value distribution of meromorphic solutions of certain difference painleve equations abstract the borel exceptional value and the exponents of convergence of poles zeros and fixed points of finite order transcendental meromorphic solutions for difference painleve i and ii equations are estimated and the forms of rational solutions of the difference painleve ii equation and the autonomous difference painleve i equation are also given it is also proved that the non autonomous difference painleve i equation has no rational solution

multiplicity of time periodic solutions and their stabilities for quasigeostrophic equations in this article we study the quasigeostrophic equation which is a prototypical geophysical fluid model we will show the existence of multiple time periodic solutions and study their stabilities we also prove the exponential decaying of solutions to the initial value problem with small initial data

growth and zeros of meromorphic solution of some linear difference equations abstract in this paper we study growth and zeros of linear difference equations p n z f z n p 1 z f z 1 p 0 z f z f z where f z p n z p 0 z are polynomials with f p n p 0 0 and satisfy deg p n p 0 max deg p j j 0 n 1 the corresponding homogeneous equation of the above equation is also investigated

growth and poles of meromorphic solutions of some difference equations abstract this paper is devoted to studying the growth property and the pole distribution of meromorphic solutions f of some complex difference equations with all coefficients being rational functions or of growth s r f we find the lower bound of the lower order or the relation between lower order and the convergence exponent of poles of meromorphic solutions of such equations

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

relationships between characteristic functions of δnf and f abstract in this paper we obtain the accurate relationships between the characteristic functions of a finite order meromorphic function f z and its forward differences n f z where f z has two borel exceptional values

a 3im 1cm result for periodic meromorphic functions abstract let f and g be two periodic meromorphic functions of hyper order less than 1 such that f and g share three values im and one value cm if f and g have a same period and the lower order of f is not equal to 1 then f g examples are given to illustrate the necessity of the conditions

conditional lie backlund symmetries and functionally separable solutions of the generalized inhomogeneous nonlinear diffusion equations abstract the functionally separable solutions of the generalized inhomogeneous nonlinear diffusion equations are studied by applying the conditional lie backlund symmetry method a complete list of canonical forms for such equations are presented exact solutions to the resulting equations are constructed the asymptotic behaviors and blow up properties of some solutions are also discussed

existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasi geostrophic model we consider a two dimensional viscous shallow water model with friction term existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi geostrophic equation with free surface term is proven in the well prepared case the ill prepared data case is also discussed

meromorphic functions sharing a set with applications to difference equations this paper is devoted to proving some uniqueness type results for an entire function f z that shares a common set with its shift f z c or its difference operator cf we also give some applications to solutions of non linear difference equations related to a conjecture proposed by c c yang

holomorphic curves with shift invariant hyperplane preimages if f c p n is a holomorphic curve of hyper order less than one for which 2n 1 hyperplanes in general position have forward invariant preimages with respect to the translation z z c then f is periodic with period c 2 c this result which can be described as a difference analogue of m green s picard type theorem for holomorphic curves follows from a more general result presented in this paper the proof relies on a new version of cartan s second main theorem for the casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives both of which are proved here finally an application to the uniqueness theory of meromorphic functions is given and the sharpness of the obtained results is demonstrated by examples

singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation abstract a short time existence theorem is proven for the initial boundary value problem for a class of quasilinear symmetric hyperbolic systems containing a constant coefficient spatial operator multiplied by a large parameter solutions u are shown to remain bounded independently of for a time independent of provided that u 0 and u l 0 are bounded independently of and certain structural conditions are satisfied this result is applied to the quasigeostrophic approximation of the shallow water equations and the constant pressure approximation in combustion and solutions for these cases are shown to converge to solutions of limiting equations as

on properties of meromorphic solutions for difference equations concerning gamma function abstract in this paper we mainly consider the properties of differences of meromorphic solutions for the difference equation a 1 z f z 1 a 0 z f z 0 concerning a gamma function where a 1 z and a 0 z are nonzero polynomials by these properties we deduce that a gamma function satisfies that for every n n n z z 0 n z has only n zeros z z z j z 1 j 0 1 n 1 z 1 z 1 n 1 z and 1 z have same zeros at most except n exceptional zeros where g denotes the order of growth of a meromorphic function g and g and g denote the exponents of convergence of fixed points and zeros of g respectively

