analytical methods for solving the time fractional swift hohenberg s h equation in this paper the most effective methods the homotopy perturbation method hpm and the differential transform method dtm are applied to obtain the approximate solutions of the nonlinear time fractional swift hohenberg s h equation the basic philosophy of these methods does not involve linearization weak nonlinearity assumptions or perturbation theory numerical solutions for various combinations of the parameters a eigenvalue parameter l length and a fractional index are obtained the solutions of the s h equation are useful for studies of shear thinning effects in non newtonian fluid flows at the end the solutions obtained are also presented graphically

clustering of linearly interacting diffusions and universality of their long time limit distribution let k d d 1 be a compact convex set and a countable abelian group we study a stochastic process x in k equipped with the product topology where each coordinate solves a sde of the form dx i t j a j i x j t x i t dt x i t db i t here a is the kernel of a continuous time random walk on and is a continuous root of a diffusion matrix w on k if x t converges in distribution to a limit x and the symmetrized random walk with kernel a s i a i a i is recurrent then each component x i is concentrated on x k x 0 and the coordinates agree i e the system clusters both these statements fail if a s is transient under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of k we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components this distribution turns out to be universal in all recurrent kernels a s on abelian groups

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

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

numerical simulation of equal width wave equation by means of the variational iteration method the solution of the equal width wave equation is numerically obtained and compared with that obtained by the adomian decomposition method numerical results show that the variational iteration method can readily be implemented with excellent accuracy to nonlinear equations

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

a comparison between cole hopf transformation and the decomposition method for solving burgers equations the equation under discussion in this paper is the non linear parabolic burgers equation given by 1 u t uu email protected x x where u u x t in some domain and n is a parameter n 0 and uu x is the non linear term eq 1 is similar in structure to navier stokes equation and has appeared in problems of aerodynamics and was first introduced by j m burgers an equation simply related to 1 appears in the approximate theory of weak non stationary shock wave in a real fluid g adomian solving frontier problems of physics the decomposition method kluwer boston 1994 a wazwaz partial differential equations methods and applications balkema publishers the netherlands 2002 a wazwaz a gorguis an analytic study of fisher s equation by using adomian decomposition methods appl math comput 154 2004 609 620 a wazwaz a gorguis exact solutions for heat like and wave like equations with variable coefficients appl math comput 149 2004 15 29 the coefficient n is a constant that defines the kinematic viscosity if the viscosity n 0 the equation is called inviscid burgers equation which governs gas dynamics the aim of this paper is to study the exact solution of eq 1 in some domain with initial values by the method of cole hopf transformation e hopf the partial differential equation u t uu x u x x comm pure appl math 3 1950 201 230 p lagestrom j d cole l trilling problems in the theory of viscous compressible fluids california institute of technology 1949 which transforms 1 from a non linear partial differential equation into linear heat equation which in turns can be solved exactly we then formally derive the exact solution of 1 with initial and boundary conditions using the adomian decomposition method the comparative study will be conducted to show how reliable sufficient and simpler is the adomian decomposition method

multiple time scale analysis of interacting diffusions we consider the questions how can the long term behavior of large systems of interacting components be described in terms of infinite systems on what time scale does the infinite system give a qualitatively correct description and what happens at large resp critical time scales lety n t be a solution y i n t i n n of the system of stochastic differential equations w i t are i i d brownian motions dy i n t left frac 1 2n 1 sum limits j n n y j n t y i n t right dt sqrt 2g y i n t dw i t in the mckean vlasov limit n we obtain the infinite independent system dy i infty t e y i infty t y i infty t dt sqrt 2g y i infty t dw i t i in z this infinite system has a one parameter set of invariant measures v theta mathop otimes limits x in z gamma theta with the equilibrium measure of dx t theta x t dt sqrt 2g x t dw t letq s be the transition kernel of the diffusion with generator u g x left frac partial partial x right 2 with u g x int g y gamma x dy then one main result is that asn mathcal l y n s 2n 1 rightarrow int q s theta d theta v theta theta e y 0 this provides a new example of a phenomenon also exhibited by the voter model and branching random walk in particular we are also able to modify our model by adding the termcn 1 a y i n t dt to obtain the first example in which the analog ofq s converges to an honest equilibrium instead of absorption in traps as in all models previously studied in the literature finally we discuss a hierarchical model with two levels from the point of view discussed above but now in two fast time scales

