification is given revealing the efficiency and convenience of the method proposed comparison with the work of others is also illustrated

a dynamic compensator for in domain control of burgers equation this paper addresses the problem of output regulation control of a burgers equation under pointwise in domain actuation a dynamic compensator that generates the in domain control is developed for

general compactons solutions for the focusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this paper we will study the focusing branch of the genuinely nonlinear dispersive k n n equation that exhibits compactons solitons with finite wavelengths the equation will be studied in one two and three dimensional spaces general formulas for compactons solutions are developed for all positive integers n n 1 two sets of entirely new formulas one set produces compactons for even integers n whereas the second set produces compactons and anticompactons for odd integers n n 1 are established

the decomposition method for approximate solution of the goursat problem abstract in this paper the adomian s decomposition method is effectively implemented to establish an analytic solution and a reliable numerical approximation to the goursat partial differential equation linear and nonlinear goursat examples are examined and the method supplied quantitatively reliable results for these types of problems the accuracy level of the results obtained indicates the superiority of the decomposition method over existing numerical methods that were applied to the goursat problem

solitary wave solutions of the improved kdv equation by vim the variational iteration method vim is applied to solve numerically the improved korteweg de vries equation ikdv a correction function is constructed with a general lagrange multiplier that can be identified optimally via the variational theory this technique provides a sequence of functions with easily computable components that converge rapidly to the exact solution of the ikdv equation propagation of single interaction of two and three solitary waves and also birth of solitons have been discussed three invariants of motion have been evaluated to determine the conservation properties of the problem this procedure is promising for solving other nonlinear equations

a new approach to the nonlinear advection problem an application of the decomposition method abstract in this paper the nonlinear advection problem is solved easily and elegantly by using the adomian s decomposition method the exact solution is quickly obtained by observing the appearance of the self canceling noise terms whose sum vanishes in the limit comparison with other existing techniques shows that the present approach is highly accurate

the modified decomposition method and pade approximants for solving the thomas fermi equation in this paper we propose a new approach to develop a nonperturbative approximate solution for the thomas fermi equation this approach rests on the recently developed modification of the adomian decomposition method the initial slope of the thomas fermi potential y 0 is computed by converting the obtained series solution into several diagonal pade approximants the proposed scheme is presented in a general way so that it can be used in applied sciences