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

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

conditional lie backlund symmetries and differential constraints for inhomogeneous nonlinear diffusion equations due to linear determining equations abstract the method of linear determining equations is developed to study conditional lie backlund symmetries for evolution equations which is more general than the classical determining equations for lie s generators as an application of this approach the complete classification is presented for the inhomogeneous nonlinear diffusion equations which admit the second order and third order conditional lie backlund symmetries several examples are given to illustrate the corresponding symmetry reductions due to the compatibility of the invariant conditions and the governing equations

exact similarity solutions to some nonlinear diffusion equations the author gives new closed form similarity solutions to n dimensional radially symmetric nonlinear diffusion equations with power law and exponential diffusivities these solutions are generalisations of known similarity solutions

solutions of nonlinear difference equations abstract the non vanishing finite order entire solutions of nonlinear difference equation f z n a n 1 f z n 1 a 1 f z q z e q z f z c p z are studied in this paper for n 2 and n 1 respectively where q z q z p z are polynomials and a n 1 a 1 c c we classified these solutions according to growth and zero distribution particularly we show that exponential polynomial solutions satisfying certain conditions must reduce to rather specific forms when n 2 and a 1 0 our results are improvements and complements of wen et al 13

the variational iteration method a powerful scheme for handling linear and nonlinear diffusion equations in this work we introduce a framework for obtaining exact solutions to linear and nonlinear diffusion equations exact solutions are developed for some diffusion processes of power law diffusitivies he s variational iteration method vim is used for analytic treatment of these equations the powerful vim method is capable of handling both linear and nonlinear equations in a direct manner

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

a review on some contributions to perturbation theory singular limits and well posedness abstract in the beginning of the 1990s we devoted a sequence of papers to perturbation theory singular limits and well posedness problems in particular the strong well posedness of the initial boundary value problem for the compressible euler equations was demonstrate for the first time our method also allowed singular limit results in the strong norm even under assumptions weaker than the current ones in the literature where the strong norm is not reached it is worth noting that until now the above method and results have not been substantially improved hence an introduction to it still looks timely actually in a forthcoming paper by returning to this method we improve in a very substantial way some important results recently appeared in the literature

on zeros and fixed points of differences of meromorphic functions abstract let f be a transcendental meromorphic function and g z f z 1 f z a number of results are proved concerning the existences of zeros and fixed points of g z or g z f z which expand results of bergweiler and langley w bergweiler j k langley zeros of differences of meromorphic functions math proc cambridge philos soc 142 2007 133 147

conditional lie backlund symmetries and solutions of inhomogeneous nonlinear diffusion equations abstract the second order conditional lie backlund symmetries of nonlinear diffusion equations with variable coefficients are studied a number of examples are considered and some exact solutions are constructed via the compatibility of conditional lie backlund symmetries and the governing equations these solutions possess the extended forms of the separation of variables including the extensions of the instantaneous source solutions of the porous medium equations

a note on partial sharing of values of meromorphic functions with their shifts abstract we introduce the notion of partial value sharing as follows let e a f be the set of zeros of f z a z where each zero is counted only once and a is a meromorphic function small with respect to f a meromorphic function f is said to share a partially with a meromorphic function g if e a f e a g we show that partial value sharing of f z and f z c involving 3 or 4 values combined with an appropriate deficiency assumption is enough to guarantee that f z f z c provided that f z is a meromorphic function of hyper order strictly less than one and c c

