classification of systems of nonlinear ordinary differential equations with superposition principles a system of n first order nonlinear ordinary differential equations x t f x t is said to admit a superposition principle if its general solution can be written as a function of a finite number m of particular solutions and n constants such a system can be associated with the nonlinear action of a lie group g on a space m we show that indecomposable systems of ode s with supersposition principles are obtained if and only if the lie algebras l0 l corresponding to the isotropy group h of a point and the group g respectively define a transitive primitive filtered lie algebra l l0 using known results from the theory of transitive primitive lie algebras we deduce that l0 must be a maximal subalgebra of l and that g must be an affine group a simple lie group or the direct product of two identical simple lie groups affine groups lead to linear equations the other types to nonlinear equations with polynomial or rational nonlinearities equations corresponding to the classical complex lie algebra

analytic study of the fifth order integrable nonlinear evolution equations by using the tanh method abstract the tanh method is used to derive explicit travelling wave solutions for two fifth order integrable nonlinear evolution equations the fifth order kdv equation fkdv and the fifth order caudrey dodd gibbon cdg equation are used as vehicles for our analysis new families of solitary wave solutions and periodic solutions are formally obtained for these equations the presented framework reveals quite a number of useful features of the method used here

generalized riccati equation expansion method and its application to the 3 1 dimensional jumbo miwa equation based on the computerized symbolic system maple and a riccati equation a generalized riccati equation expansion method for constructing soliton like solutions of non linear evolution equations nees is presented by a new general ansatz the proposed method is more powerful than most of the existing tanh methods the extended tanh function method the modified extended tanh function method and generalized hyperbolic function method by use of the method we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nees making use of the method we study the the 3 1 dimensional jumbo miwa equation and obtain rich new families of the exact solutions including the non travelling wave and coefficient functions soliton like solutions singular soliton like solutions periodic form solutions

a new generalized riccati equation rational expansion method to a class of nonlinear evolution equations with nonlinear terms of any order abstract with the aid of symbolic computation a new algebraic method named generalize riccati equation rational expansion grere method is devised for constructing travelling wave solutions for nonlinear evolution equations nees with nonlinear terms of any order compared with most existing tanh methods for finding travelling wave solutions the proposed method not only recovers the results by most known algebraic methods but also provides new and more general solutions for illustration we apply the generalized method to a class of nonlinear evolution equations with nonlinear term of any order as a result we obtain several new kinds of exact solutions this approach can also be applied to other nonlinear evolution equations with nonlinear terms of any order

a nonlinear conjugate gradient method with a strong global convergence property conjugate gradient methods are widely used for unconstrained optimization especially large scale problems the strong wolfe conditions are usually used in the analyses and implementations of conjugate gradient methods this paper presents a new version of the conjugate gradient method which converges globally provided the line search satisfies the standard wolfe conditions the conditions on the objective function are also weak being similar to those required by the zoutendijk condition

novel interacting phenomena in 2 1 dimensional akns system abstract by means of an new auxiliary equation method based on mapping method and the second order linear ordinary differential equation some types of variable separation solutions with two arbitrary lower dimensional functions of the 2 1 dimensional ablowitz kaup newell segur system are derived based on the derived variable separation excitation some new special types of non localized solutions such as doubly periodic wave and line periodic wave are constructed by choosing appropriate functions in addition we also investigate the annihilation phenomenon and non elastic interaction graphically

new exact solutions for mch and mdp equations by auxiliary equation method abstract with the aid of symbolic computation auxiliary equation method is introduced to investigate modified forms of camassa holm and degasperis procesi equations a series of new exact traveling wave solutions including smooth solitary wave solution peakons singular solution periodic wave solution jacobi elliptic solution are obtained in general form these new exact solutions will enrich previous results and help us further understand the physical structures of these two nonlinear equations