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

periodic and solitary wave solutions for a generalized variable coefficient boitileonpempinlli system in this paper a generalized variable coefficient boitileonpempinlli blp system is studied via the modified clarkson and kruskal ck direct reduction method connected with homogeneous balance hb method which can describe the water waves in fluid physics a direct similarity reduction to nonlinear ordinary differential system is obtained by solving the reduced ordinary differential system new analytical solutions including solitary and periodic types in terms of jacobi elliptic functions are given for the variable coefficient blp system

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

decomposition method for solving fractional riccati differential equations in this article we implement a relatively new analytical technique the adomian decomposition method for solving fractional riccati differential equations the fractional derivatives are described in the caputo sense in this scheme the solution takes the form of a convergent series with easily computable components the diagonal pade approximants are effectively used in the analysis to capture the essential behavior of the solution the corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations some numerical examples are presented to illustrate the efficiency and reliability of the method

numerical method for solving fractional coupled burgers equations in this paper we use the fractional variational iteration method fvim to solve a time and space fractional coupled burgers equations some numerical examples are presented to show the efficiency of considered method a comparison of the proposed method is made with the exact solution adomain decomposition method adm generalized differential transformation method gdtm and homotopy perturbation method hpm

a variety of exact travelling wave solutions for the 2 1 dimensional boiti leon pempinelli equation abstract in this paper the 2 1 dimensional boiti leon pempinelli blp equation will be studied the tanh coth method will be used to obtain exact travelling wave solutions for this equation the exp function method will also be applied to the blp equation to derive a new variety of travelling wave solutions with distinct physical structures

a collocation method for fractional diffusion equation in a long time with chebyshev functions abstract in this paper our aim is to find a new numerical method for diffusion equation with fractional derivative on time and space the employed fractional derivative is in the caputo sense also by employing a class of shifted chebyshev polynomials for the space area and a collection of rational chebyshev functions for the time domain and then using collocation method we obtain an algebraic system of equations the convergence estimate of the new scheme have been concluded finally we evaluate results of this method with other numerical methods

approximate analytical solution of the nonlinear fractional kdv burgers equation in this paper explicit and approximate solutions of the nonlinear fractional kdv burgers equation with time space fractional derivatives are presented and discussed the solutions of our equation are calculated in the form of rabidly convergent series with easily computable components the utilized method is a numerical technique based on the generalized taylor series formula which constructs an analytical solution in the form of a convergent series five illustrative applications are given to demonstrate the effectiveness and the leverage of the present method graphical results and series formulas are utilized and discussed quantitatively to illustrate the solution the results reveal that the method is very effective and simple in determination of solution of the fractional kdv burgers equation

phase transitions for the long time behavior of interacting diffusions let xi t i zd t 0 be the system of interacting diffusions on 0 defined by the following collection of coupled stochastic differential equations dxi t j zd a i j xj t xi t dt bxi t dwi t i z t 0 here a is an irreducible random walk transition kernel on z z b 0 is a diffusion parameter and wi t i zd t 0 is a collection of independent standard brownian motions on r the initial condition is chosen such that xi 0 i zd is a shift invariant and shift ergodic random field on 0 with mean 0 the evolution preserves the mean we show that the long time behaviour of this system is the result of a delicate interplay between a and b in contrast to systems where the diffusion function is subquadratic in particular let i j 12 a i j a j i i j z denote the symmetrised transition kernel we show that a if is recurrent then for any b 0 the system locally dies out b if is transient then there exist b b2 0 such that b1 the system converges to an equilibrium with mean if 0 b b3 has a finite 2 nd moment if and only if 0 b2 the equilibrium is shown to be associated and mixing for all 0 b2 we further conjecture that the system locally dies out at b b for the case where a is symmetric and transient we further show that c there exists a sequence b2 b3 b4 0 such that c1 has a finite m th moment if and only if 0 bm c3 b2 m 1 bm 2 c4 limm m 1 bm c supm 2 m 1 bm mathematisches institut universitat erlangen nurnberg bismarckstrasse 1 1 2 d 91504 erlangen germany greven mi uni erlangen de mathematical institute leiden university p o box 9512 2300 ra leiden the netherlands denholla math leidenuniv nl eurandom p o box 513 5600 mb eindhoven the netherlands