existence of finite order meromorphic solutions as a detector of integrability in difference equations abstract the existence of sufficiently many finite order in the sense of nevanlinna meromorphic solutions of a difference equation appears to be a good indicator of integrability it is shown that out of a large class of second order difference equations the only equation that can admit a sufficiently general finite order meromorphic solution is the difference painleve ii equation the proof given relies on estimates obtained by arguments related to singularity confinement the existence of meromorphic solutions of a general class of first order difference equations is also proven by a simple method based on banach s fixed point theorem

finite order solutions of second order linear differential equations on considere l equation differentielle f a z f b z f 0 ou a z et b z sont des fonctions entieres on trouve des conditions sur a z et b z qui garantissent que chaque solution f 0 de l equation a un ordre fini

hopf bifurcation in quasi geostrophic channel flow in this article we conduct a rigorous stability and bifurcation analysis for a highly idealized model of planetary scale atmospheric and oceanic flows the model is governed by the two dimensional quasi geostrophic equation for the conservation of vorticity in an east west oriented periodic channel the main result is the existence of hopf bifurcation of the flow as the reynolds number crosses a critical value the key idea in proving this result is translating the eigenvalue problem into a difference equation and treating the latter by continued fraction methods numerical results are obtained by using a finite difference scheme with high spatial resolution and these results agree closely with the theoretical predictions the spatio temporal structure of the limit cycle corresponds to a wave that propagates slowly westward and is symmetric about the midaxis of the channel for plausible paramater values that correspond to midlatitude atmospheric flows the period of this wave is 20 25 days

exact solutions of nonlinear diffusion equations by variational iteration method in this paper he s variational iteration method is applied to obtain exact solutions of some nonlinear diffusion equations the variational iteration method is used to construct correction functionals using general lagrange multipliers identified optimally via the variational theory and the initial approximations can be freely chosen with unknown constants the solutions obtained are compared with those obtained by the adomian decomposition method and homotopy perturbation method showing excellent agreement but the variational iteration method is more effective he s variational iteration method can be introduced to overcome the difficulties arising in calculating adomian polynomials

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

problems with different time scales for nonlinear partial differential equations consider a system of partial differential equations u t varepsilon 1 p 0 p 1 u here 0 varepsilon ll 1 is a small constant p 0 p 0 partial partial x is a differential operator with constant coefficients and p 1 p 1 x t u partial partial x is a quasilinear operator the coefficients of p 0 p 1 are of order o 1 it is shown how to prepare the initial data such that the solutions u x t and their derivatives can be bounded independently of varepsilon applications are discussed

vanishing viscosity limit for global attractors for the damped navier stokes system with stress free boundary conditions abstract we consider the damped and driven navier stokes system with stress free boundary conditions and the damped euler system in a bounded domain r 2 we show that the damped euler system has a strong global attractor in h 1 we also show that in the vanishing viscosity limit the global attractors of the navier stokes system converge in the non symmetric hausdorff distance in h 1 to the strong global attractor of the limiting damped euler system whose solutions are not necessarily unique

low temperature asymptotics of integrable systems in an external field an asymptotic low temperature expansion is performed for an integrable bosonic lattice model and for the critical spin 1 2 heisenberg chain in a magnetic field the results apply to the integrable bose gas as well we also comment on a high temperature expansion of the bosonic lattice model

on the value distribution of some difference polynomials abstract in this paper we continue to investigate the value distribution of some difference polynomials which can be considered as difference counterparts of some classic results of hayman

steady state bifurcation analysis of a strong nonlinear atmospheric vorticity equation the quasi geostrophic vorticity equation studied in the present paper is a simplified form of the atmospheric circulation model introduced by charney and devore j atmos sci 36 1979 1205 1216 on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking the fluid motion defined by the equation is driven by a zonal thermal forcing and an ekman friction forcing measured by it is proved that the steady state solution is globally unique for large values while multiple steady state solutions branch off the basic steady state solution for crit where the critical value crit is less than one without involvement of viscosity the equation has fully non linear property as its non linear part contains the highest order derivative term steady state bifurcation analysis is essentially based on the compactness which can be simply obtained for semilinear equations such as the navier stokes equations but is not available for the fully nonlinear quasi geostrophic vorticity equation in the euler formulation therefore the lagrangian formulation of the equation is employed to gain the required compactness