a new method to obtain either first or second order reductions for parametric polynomial odes abstract for a n th order ordinary differential equation that is either polynomial on the dependent variable and its derivatives or polynomial in the derivatives of the dependent variable a new method to obtain either first or second order reductions respectively is introduced the corresponding reduced equations are generalized polynomials in the dependent variable or in its derivative the method can be applied to equations that may have integer parameters in the exponents of the polynomial of the initial equation a procedure to determine in function of the parameters if the given equation may admit such reductions and to restrict the possible variations of the exponents in the reduced equations is also provided the introduced method strictly generalizes several methods that have recently appeared in the literature as applications of the method this study considers first order reductions for a generalized emden fowler equation and second order reductions for the travelling waves equation of a fifth order korteweg de vries equation and for a painleve chazy equation as far as we know some of the reported reductions are new several maple programs to determine if a given polynomial equation admits any of the considered reductions and if this is the case to determine the reduced equations are also included

a new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

finite element exterior calculus from hodge theory to numerical stability this article reports on the confluence of two streams of research one emanating from the fields of numerical analysis and scientific computation the other from topology and geometry in it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de rham cohomology and hodge theory are key tools for exploring the well posedness of the continuous problem the discretization methods we consider are finite element methods in which a variational or weak formulation of the pde problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace after a brief introduction to finite element methods we develop an abstract hilbert space framework for analyzing the stability and convergence of such discretizations in this framework the differential complex is represented by a complex of hilbert spaces and stability is obtained by transferring hodgetheoretic structures that ensure well posedness of the continuous problem from the continuous level to the discrete we show stable discretization is achieved if the finite element spaces satisfy two hypotheses they can be arranged into a subcomplex of this hilbert complex and there exists a bounded cochain projection from that complex to the subcomplex in the next part of the paper we consider the most canonical example of the abstract theory in which the hilbert complex is the de rham complex of a domain in euclidean space we use the koszul complex to construct two families of finite element differential forms show that these can be arranged in subcomplexes of the de rham complex in numerous ways and for each construct a bounded cochain projection the abstract theory therefore applies to give the stability and convergence of finite element approximations of the hodge laplacian other applications are considered as well especially the elasticity complex and its application to the equations of elasticity background material is included to make the presentation self contained for a variety of readers

a fourth order numerical scheme for solving the modified burgers equation a finite difference scheme based on fourth order rational approximants to the matrix exponential term in a two time level recurrence relation is proposed for the numerical solution of the modified burgers equation the resulting nonlinear system which is analyzed for stability is solved using an already known modified predictor corrector scheme the results arising from the experiments are compared with the corresponding ones known from the available literature

the extended hyperbolic functions method and new exact solutions to the zakharov equations abstract the multiple exact solutions for the nonlinear evolution equations describing the interaction of laser plasma are developed the extended hyperbolic function method are employed to reveal these new solutions the solutions include that of the solitary wave solutions of bell type for n and e the solitary wave solutions of kink type for e and bell type for n the solitary wave solutions of a compound of the bell type and the kink type for n and e the singular traveling wave solutions periodic traveling wave solutions of triangle function types and solitary wave solutions of rational function types in addition to re deriving all known solutions in a systematic way several new and more general solutions can be obtained by using our method

some special types of solutions of a class of the n 1 dimensional nonlinear wave equations some special types of solutions with arbitrary functions for the n 1 dimensional sine cosine gordon equation the n 1 dimensional double sinh gordon equation and the n 1 dimensional sinh cosinh gordon equations are constructed by means of the separation transformation approach as an illustrative sample the properties of some solutions are shown by their figures

a new conjugate gradient method with guaranteed descent and an efficient line search a new nonlinear conjugate gradient method and an associated implementation based on an inexact line search are proposed and analyzed with exact line search our method reduces to a nonlinear version of the hestenes stiefel conjugate gradient scheme for any inexact line search our scheme satisfies the descent condition bf g k sf t bf d k le frac 7 8 bf g k 2 moreover a global convergence result is established when the line search fulfills the wolfe conditions a new line search scheme is developed that is efficient and highly accurate efficiency is achieved by exploiting properties of linear interpolants in a neighborhood of a local minimizer high accuracy is achieved by using a convergence criterion which we call the approximate wolfe conditions obtained by replacing the sufficient decrease criterion in the wolfe conditions with an approximation that can be evaluated with greater precision in a neighborhood of a local minimum than the usual sufficient decrease criterion numerical comparisons are given with both l bfgs and conjugate gradient methods using the unconstrained optimization problems in the cute library