an oscillation criterion for certain second order strongly sublinear differential equations on considere l equation differentielle d ordre 2 x t q t f x t g x t 0 ou q t 0 0 f g r r sont continues q t non identiquement nulle sur un rayon de la forme t pour t t o xf x 0 pour x 0 et g y 0 pour y 0 on donne un critere d oscillation pour les solutions propres

different clustering regimes in systems of hierarchically interacting diffusions we study a system of interacting diffusions dx t a x t x t dt g x t dw t indexed by the hierarchical group as a genealogical two genotype model where x t denotes the frequency of say type a with hierarchically determined degrees of relationship between colonies in the case of short interaction range it is known that the system clusters that is locally one genotype dies out we focus on the description of the different regimes of cluster growth which is shown to depend on the interaction kernel a via its recurrent potential kernel one of these regimes will be further investigated by mean field methods for general interaction range we shall also relate the behavior of large finite systems indexed by finite subsets n of to that of the infinite system on the way we will establish relations between hitting times of random walks and their potentials

the variational iteration method a highly promising method for solving the system of integro differential equations this paper applies he s variational iteration method for solving two systems of volterra integro differential equations the solution process is illustrated and various physically relevant results are obtained comparison of the obtained results with exact solutions shows that the used method is an effective and highly promising method for various classes of both linear and nonlinear integro differential equations

variational iteration method new development and applications variational iteration method has been favourably applied to various kinds of nonlinear problems the main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper recent trends and developments in the use of the method are reviewed major applications to nonlinear wave equation nonlinear fractional differential equations nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed the confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods variational iteration method is uniquely qualified to address this challenge the flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions this paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations a new iteration formulation is suggested to overcome the shortcoming a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested examples are given to illustrate the solution procedure

variational iteration method for solving burger s and coupled burger s equations by means of variational iteration method the solutions of burger s equation and coupled burger s equations are exactly obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

on the solution of nonlinear fractional order differential equation abstract this paper investigates the uniqueness and dependence of the solution of nonlinear differential equation with fractional order by the fixed points theorem the obtained results include naturally those in open literature for some special cases and a numerical approach to the discussed problem is suggested

numerical solution of a biological population model using he s variational iteration method this paper presents numerical solution of a degenerate parabolic equation arising in the spatial diffusion of biological populations the variational iteration method and adomian decomposition method are used for solving this equation and then numerical results are compared with each other showing that the variational iteration method leads to more accurate results furthermore variational iteration method overcomes the difficulty arising in calculating the adomian s polynomials which is an important advantage over the adomian decomposition method

application of the variational iteration method to the regularized long wave equation this paper applies the variational iteration method to the regularized long wave equation in this method a generalized lagrange multiplier is introduced to construct a correction functional for the discussed problem the multiplier can be identified optimally via variational theory three invariants of the motion are evaluated to determine the conservation properties of the system

application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives two nonlinear anelastic models with fractional derivatives describing the properties of a series of materials as polymers and polycrystalline materials are presented in this paper these models are studied analytically using a variational iteration method the paper clarifies the different ways in which the fractional differentiation operator can be defined a volterra series method of model parameters identification from the experimental data is also presented

renormalization of hierarchically interacting isotropic diffusions we study a renormalization transformation arising in an infinite system of interacting diffusions the components of the system are labeled by the n dimensional hierarchical lattice n 2 and take values in the closure of a compact convex set bar d subset mathbb r d d geqslant 1 each component starts at some d and is subject to two motions 1 an isotropic diffusion according to a local diffusion rate g bar d to 0 infty chosen from an appropriate class 2 a linear drift toward an average of the surrounding components weighted according to their hierarchical distance in the local mean field limit n block averages of diffusions within a hierarchical distance k on an appropriate time scale are expected to perform a diffusion with local diffusion rate f k g where f k g f c k circ circ f c 1 g is the kth iterate of renormalization transformations fc c 0 applied to g here the ck measure the strength of the interaction at hierarchical distance k we identify fc and study its orbit f k g k 0 we show that there exists a fixed shape g such that limk kf k g g for all g where the k are normalizing constants in terms of the infinite system this property means that there is complete universal behavior on large space time scales our results extend earlier work for d 1 and bar d 0 1 resp 0 the renormalization transformation fc is defined in terms of the ergodic measure of a d dimensional diffusion in d 1 this diffusion allows a yamada watanabe type coupling its ergodic measure is reversible and the renormalization transformation fc is given by an explicit formula all this breaks down in d 2 which complicates the analysis considerably and forces us to new methods part of our results depend on a certain martingale problem being well posed