value sharing results for shifts of meromorphic functions and sufficient conditions for periodicity abstract this research is a continuation of a recent paper due to the first four authors shared value problems related to a meromorphic function f z and its shift f z c where c c are studied it is shown for instance that if f z is of finite order and shares two values cm and one value im with its shift f z c then f is a periodic function with period c the assumption on the order of f can be dropped if f shares two shifts in different directions leading to a new way of characterizing elliptic functions the research findings also include an analogue for shifts of a well known conjecture by bruck concerning the value sharing of an entire function f with its derivative f

hopf bifurcation and transition of three dimensional wind driven ocean circulation problem abstract the purpose of this article is three fold the first is to investigate the linear stability of the basic flow governed by the three dimensional 3d quasi geostrophic qg equation with a steady wind force we show that the basic flow will become unstable as the reynolds number crosses a critical value the second is to study the dynamic transition associated with the instability of the basic flow at the critical reynolds number it is proved that the type of the phase transition is determined by a parameter called transition number which is mainly determined by the reynolds number and the aspect ratio more precisely the negative positive sign of the real part of the transition number corresponds to the continuous catastrophic transition the last objective is to apply numerical experiments to examine the specific transition type for specified control parameters we show that the real part of the transition number is always negative leading to only the continuous transition

value sharing results for shifts of meromorphic functions abstract in this paper we deal with the value distribution of difference products of entire functions and present some result on two difference products of entire functions sharing one value with the same multiplicities the research findings also include an analogue for shift of a well known conjecture by bruck our theorems improve the results of i laine and c c yang i laine c c yang value distribution of difference polynomials proc japan acad ser a math sci 83 2007 148 151 k liu and l z yang k liu l z yang value distribution of the difference operator arch math 92 2009 270 278 and j heittokangas et al j heittokangas r korhonen i laine j rieppo j l zhang value sharing results for shifts of meromorphic function and sufficient conditions for periodicity j math anal appl 355 2009 352 363 moreover we show by illustrating a number of examples that our results are best possible in certain senses

a two level finite element discretization of the streamfunction formulation of the stationary quasi geostrophic equations of the ocean in this paper we proposed a two level finite element discretization of the nonlinear stationary quasi geostrophic equations which model the wind driven large scale ocean circulation optimal error estimates for the two level finite element discretization were derived numerical experiments for the two level algorithm with the argyris finite element were also carried out the numerical results verified the theoretical error estimates and showed that for the appropriate scaling between the coarse and fine mesh sizes the two level algorithm significantly decreases the computational time of the standard one level algorithm

difference analogue of the lemma on the logarithmic derivative with applications to difference equations the lemma on the logarithmic derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary dierential equa tions in this paper a dierence analogue of the logarithmic derivative lemma is presented and then applied to prove a number of results on meromorphic solutions of complex dierence equations these results include a dierence analogue of the clunie lemma as well as other results on the value distribution of solutions

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

exact solutions of a class of nonlinear boundary value problems with moving boundaries presents a method for the construction of the exact solutions of nonlinear bvps in melting and evaporation processes the case of smooth functions of thermal conductivity is studied and illustrations of some specific nonlinearities given

the compressible euler equations in a bounded domain existence of solutions and the incompressible limit a short time existence theorem is proven for the euler equations for nonisentropic compressible fluid flow in a bounded domain and solutions with low mach number and almost incompressible initial data are shown to be close to corresponding solutions of the equations for incompressible flow

from continuous to discrete painleve equations abstract discrete analogues of the painleve equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method here we analyze additional methods for deriving integrable non autonomous discrete equations these methods are based on the discrete akns approach on the use of backlund and schlesinger transforms of the continuous painleve equations and on discrete analogues of miura transformations