exact solutions to the two dimensional korteweg de vries burgers equation two methods are described for obtaining exact travelling monotone shock wave solutions to the two dimensional korteweg de vries burgers equation they are compared with two methods that have been reported recently by other authors

a note about wyl s conjugate gradient method and its applications this paper reviews the development of different versions of nonlinear conjugate gradient methods shortly the special attentions were given to the wyl method which was proposed by zengxin wei et al the convergence properties of some conjugate gradient methods applied mathematics and computation 183 2006 1341 1350 and its applications

link between solitary waves and projective riccati equations many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective hence linearizable riccati system from that property the authors deduce a method for building these solutions by determining only a finite number of coefficients this method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation they handle several examples for the henon heiles hamiltonian system they obtain several exact solutions one of them defines a new solitary wave solution for a coupled system of boussinesq and nonlinear schrodinger equations for a third order dispersive equation with two monomial nonlinearities they isolate all cases where the general solution is single valued

the proof of the sufficient descent condition of the wei yao liu conjugate gradient method under the strong wolfe powell line search abstract in z wei et al the convergence properties of some conjugate gradient methods appl math comput 183 2006 1341 1350 wei et al proposed a new conjugate gradient method called wyl method which has good numerical experiments under the sufficient descent condition and wwp line search the given method s convergent results have been established by wei et al so the sufficient descent condition is important for the wyl method in this paper we prove that by restricting the parameter 1 4 under the swp line search condition the wyl method possess the sufficient descent condition which deduces the global convergent results of the wyl method under swp line search

superposition principles for matrix riccati equations a superposition rule is obtained for the matrix riccati equation mre w a wb cw wdw where w t a t b t c t and d t are real n n matrices expressing the general solution in terms of five known solutions for all n 2 the symplectic mre w wt a at d dt c bt is treated separately and a superposition rule is derived involving only four known solutions for the unitary and gl n r subcases with d a and c bt or d a and c bt respectively superposition rules are obtained involving only two solutions the derivation of these results is based upon an interpretation of the mre in terms of the action of the groups sl 2n r sp 2n r u n and gl n r on the grassman manifold gn r2n

local bounded cochain projections we construct projections from h k the space of differential k forms on which belong to l2 and whose exterior derivative also belongs to l2 to finite dimensional subspaces of h k consisting of piecewise polynomial differential forms defined on a simplicial mesh of thus their definition requires less smoothness than assumed for the definition of the canonical interpolants based on the degrees of freedom moreover these projections have the properties that they commute with the exterior derivative and are bounded in the h k norm independent of the mesh size h unlike some other recent work in this direction the projections are also locally defined in the sense that they are defined by local operators on overlapping macroelements in the spirit of the clement interpolant a double complex structure is introduced as a key tool to carry out the construction

systems of ordinary differential equations with nonlinear superposition principles abstract this work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations x x t x r n t r in terms of a finite number of particular solutions the point of departure is lie s criterion according to which such a rule exists if and only if the vector fields x t generate a finite dimensional lie algebra we provide three different constructive methods for deriving superposition rules and apply them to systems of coupled riccati equations of the projective and conformal types based respectively on the lie algebra sl n 1 r and o p 1 n p 1

the pinney equation and its discretization the pinney equation is part of the original ermakov system which has been the subject of intensive study recently here we show that it may be related to a two dimensional conformal riccati equation leading to a new method for its linearization a discrete analogue of the pinney equation is constructed using the above connection with the conformal group an alternative discretization is obtained by using a discrete schwarz derivative both of these nonlinear difference equations are linearizable

