explicit series solutions to nonlinear evolution equations the sine cosine method in this paper a sine cosine method is used to construct many periodic and solitary wave solutions to two nonlinear evolution systems the coupled quadratic nonlinear equations and the coupled klein gordon schrodinger equations under different parameter conditions explicit formulas for some new periodic and solitary wave solutions are successfully obtained the proposed solutions are found to be important for the explanation of some practical physical problems

the sine cosine method for obtaining solutions with compact and noncompact structures in this paper we establish compact and noncompact solutions for nonlinear dispersive equations a sine cosine method is used to demonstrate this work the different physical structures of the focusing branch and the defocusing branch are emphasized many models are approached to illustrate the usage of our main results

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

compact and noncompact structures for a variant of kdv equation in higher dimensions in this paper we study two mathematical variants of kdv equation we show that the focusing branches of these variants exhibit compactons solitons with finite wavelength whereas the defocusing branches support solitary patterns solutions with cusps or infinite slopes a framework is developed for studying these problems in higher dimensions two distinct sets of general formulas for each type of variants are developed to present a fairly complete understanding of the compact and noncompact dispersive structures

patterns on liquid surfaces cnoidal waves compactons and scaling abstract localized patterns and nonlinear oscillation formations on the bounded free surface of an ideal incompressible liquid are investigated cnoidal modes solitons and compactons as traveling non axially symmetric shapes are discussed a finite difference differential generalized korteweg de vries kdv equation is shown to describe the three dimensional motion of the fluid surface and in the limit of long and shallow channels one recovers the well known kdv equation a tentative expansion formula for the representation of the general solution of a nonlinear equation for given initial conditions is introduced the model is useful in multilayer fluid dynamics cluster formation and nuclear physics since up to an overall scale these systems display a free liquid surface behavior

on a class of nonlinear dispersive dissipative interactions abstract we study a prototype dissipative dispersive equation ut a um x un xxx uk xx a consts which represents a wide variety of interactions at the critical value k m n 2 which separates dispersive and dissipation dominated phenomena these effects are in a detailed balance and the patterns formed do not depend on the amplitude in particular when m n 2 k 1 the equation can be transformed into a form free of convection and dissipation making it accessible to analysis both bounded and unbounded oscillations as well as solitary waves are found a variety of exact solutions are presented with a notable example being a solitary doublet for n 1 and a 2 3 2 the problem may be mapped into a linear equation leading to rational periodic or aperiodic solutions among others

the effect of the order of nonlinear dispersive equation on the compact and noncompact solutions the effect of the order of nonlinear dispersive partial differential equation on the compact and noncompact solutions is studied in this paper two variants of the kdv equation of odd orders are examined our analysis reveals that the physical nature of these nonlinear kdv type of equations are different two distinct sets of general formulas for each type of variants are performed to present a fairly complete understanding of the compact and noncompact structures

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

a numerical study of adding an artificial dissipation term for solving the nonlinear dispersive equations k n n a linearized implicit finite difference method is devised for k n n the stability and accuracy of the proposed methods are discussed a compacton wave solution of the equation k n n is used to examine the accuracy and efficiency of the proposed methods and study the effect of the added artificial dissipation term to solve the k n n equation using finite difference method the dynamics of waves having various initial wavepackets are discussed

the tanh method for a reliable treatment of the k n n and the kp n n equations and its variants abstract in this work the nonlinear k n n and kp n n equations and its variants are analytically studied we show that these equations may exhibit compactons solitons or periodic solutions by using the tanh method the analysis reveals the change of the physical structure of the solutions as a result of the exponents and the coefficients of the derivatives

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

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

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

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

new compact and noncompact solutions for two variants of a modified camassa holm equation in this paper exact solutions of two variants of a modified camassa holm equation are determined by using direct ansatze as a result compactons solutions solitons with the absence of infinite tails solitons nonlinear localized waves of infinite support solitary patterns solutions having infinite slopes or cusps and plane periodic solutions are established it is also found that the qualitative change in the physical structure of solutions depends mainly on whether the exponent of the wave function u x t and the expression 2k c are positive or negative

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

the sine cosine and the tanh methods reliable tools for analytic treatment of nonlinear dispersive equations the sine cosine method and the tanh method are used for analytic treatment of nonlinear dispersive equations nonlinear variants of boussinesq equation are used as vehicles to show the strength of these methods solutions of distinct physical structures solitons compactons solitary patterns solutions and periodic solutions are formally derived the results show that the change in physical structures of the obtained solutions depends mainly on exponents and on the coefficients of the derivatives involved

nonlinear dispersive special type of the zakharov kuznetsov equation zk n n with compact and noncompact structures in this paper we study nonlinear dispersive special type of the zakharov kuznetsov equation with positive and negative exponents the approach depends mainly on the sine cosine algorithm compactons solitary patterns solitons and periodic solutions are formally derived

exact travelling wave solutions to seventh order and ninth order kdv like equations new ansatze that involve hyperbolic functions are introduced to handle seventh order and ninth order kdv like equations a set of entirely new exact travelling wave solutions is established the work introduces useful guidance for other related nonlinear evolution equations

compact and noncompact solutions for nonlinear dispersive variants of the generalized kdv equation in this paper we examine three distinct nonlinear dispersive variants of the generalized kdv equation we study the compact and the noncompact structures for these variants we also emphasize the fact that purely nonlinear dispersion can cause a deep qualitative change in the genuinely nonlinear phenomenon

iterative wiener filters for image restoration the iterative wiener filter which successively uses the wiener filtered signal as an improved prototype to update the covariance estimates is investigated the convergence properties of this iterative filter are analyzed it has been shown that this iterative process converges to a signal which does not correspond to the minimum mean squared error solution based on the analysis an alternate iterative filter is proposed to correct for the convergence error the theoretical performance of the filter has been shown to give minimum mean squared error in practical implementation when there is unavoidable error in the covariance computation the filter may still result in undesirable restoration its performance has been investigated and a number of experiments in a practical setting were conducted to demonstrate its effectiveness