periodic solutions of parabolic systems with nonlinear boundary conditions this paper is concerned with the existence and stability of periodic solutions for a coupled system of nonlinear parabolic equations under nonlinear boundary conditions the approach to the problem is by the method of upper and lower solutions and its associated monotone iterations this method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for parabolic initial boundary value problems a sufficient condition for the stability of a periodic solution is also given these results are applied to three model problems arising from chemical kinetics ecology and population biology

the road to the discrete analogue of the painleve property nevanlinna meets singularity confinement the question of integrability of discrete systems is analysed in the light of the recent findings of ablowitz et al who have conjectured that a fast growth of the solutions of a differ ence equation is an indication of nonintegrability the study of the behaviour of the solutions of a mapping is based on the theory of nevanlinna in this paper we show how this approach can be implemented in the csse of second order mappings which include the discrete painlevk equations since the nevanlinna approach does offer only a necessary condition which is not restrictive enough we complement it by the singularity confinement requirement first in an autonomous setting and then for deautonomisation we believe that this three tiered approach is the closest one can get to a discrete analogue of the painlevb property 2003 elsevier science ltd all rights reserved

waiting time solutions of a nonlinear diffusion equation similarity solutions of the equation partial c partial t partial partial x c n partial c partial x n 0 are studied for both t 0 and t 0 and there is a stationary free boundary at x 0 and for t 0 have support in x at beta where beta 1 a 0 i e there is a moving free boundary x at beta by prescribing appropriate initial data c c 0 x x 0 at t t 0 these solutions demonstrate the existence of a waiting time t 0 before the free boundary moves a comparison theorem then enables a number of results to be obtained and conjectures to be made about the existence and length of waiting times for general initial data a three parameter family of solutions which exhibit blow up at t 0 is also constructed applications of these results to models for the spreading of a viscous blob on a hor

on the derivative of meromorphic functions with multiple zeros abstract let f be a transcendental meromorphic function and let r be a rational function r 0 we show that if all zeros and poles of f are multiple except possibly finitely many then f r has infinitely many zeros if f has finite order and r is a polynomial then the conclusion holds without the hypothesis that poles be multiple

uniqueness and periodicity of meromorphic functions concerning the difference operator in this paper we investigate the uniqueness problems of difference polynomials of meromorphic functions that share a value or a fixed point we also obtain several results concerning the shifts of meromorphic functions and the sufficient conditions for periodicity which improve some recent results in heittokangas et al 2009 10 and liu 2009 11

meromorphic solutions of certain type of differential difference equations abstract we consider meromorphic solutions to general differential difference equation of type l z f j 1 q c j z f n j z j j 1 r j e j z where l z f 0 is a linear differential difference polynomial of f with small function coefficients and 1 n 1 n q j 1 j r are nonvanishing constants c j z 1 j q are nonvanishing small functions if the equation admits a transcendental meromorphic solution f of finite order satisfying that f f 1 f f then representation of f and relations between coefficients in z f and i j can be obtained

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 growth of meromorphic solutions for q difference painleve iv equation abstract we consider meromorphic solutions of q difference painleve iv equation f q z f z f z f z q p z f z q z f z where p z f z and q z f z are polynomials in f having polynomial coefficients and no common roots and q c 0 we investigate the growth of transcendental meromorphic solutions of q difference painleve iv equation and find lower bounds for their characteristic functions for transcendental meromorphic solutions of such equation for the case m deg f p deg f q 3

some properties of meromorphic solutions of q difference equations abstract we investigate the growth of transcendental meromorphic solutions of some complex q difference equations and find lower bounds for nevanlinna lower order for meromorphic solutions of such equations we also obtain a q difference version of tumura clunie theorem