on the integrability of systems of nonlinear ordinary differential equations with superposition principles a new class of solvable nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations ode s describing each member of this class possess nonlinear superposition principles these systems of ode s are generally not derived from a hamiltonian and are classified by associated pairs of lie algebras of vector fields in this paper all such systems of n 3 ode s are integrated in a unified way by finding explicit integrals for them and relating them all to a pivotal member of their class the projective riccati equations moreover by perturbing two parametrically driven projective riccati equations thus making them nonsolvable in the above sense evidence is discovered of chaotic behavior on the poincare surface of section in the form of sensitive dependence on initial conditions near a boundary separating bounded from unbounded motion

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

generalized tanh method with symbolic computation and generalized shallow water wave equation abstract we report that symbolic computation with the generalized tanh method leads to new soliton like solutions for a 2 1 dimensional generalization of the shallow water wave equations

new exact traveling wave solutions for the klein gordon zakharov equations based on the extended hyperbolic functions method we obtain the multiple exact explicit solutions of the klein gordon zakharov equations the solutions obtained in this paper include a the solitary wave solutions of bell type for u and n b the solitary wave solutions of kink type for u and bell type for n c the solitary wave solutions of a compound of the bell type and the kink type for u and n d the singular traveling wave solutions e periodic traveling wave solutions of triangle function types and solitary wave solutions of rational function types we not only rederive all known solutions of the klein gordon zakharov equations in a systematic way but also obtain several entirely new and more general solutions the variety of structures of the exact solutions of the klein gordon zakharov equations is illustrated

global convergence properties of conjugate gradient methods for optimization this paper explores the convergence of nonlinear conjugate gradient methods without restarts and with practical line searches the analysis covers two classes of methods that are globally convergent on smooth nonconvex functions some properties of the fletcher reeves method play an important role in the first family whereas the second family shares an important property with the polak ribiere method numerical experiments are presented

a bounded linear integrator for some diffusive nonlinear time dependent partial differential equations we propose a numerical method to approximate the solutions of generalized forms of two bi dimensional models of mathematical physics namely the burgers fisher and the burgers huxley equations in one dimensional form the literature in the area gives account of the existence of analytical solutions for both models in the form of traveling wave fronts bounded within an interval i of the real numbers motivated by this fact we propose a finite difference methodology that guarantees that under certain analytical conditions on the model and computer parameters estimates within i will evolve discretely into new estimates which are likewise bounded within i additionally we establish the preservation in the discrete domain of the skew symmetry of the solutions of the models under study our computational implementation of the method confirms numerically that the properties of positivity and boundedness are preserved under the analytical constraints derived theoretically our simulations show a good agreement between the analytical solutions derived in the present work and the corresponding numerical approximations

the convergence properties of some new conjugate gradient methods abstract in this paper a new conjugate gradient formula k is given to compute the search directions for unconstrained optimization problems general convergence results for the proposed formula with some line searches such as the exact line search the wolfe powell line search and the grippo lucidi line search are discussed under the above line searches and some assumptions the global convergence properties of the given methods are discussed the given formula k 0 and has the similar form with k prp preliminary numerical results show that the proposed methods are efficient

auto backlund transformation and similarity reductions for coupled burger s equation in this work we applied backlund transformation and similarity reduction for coupled burger s equation clarkson and kruskal developed a direct and simple method to obtain more similarity solutions of nonlinear partial differential equation we received our inspiration from fan s article as far as we think that our work is an extended one from the fan s has done in his paper as a result of this study we obtained solitary wave solutions and traveling wave solutions of coupled burger s equation

soliton solutions and doubly periodic wave solutions for a new generalized hirota satsuma coupled system in this paper a new algebraic method with symbolic computation is presented to uniformly construct a series of exact solutions for general nonlinear equations compared with most existing tanh methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the solutions according to the values of some parameters for illustration we apply the proposed method to solve a new generalized hirota satsuma coupled system and explicitly construct a series of exact solutions which include the soliton solutions and elliptic doubly periodic solutions as special cases in addition the links among our proposed method the tanh method and the extended method are also clarified generally