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

traveling wave solutions of the bbm like equations in this work we apply the factorization technique to the benjamin bona mahony like equations b m n in order to get traveling wave solutions we will focus on some special cases for which m n and we will obtain these solutions in terms of the special forms of weierstrass functions

new exact explicit peakon and smooth periodic wave solutions of the k 3 2 equation abstract based on an independent variable transformation the k 3 2 equation is investigated by the bifurcation method of planar systems and qualitative theory of polynomial differential system in different regions of the parametric space some new exact explicit peakon and smooth periodic wave solutions are obtained the results presented in this paper improve the previous results

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

the tanh method exact solutions of the sine gordon and the sinh gordon equations in this work we use the tanh method for traveling wave solutions of the sine gordon and the sinh gordon equations several exact solutions of distinct physical structures are obtained the method is powerful with minimal algebra work and is demonstrated for four models

an analytic study of compacton solutions for variants of kuramoto sivashinsky equation in this paper two mathematical variants of kuramoto sivashinsky equation are examined a framework is developed for studying these variants in one and higher dimensions two distinct sets of general formulae for each variant are developed compactons solitons with compact support and noncompact dispersive structures are formally established

solutions of compact and noncompact structures for nonlinear klein gordon type equation the focusing and the defocusing branches of nonlinear klein gordon type equation kg n n are considered a framework is implemented to show that the first model exhibits compactons solitons that do not have exponential tails whereas the second model demonstrates solitary patterns solutions the two variants of kg n n equation will be examined in one and higher dimensions general formulas will be developed to present a fairly complete understanding of the solutions of compact and noncompact structures

the camassa holm kp equations with compact and noncompact travelling wave solutions abstract in this paper two variants of camassa holm kp equations are investigated compactons solitons with the absence of infinite tails solitons nonlinear localized waves of infinite support solitary patterns having infinite slopes or cusps and plane periodic solutions are formally derived the work highlights the qualitative change in the physical structures of the obtained solutions

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

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

ignorability and coarse data some biomedical examples heitjan and rubin 1991 annals of statistics 19 2244 2253 define data to be coarse when one observes not the exact value of the data but only some set a subset of the sample space that contains the exact value this definition covers a number of incomplete data problems arising in biomedicine including rounded heaped censored and missing data in analyzing coarse data it is common to proceed as though the degree of coarseness is fixed in advance in a word to ignore the randomness in the coarsening mechanism when coarsening is actually stochastic however inferences that ignore this randomness may be seriously misleading heitjan and rubin 1991 have proposed a general model of data coarsening and established conditions under which it is appropriate to ignore the stochastic nature of the coarsening the conditions are that the data be coarsened at random a generalization of missing at random rubin 1976 biometrika 63 581 592 and that the parameters of the data and the coarsening process be distinct this article presents detailed applications of the general model and the ignorability conditions to a variety of coarse data problems arising in biomedical statistics a reanalysis of the stanford heart transplant data crowley and hu 1977 journal of the american statistical association 72 27 36 reveals significant evidence that censoring of pretransplant survival times by transplantation was nonignorable suggesting a greater benefit from cardiac transplantation than previous analyses had found

variants of the generalized kdv equation with compact and noncompact structures abstract in this paper nonlinear dispersive variants of the generalized kdv equation are investigated the compact and the noncompact physical structures are studied two general formulas for the focusing branch and the defocusing branch that are of substantial interest are formally derived

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

a construction of compact and noncompact solutions for nonlinear dispersive equations of even order a construction of compact and noncompact solutions for nonlinear dispersive partial differential equations of even order is considered two model equations of even orders k 4 are adopted to carry out this study the analysis reveals the presence of distinct roots of different indices that play a significant role in the solutions

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

a class of nonlinear fourth order variant of a generalized camassa holm equation with compact and noncompact solutions abstract in this paper we study a class of nonlinear fourth order analogue of a generalized camassa holm equation it is shown that this class gives compactons conventional solitons solitary patterns and periodic solutions it is also found that the qualitative change in the physical structure of solutions depends mainly on the exponent of the wave function u x t positive or negative and on the coefficient of u n as well

compactons structures for fifth order kdv like equations in higher dimensions in this paper we study a variant of the fifth order kdv equation fkdv that exhibits compactons solitons with finite wave lengths the work formally shows how to construct compact dispersive structures in higher dimensions two sets of general formulas for compactons solutions that are of substantial interest are developed for this variant fk n n for all positive integers n n 1

computations of soliton solutions and periodic solutions for the focusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in the present paper the focusing branch of the genuinely nonlinear dispersive k n n equation is studied in one two and three dimensional spaces in the case of 0