nevanlinna theory of the askey wilson divided difference operator abstract this paper establishes a version of nevanlinna theory based on askey wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane c a second main theorem that we have derived allows us to define an askey wilson type nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero pole scarce that is their zeros poles are indeed deficient in the sense of difference nevanlinna theory a natural consequence is a version of askey wilson type picard theorem we also give an alternative and self contained characterisation of the kernel functions of the askey wilson operator in addition we have established a version of unicity theorem in the sense of askey wilson this paper concludes with an application to difference equations generalising the askey wilson second order divided difference equation

the barotropic vorticity equation under forcing and dissipation bifurcations of nonsymmetric responses and multiplicity of solutions a simplified model of barotropic flow in a bounded domain has been used and stationary responses due to the balance between dissipation and external forcing have been studied the forcing as well as the domain is assumed to admit some symmetry group sufficient conditions are deduced by which a branch of nonsymmetric stationary solutions bifurcate from a symmetric one and those results are applied to a specific problem of zonally symmetric forcing in an open channel

global well posedness and finite dimensional global attractor for a 3 d planetary geostrophic viscous model in this paper we consider a three dimensional planetary geostrophic viscous model of the gyre scale mid latitude ocean we show the global existence and uniqueness of the weak and strong solutions to this model moreover we establish the existence of a nite dimensional global attractor to this dissipative evolution system c 2003 wiley periodicals inc

the incompressible limit in nonlinear elasticity the incompressible limit in nonlinear elasticity is shown to fall under the theory of singular limits of quasilinear symmetric hyperbolic systems developed by klainerman and majda specifically initial value problems for a family of hyperelastic materials with stored energy functions w left frac partial x partial x right w infty left frac partial x partial x right lambda 2 w left det frac partial x partial x right are considered wherex andx are reference and deformed coordinates respectively under the assumption that the elasticity tensor a kl ij equiv frac partial 2 w infty partial left frac partial x i partial x k right partial left frac partial x j partial x l right is positive definite near the identity matrix and thatw 1 0 the following results are proven for appropriate initial data

how to detect the integrability of discrete systems several integrability tests for discrete equations will be reviewed all tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as lax pairs specifically singularity confinement algebraic entropy nevanlinna theory diophantine integrability and discrete systems over finite fields will be described

global well posedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics in this paper we prove the global existence and uniqueness regularity of strong solutions to the three dimensional viscous primitive equations which model large scale ocean and atmosphere dynamics 1 introduction large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the navier stokes equations with rotation coupled to thermodynamics and salinity diffusion transport equations which account for the buoyancy forces and stratification effects under the boussinesq approximation moreover and due to the shallowness of the oceans and the atmosphere i e the depth of the fluid layer is very small in comparison to the radius of the earth the vertical large scale motion in the oceans and the atmosphere is much smaller than the horizontal one which in turn leads to modeling the vertical motion by the hydrostatic balance as a result

symmetric positive systems with boundary characteristic of constant multiplicity the theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity no such hypothesis is needed on a neighborhood of the boundary both regularity theorems and mixed initial boundary value problems are discussed many classical ideas are sharpened in the process

on the growth of logarithmic differences difference quotients and logarithmic derivatives of meromorphic functions a crucial ingredient in the recent discovery by ablowitz halburd herbst and korhonen 2000 2007 that a connection exists between discrete painleve equations and finite order nevanlinna theory is an estimate of the integrated average of log f z 1 f z on z r we obtained essentially the same estimate in our previous paper 2008 independent of halburd et al 2006 we continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences difference quotients and logarithmic derivatives for finite order meromorphic functions in addition to the potential applications of our new estimates in integrable systems they are also of independent interest in particular our findings show that there are marked differences between the growth of meromorphic functions with nevanlinna order less than and greater than one we have established a difference analogue of the classical wiman valiron type estimates for meromorphic functions with order less than one which allow us to prove that all entire solutions of linear difference equations with polynomial coefficients of order less than one must have positive rational order of growth we have also established that any entire solution to a first order algebraic difference equation with polynomial coefficients must have a positive order of growth which is a difference analogue of a classical result of polya