variational iteration method for solving the epidemic model and the prey and predator problem in this article two problems are considered the first one is the problem of the spread of a non fatal disease in a population which is assumed to have constant size over the period of the epidemic and the other one is the problem of prey and predator the variational iteration method is employed to compute an approximation to the solutions of the systems of nonlinear ordinary differential equations governing on these problems the results are compared with the results obtained by adomian decomposition and power series methods some plots are presented to show the reliability and simplicity of the method

application of he s variational iteration method to nonlinear jaulent miodek equations and comparing it with adm instead of finding a small parameter for solving nonlinear problems through perturbation method a new analytical method called he s variational iteration method vim is introduced to be applied to solve nonlinear jaulent miodek coupled kdv and coupled mkdv equations in this article in this method general lagrange multipliers are introduced to construct correction functionals for the problems the multipliers can be identified optimally via the variational theory the results are compared with exact solutions

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

oscillation theorems for nth order differential equations with deviating arguments abstract this paper is concerned with the study of the oscillatory behavior of solutions of the n th order nonlinear differential equation a t x n v t v q t f x g t 0 where n is even and 1 v n 1 a systematic study is attempted which extends and correlates a number of existing results

variational iteration method for solving nonlinear boundary value problems abstract in this paper he s variational iteration method is implemented to give approximate and analytical solutions for a class of boundary value problems the variational iteration method which produces the solutions in terms of convergent series requiring no linearization or small perturbation numerical examples are given and comparisons are made with the adomian decomposition method the fact that this method solves nonlinear equations without using adomian polynomials can be considered as an advantage of this method over adomian decomposition method

the tanh method and adomian decomposition method for solving the foam drainage equation foaming occurs in many distillation and absorption processes the drainage of liquid foams involves the interplay of gravity surface tension and viscous forces in this paper we use a semi analytic method the adomian decomposition method and an analytic method the tanh method to handle the foam drainage equation the powerful tanh method gives the solution in a closed form however adomian decomposition method computes the solution in a rapidly convergent infinite series the comparison between the two approaches is conducted to illustrate the performance of each method

variational approach to solitary wave solution of the generalized zakharov equation he s semi inverse method is applied to search for the solitary wave solution of the generalized zakharov equation gze the solution process reveals that the method is easy and straightforward

lie symmetry analysis backlund transformations and exact solutions of a 2 1 dimensional boiti leon pempinelli system in this paper the lie symmetry analysis method is employed to investigate the lie point symmetries and the one parameter transformation groups of a 2 1 dimensional boiti leon pempinelli system by using ibragimov s method the optimal system of one dimensional subalgebras of this system is constructed truncated painleve analysis is used for deriving the backlund transformation the method of constructing lump type solutions of integrable equations by means of backlund transformation is first presented meanwhile the lump type solutions of the 2 1 dimensional boiti leon pempinelli system are obtained the lump type wave is one kind of rogue wave the fusion type n solitary wave solutions are also constructed in addition this system is integrable in terms of the consistent riccati expansion method

generalized continuous time random walks master equations and fractional fokker planck equations continuous time random walks which generalize random walks by adding a stochastic time between jumps provide a useful description of stochastic transport at mesoscopic scales the continuous time random walk model can accommodate certain features such as trapping which are not manifest in the standard macroscopic diffusion equation the trapping is incorporated through a waiting time density and a fractional diffusion equation results from a power law waiting time a generalized continuous time random walk model with biased jumps has been used to consider transport that is also subject to an external force here we have derived the master equations for continuous time random walks with space and time dependent forcing for two cases when the force is evaluated at the start of the waiting time and at the end of the waiting time the differences persist in low order spatial continuum approximations however the two processes are shown to be governed by the same fokker planck equations in the diffusi

numerical treatment for solving the perturbed fractional pdes using hybrid techniques in this article an accurate numerical approach is introduced in this approach we mixed between the fractional finite difference method and the restrictive taylor approximation rta the proposed method is implemented to solve numerically the perturbed fractional partial differential equations fpdes special attention is given to study the stability and consistency of the method by means of gerschgorin theorem and using the stability matrix analysis two numerical examples are given and compared with the exact solution