persistence of travelling wave solutions of a fourth order diffusion system in this paper the extended burgers huxley equation with the fourth order derivative is considered first the convergence to the uniform steady state is proved which means the solution of the equation with positive initial data will remain positive for time t sufficiently large then the persistence of the travelling wave solution for the extended equation on the unbounded domain is investigated we have proved that this solution will persist under small perturbation of the equation

traveling wave solutions for nonlinear equations using symbolic computation abstract in this paper a direct and unified algorithm for constructing multiple travelling wave solutions of nonlinear partial differential equations pdes is presented and implemented in a computer algebraic system the key idea of this method is to take full advantage of a riccati equation involving a parameter and use its solutions to replace the tanh function in the tanh method it is quite interesting that the sign of the parameter can be used to exactly judge the number and types of such travelling wave solutions in this way we can successfully recover the previously known solitary wave solutions that had been found by the tanh method and other more sophisticated methods more importantly for some equations with no extra effect we also find other new and more general solutions at the same time by introducing appropriate transformations our method is further extended to the nonlinear pdes whose balancing numbers may be any nonzero real numbers the efficiency of the method can be demonstrated on a large variety of nonlinear pdes such as those considered in this paper burgers huxley equation coupled korteweg de vries equation caudrey dodd gibbon kawada equation active dissipative dispersive media equation generalized fisher equation and nonlinear heat conduction equation

global convergence of the polak ribiere polyak conjugate gradient method with an armijo type inexact line search for nonconvex unconstrained optimization problems we propose two algorithms for nonconvex unconstrained optimization problems that employ polak ribiere polyak conjugate gradient formula and new inexact line search techniques we show that the new algorithms converge globally if the function to be minimized has lipschitz continuous gradients preliminary numerical results show that the proposed methods for particularly chosen line search conditions are very promising

on a conditionally stable nonlinear method to approximate some monotone and bounded solutions of a generalized population model in this work we design a numerical method to approximate some solutions of a generalization of a nonlinear diffusion reaction model which appears in the context of population dynamics the existence of traveling wave solutions for the equation under consideration is a well known fact some of such solutions are positive bounded from above and monotone in both space and time motivated by these facts we propose an explicit nonlinear finite difference methodology to approximate consistently the solutions of the model under investigation in the linear regime the method is consistent of first order in time and of second order in space under certain flexible parameter conditions the method is capable of preserving the positivity the boundedness and the spatial and the temporal monotonicity of the traveling wave solutions moreover we establish analytically and numerically that the nonlinear method is conditionally stable a computational implementation of our technique shows that the method preserves in practice the mathematical features of interest of the exact solutions considered

an algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations an algebraic method is devised to uniformly construct a series of exact solutions for general integrable and nonintegrable nonlinear evolution equations compared with most existing tanh methods the jacobi function expansion method or other sophisticated methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the solutions according to the values of some parameters the solutions obtained in this paper include a polynomial solutions b exponential solutions c rational solutions d triangular periodic wave solutions e hyperbolic and solitary wave solutions and f jacobi and weierstrass doubly periodic wave solutions the efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper new 2 1 dimensional calogero kdv equation 3 1 dimensional jimbo miwa equation symmetric regular long wave equation drinfel d sokolov wilson equation 2 1 dimensional generalized dispersive long wave equation double sine gordon equation calogero degasperis fokas equation and coupled schrodinger boussinesq equation in addition the links among our proposed method the tanh method the extended method and the jacobi function expansion method are also clarified generally