value distribution and shared sets of differences of meromorphic functions we investigate value distribution and uniqueness problems of difference polynomials of meromorphic functions in particular we show that for a finite order transcendental meromorphic function f with 1 f f and a non zero complex constant c if n 2 then f z nf z c assumes every non zero value a c infinitely often this research also shows that there exist two sets s1 with 9 resp 5 elements and s2 with 1 element such that for a finite order nonconstant meromorphic resp entire function f and a non zero complex constant c ef z sj ef z c sj j 1 2 imply f z f z c this gives an answer to a question of gross concerning a finite order meromorphic function f and its shift

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

an extension of picard s theorem for meromorphic functions of small hyper order abstract a version of the second main theorem of nevanlinna theory is proved where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper order as a corollary of this result it is shown that if n n and three distinct values of a meromorphic function f of hyper order less than 1 n 2 have forward invariant pre images with respect to a fixed branch of the algebraic function z z n 1 z 1 1 n 1 z 1 n 0 with constant coefficients then f f this is a generalization of picard s theorem for meromorphic functions of small hyper order since the empty pre images of the usual picard exceptional values are special cases of forward invariant pre images

numerical modelling of linear and nonlinear diffusion equations by compact finite difference method in this work accurate solutions to linear and nonlinear diffusion equations were introduced a combination of a sixth order compact finite difference scheme in space and a low storage third order total variation diminishing runge kutta scheme in time have been used for treatment of these equations the computed results with the use of this technique have been compared with the exact solution to show the accuracy of it here the approximate solution to the diffusion equations has been obtained easily and elegantly with neither transforming nor linearizing the equation the present method is seen to be a very good alternative method to some existing techniques for realistic problems

several new exact solutions for a fast diffusion equation by the differential constraints of the linear determining equations in this paper we propose a new approach to develop several new exact solutions to a fast nonlinear diffusion equation this approach rests on the differential constraints obtained by the method of linear determining equations the method is capable of reducing the nonlinear diffusion equation to systems of nonlinear ordinary equations the proposed scheme is presented in a general way so that it can be used in other diffusion processes

growth of meromorphic solutions of linear difference equations abstract in this paper the authors continue to study the growth of meromorphic solutions of homogeneous or non homogeneous linear difference equations with entire coefficients and obtain some results which are improvement and extension of previous results in chiang and feng 2008 7 and laine and yang 2007 19 examples are also given to illustrate the sharpness of our results

the fixed boundary value problems for the equations of ideal magneto hydrodynamics with a perfectly conducting wall condition the equations of ideal magneto hydrodynamics are investigated concerning initial boundary value problems with a perfectly conducting wall condition the local in time solution is proved to exist uniquely provided that the normal component of the initial magnetic field vanishes everywhere or nowhere on the boundary

nevanlinna theory for the q difference operator and meromorphic solutions of q difference equations it is shown that if f is a meromorphic function of order zero and q in mathbb c then begin equation label abstid m bigg r frac f qz f z bigg o t r f tag ddag end equation for all r on a set of logarithmic density 1 the remainder of the paper consists of applications of identity eqref abstid to the study of value distribution of zero order meromorphic functions and in particular zero order meromorphic solutions of q difference equations the results obtained include q shift analogues of the second main theorem of nevanlinna theory picard s theorem and clunie and mohon ko lemmas

finite order meromorphic solutions and the discrete painleve equations let w z be a finite order meromorphic solution of the second order difference equation w z 1 w z 1 r z w z where r z w z is rational in w z and meromorphic in z then either w z satisfies a difference linear or riccati equation or else equation can be transformed to one of a list of canonical difference equations this list consists of all known difference painleve equation of the form together with their autonomous versions this suggests that the existence of finite order meromorphic solutions is a good detector of integrable difference equations