a fractional variational iteration method for solving fractional nonlinear differential equations recently fractional differential equations have been investigated by employing the famous variational iteration method however all the previous works avoid the fractional order term and only handle it as a restricted variation a fractional variational iteration method was first proposed in g c wu e w m lee fractional variational iteration method and its application phys lett a 374 2010 2506 2509 and gave a generalized lagrange multiplier in this paper two fractional differential equations are approximately solved with the fractional variational iteration method

the tanh method a tool for solving certain classes of nonlinear evolution and wave equations the tanh or hyperbolic tangent method is a powerful technique to search for travelling waves coming out from one dimensional nonlinear wave and evolution equations in particular in those problems where dispersive effects reaction diffusion and or convection play an important role to show the strength of the method an overview is given to find out which kind of problems are solved with this technique and how in some nontrivial cases this method adapted to the problem at hand still can be applied single as well as coupled equations arising from wave phenomena which appear in different scientific domains such as physics chemical kinetics geochemistry and mathematical biology will be treated next attention is focussed towards approximate solutions as a result solitary and shock wave profiles are derived together with the associated width and velocity the same method can be easily extended so that difference differential equations can be similarly solved finally some extensions of the method are discussed

some more similarity solutions of the 2 1 dimensional blp system some more similarity solutions of the coupled boiti leon pempinelli system have been obtained by using similarity transformations method via lie group theory the solutions are attained under the restrictions imposed on the functions involved in infinitesimals then solutions are investigated through the graphical representation in different cases the cases depend upon some suitable choice of arbitrary functions involved there i e in the solutions to get physically meaningful structures analytical solutions are supplemented by numerical simulations with appropriate choice of arbitrary functions and constants eventually profiles show single soliton elastic multisolitons quasi periodic parabolic and asymptotic nature of solutions

oscillation theorems for nth order delay differential equations abstract oscillation criteria are obtained for a class of functional differential equations including the generalized emden fowler equation x n q t x g t sgn x g t 0 0 as a special case the purpose of this paper is to extend and improve known fundamental oscillation criteria for superlinear sublinear and linear differential equations with retarded arguments

ergodic theorems for infinite systems of locally interacting diffusions let x t x i t i in mathbb z d be the solution of the system of stochastic differential equations dx i t bigg sum j in mathbb z d a i j x j t x i t bigg dt sqrt 2g x i t dw i t quad i in mathbb z d here g lbrack 0 1 rbrack rightarrow mathbb r satisfies g 0 on 0 1 g 0 g 1 0 g is lipschitz a i j is an irreducible random walk kernel on mathbb z d and w i t i in mathbb z d is a family of standard independent brownian motions on mathbb r x t is a markov process on x lbrack 0 1 rbrack mathbb z d this class of processes was studied by notohara and shiga the special case g v v 1 v has been studied extensively by shiga we show that the long term behavior of x t depends only on hat a i j a i j a j i 2 and is universal for the entire class of g considered if hat a i j is transient then there exists a family nu theta theta in lbrack 0 1 rbrack of extremal translation invariant equilibria each nu theta is mixing and has density theta int x 0 d nu theta if hat a i j is recurrent then the set of extremal translation invariant equilibria consists of the point masses delta 0 delta 1 the process starting in a translation invariant shift ergodic measure mu on x with int x 0 d mu theta converges weakly as t rightarrow infty to nu theta if hat a i j is transient and to 1 theta delta 0 theta delta 1 if hat a i j is recurrent our results in the recurrent case remove a mild assumption on g imposed by notohara and shiga for the case hat a i j transient we use methods developed for infinite particle systems by liggett and spitzer for the case hat a i j recurrent we use a duality comparison argument

some oscillation criteria for delay differential equations of even order abstract some new oscillation criteria for the delay differential equations of the form a t x n v v q t x g 1 t h x g 2 t 0 d dt where n is even and 1 v n 1 are established

hierarchical models of interacting diffusions multiple time scale phenomena phase transition and pattern of cluster formation the purpose of this paper is to explore the connection between multiple space time scale behaviour for block averages and phase transitions respectively formation of clusters in infinite systems with locally interacting components the essential object is the associated markov chain which describes the joint distribution of the block averages at different time scales a fixed point and stability property of a particular dynamical system under a renormalisation procedure is used to explain this pattern of cluster formation and the fact that the longtime behaviour is universal in entire classes of evolutions