a skew symmetry preserving computational technique for obtaining the positive and the bounded solutions of a time delayed advection diffusion reaction equation abstract in this work we consider a one dimensional time delayed advective version of the well known fisher kolmogorov petrovsky piscounov equation from population dynamics which extends several models from mathematical physics including the classical wave equation the nonlinear klein gordon equation a fitzhugh nagumo equation from electrodynamics and the burgers huxley equation and the newell whitehead segel equation from fluid mechanics we propose a skew symmetry preserving finite difference scheme for approximating the solutions of the model under investigation and establish conditions on the model coefficients and the numerical parameters under which the method provides positive or bounded approximations for initial data which are likewise positive or bounded respectively the derivation of the conditions under which the positivity and the boundedness of the approximations is guaranteed is based on the properties of the inverses of m matrices in fact the conditions obtained here assure that the iterative method is described in vector form through the multiplication by a matrix of this type we provide simulations in order to show that the technique is indeed conditionally positivity preserving and boundedness preserving

a posteriori error estimates for maxwell equations maxwell equations are posed as variational boundary value problems in the function space h curl and are discretized by nedelec finite elements in beck et al 2000 a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain in the present paper we prove the reliability of that error estimator on lipschitz domains the key is to establish new error estimates for the commuting quasi interpolation operators recently introduced in j schoberl commuting quasi interpolation operators for mixed finite elements similar estimates are required for additive schwarz preconditioning to incorporate boundary conditions we establish a new extension result

a unified rational expansion method to construct a series of explicit exact solutions to nonlinear evolution equations a unified basic frame of rational expansion methods is presented which leads to closed form solutions of nonlinear evolution equations nlees the new unified algorithms are given to find exact rational formal polynomial solutions of nlees in terms of jacobi elliptic functions solutions of the riccati equation and solutions of the generalized riccati equation they can be implemented in symbolic computation system maple as applications of the methods we choose some physical significance nlees to illustrate the methods as a consequence we not only can successfully obtain the solutions found by most existing jacobi elliptic function methods and tanh methods but also find other new and more general solutions at the same time

exact travelling wave solutions of nonlinear evolution equations in 1 1 and 2 1 dimensions abstract based upon the symbolic computation system maple the projective riccati equation method is improved to seek for travelling wave solutions in 1 1 and 2 1 dimensional nonlinear evolution equations by introducing two new elementary bell shaped and kink shaped functions the validity and reliability of the method is tested by applying it to the 2 1 dimensional generalized of the shallow water wave equation potential kdv equation and burgers kdv equation

solitary solutions of some nonlinear evolution equations in this paper the traveling wave reduction combined with the homogeneous balance method hbm is used to find new exact solutions by using less studied nonlinear partial differential equations npdg of higher order the usual starting point is a special transformation converting the npdg in its two variables x and t into an nonlinear ordinary differential equation node in the single variable x using the hyperbolic tangent method new exact solutions for the three npdgs are studied the new feature of this paper is of course the fact that we are dealing with npdgs which cannot be found by studying the relevant literature therefore we believe it is time for publishing some special solutions of this interesting kind of equations clearly we do not use a new formalism but with the aid of the tanh method we are able to calculate at least exact traveling wave solutions solitary waves which can physically interpreted as an action of wave propagation simultaneous we point out the necessity of such sophisticated methods since a general theory of npdgs does not exist the existence of nontrivial noncomplex valued solutions of the ramani equation enables us to generate families of real valued solutions of a nonlinear sixth order system special differential transformations are leading to connections between traveling wave solutions of associated nonlinear evolution equations

application of he s exp function method for nonlinear evolution equations in this paper he s exp function method is used to construct solitary and soliton solutions of nonlinear evolution equations modified equal width burger huxley and nonlinear heat conduction equations are chosen to illustrate the effectiveness of the method the method is straightforward and concise and its applications are promising the exp function method presents a wider applicability for handling nonlinear wave equations

generalized algebraic method and new exact traveling wave solutions for 2 1 dimensional dispersive long wave equation abstract with the help of the symbolic computation system maple a new generalized algebraic method to uniformly construct solutions in terms of special function of nonlinear partial differential equations is presented by means of a more general ansatz as an application of the method we choose a 2 1 dimensional dispersive long wave equation to illustrate the method as a result we can successfully obtain the solutions found by the method proposed by fan e g fan phys lett a 300 2002 243 and find other new and more general solutions at the same time which include polynomial solutions exponential solutions rational solutions triangular periodic wave solutions hyperbolic and soliton solutions jacobi and weierstrass doubly periodic wave solutions

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

non classical symmetries and the singular manifold method the burgers and the burgers huxley equations in this paper a generalization of the direct method of clarkson and kruskal 1989 for finding similarity reductions of partial differential equations is found and discussed for the burgers and burgers huxley equations the generalization incorporates the singular manifold method largely based upon the painleve property this singular manifold can be used as a reduced variable furthermore a sort of inverse procedure is hereby developed through which we find the equations that yield the vector field components associated to the symmetries of the pde this procedure also displays the profound relationship among the symmetries and the singular manifold as a reduced variable the symmetries found in this way are shown to be those corresponding to the so called non classical symmetries by bluman and cole 1974 and olver and rosenau 1986

a rational generalization of fan s method and its application to generalized shallow water wave equations in this paper we employ a rational expansion to generalize fan s method for exact travelling wave solutions for nonlinear partial differential equations pdes to verify the reliability of the proposed method the generalized shallow water wave gsww equation has been investigated as an example kinds of new exact travelling wave solutions of a rational form have been obtained this indicates that the proposed method provides a more general result for exact solution of nonlinear equations

positivity preserving finite volume scheme for the nagumo type equations on distorted meshes abstract in this paper we present a nonlinear positivity preserving finite volume scheme for the nagumo type equations with anisotropic tensor diffusion coefficient for the diffusion term we use the positivity preserving finite volume scheme for the time direction we use the backward euler approximation we deal with nonlinear reaction term implicitly and decompose nonlinear reaction coefficient into two nonnegative functions thus we get a system of nonlinear algebraic equations the advantages of our scheme are that it can be applied to distorted meshes and has no severe constraint on the time step the numerical results verify the theoretical result

the numerical solution of a generalized burgers huxley equation through a conditionally bounded and symmetry preserving method in this article we propose a non standard finite difference scheme to approximate the solutions of a generalized burgers huxley equation from fluid dynamics our numerical method preserves the skew symmetry of the partial differential equation under study and under some analytical constraints of the model constants and the computational parameters involved it is capable of preserving the boundedness and the positivity of the solutions in the linear regime the scheme is consistent to first order in time due partially to the inclusion of a tuning parameter in the approximation of a temporal derivative and to second order in space we compare the results of our computational technique against the exact solutions of some particular initial boundary value problems our simulations indicate that the method presented in this work approximates well the theoretical solutions and moreover that the method preserves the boundedness of solutions within the analytical constraints derived here in the problem of approximating solitary wave solutions of the model under consideration we present numerical evidence on the existence of an optimum value of the tuning parameter of our technique for which a minimum relative error is achieved finally we linearly perturb a steady state solution of the partial differential equation under investigation and show that our simulations still converge to the same constant solution establishing thus robustness of our method in this sense

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

convergence of adaptive mixed finite element methods for the hodge laplacian equation without harmonic forms finite element exterior calculus feec has been developed as a systematic framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes this paper is devoted to analyzing the convergence of adaptive mixed finite element methods for the hodge laplacian equations based on feec without considering harmonic forms a residual type a posteriori error estimate is obtained by using the hodge decomposition the regular decomposition and bounded commuting quasi interpolants an additional marking strategy is added to ensure the quasi orthogonality based on which the convergence of adaptive mixed finite element methods is obtained without assuming the initial mesh size is small enough

a modified wei yao liu conjugate gradient method for unconstrained optimization in this paper we give a modified wei yao liu conjugate gradient method wei et al 2006 18 which will reduce to the original wei yao liu method and possess the sufficient descent property without any line search furthermore we prove that the presented method is globally convergent for nonconvex functions with the weak wolfe powell line search in a similar way we also extend these results to the modified liu storey method preliminary numerical results show that the proposed methods are effective for the given test problems

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

new exact travelling wave solutions of two nonlinear physical models abstract in this paper an improved tanh function method is used with a computerized symbolic computation for constructing new exact travelling wave solutions on two nonlinear physical models namely the quantum zakharov equations and the 2 1 dimensional broer kaup kupershmidt bkk system the main idea of this method is to take full advantage of the riccati equation which has more new solutions the exact solutions are obtained which include new soliton like solutions trigonometric function solutions and rational solutions the method is straightforward and concise and its applications are promising

new nonlinear conjugate gradient formulas for large scale unconstrained optimization problems abstract we propose new conjugate gradient formulas for computing the search directions for unconstrained optimization problems the new formulas turn out to be the conjugate descent formula if exact line searches are made some formulas possess the sufficient descent property without any line searches general convergence results for the proposed formulas with the weak wolfe powell conditions are studied we prove that some of the formulas with the steplength technique which ensures the zoutendijk condition to be held are globally convergent in addition the global convergence results for some other formulas with the standard armijo line search are also given preliminary numerical results show that the proposed methods are very promising

backlund transformation lax pairs and explicit exact solutions for the shallow water waves equation in this paper we consider the hirota satsuma equation for shallow water waves we first obtain the backlund transformation and lax pairs by using the extended homogeneous balance method then we find some explicit exact solutions by means of backlund transformation and the extended hyperbolic function method these solutions include the solitary wave solution of rational function soliton solutions double soliton solutions n soliton solutions the multiple solitary wave solutions singular solutions and the periodic wave solutions of triangle function type

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

an improved wei yao liu nonlinear conjugate gradient method for optimization computation in this paper we take a little modification to the wei yao liu nonlinear conjugate gradient method proposed by wei et al z wei s yao l liu the convergence properties of some new conjugate gradient methods appl math comput 183 2006 1341 1350 such that the modified method possesses better convergence properties in fact we prove that the modified method satisfies sufficient descent condition with greater parameter email protected 0 12 in the strong wolfe line search and converges globally for nonconvex minimization we also extend these results to the hestenes stiefel method and prove that the modified hs method is globally convergent for nonconvex functions with the standard wolfe conditions numerical results are reported by using some test problems in the cute library

application of the method of simplest equation for obtaining exact traveling wave solutions for the extended korteweg de vries equation and generalized camassa holm equation a version of the method of the simplest equation called modified method of simplest equation is applied to the extended korteweg de vries equation and to generalized camassa holm equation exact traveling wave solutions of these two nonlinear partial differential equations are obtained the equations of bernoulli riccati and the extended tanh equation are used as simplest equations some of the obtained solutions correspond to surface water waves

a new riccati equation expansion method with symbolic computation to construct new travelling wave solution of nonlinear differential equations abstract in this paper based on a new system of riccati equations we present a triple riccati equation expansion method the nonlinear klein gordon equation and the combined kdv mkdv equation are chosen to illustrate the method as a result many new travelling solutions which include bell profile solitary wave solution singular solitary solutions periodic solutions are obtained this method can be used to the other nonlinear evolution equations in mathematics physics

a descent nonlinear conjugate gradient method for large scale unconstrained optimization in this paper a new nonlinear conjugate gradient method was proposed for large scale unconstrained optimization which possesses the following three properties i the sufficient descent property holds without any line searches ii employing some steplength technique which ensures the zoutendijk condition to be held this method is globally convergent iii this method inherits an important property of the polak ribiere polyak prp method the tendency to turn towards the steepest descent direction if a small step is generated away from the solution preventing a sequence of tiny steps from happening preliminary numerical results show that this method is very promising

superposition formulas for pseudounitary matrix riccati equations the purpose of this article is to derive a superposition formula for the pseudounitary matrix riccati equation of dimension n 2 the superposition formula will be written in closed form in terms of five particular solutions satisfying certain well specified conditions defining a fundamental set examples will be studied in order to show how the superposition formula works and how it can be used in numerical calculations