relative controllability of fractional dynamical systems with distributed delays in control this paper is concerned with the global relative controllability of fractional dynamical systems with multiple delays in control for finite dimensional spaces sufficient conditions for controllability results are obtained using schauder s fixed point theorem and the controllability grammian matrix which is defined by the mittag leffler matrix function an example is provided to illustrate the theory

exact solutions of some nonlinear systems of partial differential equations by using the first integral method abstract in recent years many approaches have been utilized for finding the exact solutions of nonlinear systems of partial differential equations in this paper the first integral method introduced by feng is adopted for solving some important nonlinear systems of partial differential equations including kdv kaup boussinesq and wu zhang systems analytically by means of this method some exact solutions for these systems of equations are formally obtained the results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations

resolvent estimates for fleming viot operators and uniqueness of solutions to related martingale problems we define a two parameter scale of banach spaces contained in the contininuous functions on the space of probability measures on a compact metric space x and show that the resolvent of the fleming viot process is a bounded operator in the scale we use this result to prove uniqueness of solutions to the martingale problem for fleming viot operators whose mutation operator a a and sampling rate x y may depend on the measure variable m1 x provided they are sufficiently close to constant in a sense made precise by the norms defining the scale we prove an existence result with no such restriction on the coefficients

oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments the oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations with deviating arguments of the form e lnx t q t f x g1 t x gm t 0 where 1 and l0x t x t lkx t ak t lk 1x t k 1 2 n ddt is examined a classification of solutions of e with respect to their behavior as t and their oscillatory character is obtained the comparisons of e 1 and e 1 with first and second order equations of the form y t c1 t f y g1 t y gm t 0 and an 1 t z t c2 t f z g1 t z gm t 0 respectively are presented the obtained results unify extend and improve some of the results by graef grammatikopoulos and spikes philos and staikos

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

multiple soliton solutions for the kp equation by hirota s bilinear method and by the tanh coth method abstract in this work we use the hirota s direct method combined with the simplified version of this method to determine the n soliton solutions n 1 2 3 4 5 for the kadomtsev petviashvili kp equation the one soliton solutions will be handled by the tanh coth method the work emphasizes the power of the two methods that will be used

on the oscillatory and asymptotic behavior of even order nonlinear differential equations with retarded arguments abstract some new oscillation criteria for the retarded differential equations of the form d dt a n 1 t d dt a 2 t d dt a 1 t dx t dt q t x g 1 t h dx g 2 t dt 0 n is even are established

an improvement on the exp function method when balancing the highest order linear and nonlinear terms abstract the exp function method has been widely used to solve different kinds of nonlinear partial differential equations these nonlinear partial differential equations are transformed first into nonlinear ordinary differential equations and then the ansatz of the exp function method u n c p a n exp n m d q b m exp m is applied to obtain the solution however a part of the solution using this method is to construct the relations between c p d and q by balancing the highest order linear term with the highest order nonlinear term it is proved in the present paper that c d and p q are the only relations that can be obtained by applying this method to any nonlinear ordinary differential equation therefore the additional calculations of balancing the highest order linear term with the highest order nonlinear term are not longer required in future hence the method becomes more straightforward

on the attracting orbit of a nonlinear transformation arising from renormalization of hierarchically interacting diffusions abstract this paper analyzes the n fold composition of a non linear integral operator acting on a class of functions on 0 attracting orbits and attracting fixed points are identified various results of convergence to these orbits and to these fixed points are derived the proofs are based on order preserving properties and comparison techniques a key role is played by the eigenfunctions of the operator which are used as comparison objects the results imply that the space time scaling limit of an infinite system of interacting diffusions has universal behavior independent of model parameters the paper can be read independently of part i

application of he s methods to nonlinear chemistry problems in this paper he s variational iteration method vim and homotopy perturbation method hpm are implemented for solving analytically two systems of ordinary differential equations that often appear in chemical applications in both methods the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions the solution procedure is of utter simplicity and the obtained solution is of high accuracy

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

fourth order integro differential equations using variational iteration method in this paper the variational iteration method proposed by ji huan he is applied to solve both linear and nonlinear boundary value problems for fourth order integro differential equations the numerical results obtained with minimum amount of computation are compared with the exact solutions to show the efficiency of the method the results show that the variational iteration method is of high accuracy more convenient and efficient for solving integro differential equations

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations