generalized conditional symmetries of evolution equations we analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution earlier results on the connection between generalized conditional invariance and generalized reduction of evolution equations are revisited this leads to a no go theorem on determining equations for operators of generalized conditional symmetry it is also shown that up to certain equivalences there exists a one to one correspondence between generalized conditional symmetries of an evolution equation and parametric families of its solutions

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

reduction of the n dimensional static vacuum einstein equation and generalized schwarzschild solutions abstract we consider the static vacuum einstein spacetime when the spatial factor is conformal to a n dimensional pseudo euclidean space the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described we obtain the entire set of solutions of the reduced system where the classical schwarzschild solution arises as a particular solution in addition we show that the riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature

symmetry classification and exact solutions of the kramers equation the complete classification of lie symmetries for the kramers equation is described group invariance under infinitesimal transformations is used to generate a wide class of exact solutions some of the known solutions are obtained as particular cases the general results can also be used to solve some types of moving boundary problems we researched some q conditional symmetry properties of this equation

conditional symmetry of a porous medium equation abstract we obtain an exhaustive description of conditional symmetries admitted by a nonlinear porous medium equation in one spatial dimension it is proved that solving the determining equations for the conditional symmetries either yields classical lie symmetries or is equivalent to solving the porous medium equation itself the second possibility is shown to be typical for an arbitrary evolution type partial differential equation in one spatial variable we show that this very property is responsible for the well known phenomenon when using non classical symmetries yield exact solutions which prove to be invariant ones

higher symmetries and exact solutions of linear and nonlinear schrodinger equation a new approach for the analysis of partial differential equations is developed which is characterized by a simultaneous use of higher and conditional symmetries higher symmetries of the schrodinger equation with an arbitrary potential are investigated nonlinear determining equations for potentials are solved using reductions to weierstrass painleve and riccati forms algebraic properties of higher order symmetry operators are analyzed combinations of higher and conditional symmetries are used to generate families of exact solutions of linear and nonlinear schrodinger equations

non classical reductions of initial value problems for a class of nonlinear evolution equations we classify initial value problems for a class of one dimensional evolution equations which can be reduced to cauchy problems for some systems of first order ordinary differential equations the technique applied relies heavily on higher conditional symmetries of the equations under study which means in particular that the obtained reductions cannot be derived within the framework of the standard lie approach

nonlocal ansatze reduction and some exact solutions for the system of the van der waals equations abstract in this paper two types of nonlocal equivalence transformations of the system of convection diffusion equations are found using these transformations we obtained two images of the van der waals system lie symmetries of these images are used to construct nonlocal ansatze and reduce that system owing to this it has become possible to have built classes of exact solutions of the van der waals system lie symmetry of this system is used to construct nonlocal ansatze and reduce its both images

symmetries of the fisher kolmogorov petrovskii piskunov equation with a nonlocal nonlinearity in a semiclassical approximation abstract the classical group analysis approach used to study the symmetries of integro differential equations in a semiclassical approximation is considered for a class of nearly linear integro differential equations in a semiclassical approximation an original integro differential equation leads to a finite consistent system of differential equations whose symmetries can be calculated by performing standard group analysis the approach is illustrated by the calculation of the lie symmetries in explicit form for a special case of the one dimensional nonlocal fisher kolmogorov petrovskii piskunov population equation

on the reduction of the nonlinear multi dimensional wave equations and compatibility of the d alembert hamilton system abstract the necessary conditions of the compatibility of the d alembert hamilton system in minkowsky space r 1 n are established the problem of reduction of p 1 n invariant wave equations to ordinary differential equations is discussed

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

group properties of generalized quasi linear wave equations abstract in this paper complete group classification of a class of 1 1 dimensional generalized quasi linear wave equations is performed by using the lie ovsiannikov method additional equivalent transformation and furcate split method lie reductions of some truly variable coefficient wave equations which are singled out from the classification results are investigated some classes of exact solutions of these variable coefficient wave equations are constructed by means of both the reductions and the additional equivalent transformations the nonclassical symmetries to the generalized quasi linear wave equation are also studied this enabled to obtain some exact solutions of the wave equations which are invariant under certain conditional symmetries

on new ways of group methods for reduction of evolution type equations new exact solutions of the evolution type equations are constructed by means of a non point contact symmetries also we analyzed the discrete symmetries of maxwell equations in vacuum and decoupled ones to the four independent equations that can be solved independently

on some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula some inequalities are presented here for differentiable convex mappings using hermite hadamard s integral inequality holding for convex functions a parallel development is deduced for differentiable concave functions some applications to special means of real numbers are also given finally some error estimates for the midpoint formula are obtained

new scale invariant nonlinear differential equations for a complex scalar field abstract we describe all complex wave equations of the form u f u u invariant under the extended poincare group as a result we have obtained the five new classes of p 1 3 invariant nonlinear partial differential equations for the complex scalar field

an iterative method for finding common solutions of equilibrium and fixed point problems we introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a hilbert space we prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality which is the optimality condition for a minimization problem

reduction operators of linear second order parabolic equations the reduction operators i e the operators of nonclassical conditional symmetry of 1 1 dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described this problem proves to be equivalent in some sense to solving initial equations the no go result is extended to the investigation of point transformations admissible transformations equivalence transformations lie symmetries and lie reductions of the determining equations for the nonclassical symmetries transformations linearizing the determining equations are obtained in the general case and under different additional constraints a nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented an observed connection between reduction operators and darboux transformations is discussed

lie and conditional symmetries of the three component diffusive lotka volterra system lie and q conditional symmetries of the classical three component diffusive lotka volterra system in the case of one space variable are studied the group classification problems for finding lie symmetries and q conditional symmetries of the first type are completely solved notably non lie symmetries q conditional symmetry operators for a multi component nonlinear reaction diffusion system are constructed for the first time the results are compared with those derived for the two component diffusive lotka volterra system the conditional symmetry obtained for the non lie reduction of the three component system used for modeling competition between three species in population dynamics is applied and the relevant exact solutions are found particularly the exact solution describing different scenarios of competition between three species is constructed

amplitude phase representation for solutions of nonlinear d alembert equations we consider the nonlinear complex d alembert equation square operator psi f mod psi mod psi with psi represented in terms of amplitude and phase in 1 n dimensional minkowski space we exploit a compatible d alembert hamilton system to construct new types of exact solutions for some nonlinearities

new conditional symmetries and exact solutions of the nonlinear wave equation we have constructed the new class of conditional symmetries of the nonlinear wave equation in 1 3 dimensions these symmetries enable us to obtain broad families of new exact solutions of the nonlinear wave equation containing up to the four arbitrary functions

a hermite hadamard type inequality with applications to the estimation of moments of continuous random variables we present a new hadamard type inequality for quasi convex functions our results have applications to the estimation of moments of random variables our work generalizes some recent results published in 2014 in amc in this paper we present a hadamard type inequality for quasi convex functions with applications to the estimation of moments of continuous random variables with compact support the main result obtained here generalizes the inequality obtained by alomari et al 2010

viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces abstract in this paper we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space then we prove a strong convergence theorem which is connected with combettes and hirstoaga s result p l combettes s a hirstoaga equilibrium programming in hilbert spaces j nonlinear convex anal 6 2005 117 136 and wittmann s result r wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491 using this result we obtain two corollaries which improve and extend their results

on subalgebras of the lie algebra of the extended poincare group p 1 n some general results on the subalgebras of the lie algebra ap 1 n of the extended poincare group p 1 n n 2 with respect to p 1 n conjugation have been obtained all subalgebras of ap 1 4 that are nonconjugate to the subalgebras of ap 1 4 are classified with respect to p 1 4 conjugation the list of representatives of each conjugacy class is presented

approximation of common fixed points of a countable family of nonexpansive mappings in a banach space in this paper to find a common fixed point of a family of nonexpansive mappings we introduce a halpern type iterative sequence then we prove that such a sequence converges strongly to a common fixed point of nonexpansive mappings moreover we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings and the problem of finding a zero of an accretive operator

a viscosity approximation scheme for system of equilibrium problems nonexpansive mappings and monotone mappings abstract in this paper we introduce a new viscosity approximation scheme based on the extragradient method for finding a common element of the set of solutions to a system of equilibrium problems the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions to the variational inequality for a monotone lipschitz continuous mapping several convergence results for the sequences generated by these processes in hilbert spaces were derived

on a reduction and solutions of nonlinear wave equations with broken symmetry a generalised definition for invariance of partial differential equations is proposed exact solutions of the equations with broken symmetry are obtained

equilibrium programming using proximal like algorithms we compute constrained equilibria satisfying an optimality condition important examples include convex programming saddle problems noncooperative games and variational inequalities under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization in the main algorithm each stage of computation requires two proximal steps possibly using bregman functions one step serves to predict the next point the other helps to correct the new prediction to enhance practical applicability we tolerate numerical errors

some inequalities for differentiable convex and concave mappings abstract several inequalities are obtained for some differentiable convex mappings that are connected with the celebrated hermite hadamard integral inequality also a parallel development is made for concave functions

similarity reductions of the 1 3 dimensional burgers equation we consider the 1 3 dimensional burgers equation u t u x x u y y u z z uu x which has considerable interest in mathematical physics we complete the list of similarity reductions that are obtained from the lie symmetries admitted by this equation to achieve this goal we employ two and three dimensional subalgebras of the lie symmetry algebra in addition to the one dimensional subalgebras that appears in the literature we solve analytically where it is possible the reduced ordinary differential equations and hence we obtain closed form solutions for the original equation finally we derive certain non lie solutions

singular reduction operators in two dimensions the notion of singular reduction operators i e of singular operators of nonclassical conditional symmetry of partial differential equations in two independent variables is introduced all possible reductions of these equations to first order odes are exhaustively described as examples properties of singular reduction operators of 1 1 dimensional evolution and wave equations are studied it is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in depth prior study of the corresponding singular vector fields

continuous subgroups of the poincare group p 1 4 an exhaustive description of the nonsplitting subalgebras of the lp 1 4 algebra with respect to p 1 4 conjugation is presented

special conformal groups of a riemannian manifold and lie point symmetries of the nonlinear poisson equation abstract we obtain a complete group classification of the lie point symmetries of nonlinear poisson equations on generic pseudo riemannian manifolds m using this result we study their noether symmetries and establish the respective conservation laws it is shown that the projection of the lie point symmetries on m are special subgroups of the conformal group of m in particular if the scalar curvature of m vanishes the projection on m of the lie point symmetry group of the poisson equation with critical nonlinearity is the conformal group of the manifold we illustrate our results by applying them to the thurston geometries

initial value problems for evolutionary partial differential equations and higher order conditional symmetries we suggest a new approach to the problem of dimensional reduction of initial boundary value problems for evolution equations in one spatial variable the approach is based on higher order generalized conditional symmetries of the equations involved it is shown that reducibility of an initial value problem for an evolution equation to a cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question we also give some examples of the solution of initial value problems for second and third order nonlinear differential equations by reduction by their conditional symmetries we give a systematic classification of general second order partial differential equations admitting second order conditional symmetries based on lie s classification of invariant second order ordinary differential equations this yields five classes of principally new initial value problems for nonlinear evolution equations which admit no lie symmetries and are reducible via second order conditional symmetries

on the general solution of the d alembert equation with a nonlinear eikonal constraint and its applications we construct the general solutions of the system of nonlinear differential equations nu 0 u u 0 in the four and five dimensional complex pseudo euclidean spaces the results obtained are used to reduce the multidimensional nonlinear d alembert equation 4u f u to ordinary differential equations and to construct its new exact solutions

a complete basis set of functionally independent invariants of continuous lie groups using the methods of differential equations the author has obtained the complete set of functionally independent bases of invariants for some continuous lie groups of mathematical physics i the conformal group ii the semi direct product of the weyl group and the one parameter group of psi translation iii the direct product of the poincare group and the one parameter group of psi scaling iv the weyl group v the 14 parameter maximal symmetry group of the schrodinger equation for the coulombic system of atoms and molecules and vi su 3 of elementary particle physics the first four groups are the only groups which together with the poincare group are the maximal symmetry groups of all forms of the non linear klein gordon equation

a certain class of solutions of the nonlinear wave equation in this paper are investigated some differential geometry methods in the theory of the nonlinear wave equation 2u u u u a special class of solutions is discussed for which u u is constant on each level of the function u it is proved that levels of such solutions form in the space of independent variable s hypersurfaces with all principal curvatures constant the general form of such hypersurfaces is given then it is proved that via the method of characteristics it is possible to construct in principle all the solutions of the discussed class they may be obtained by integration of an ode of second order using a special class of the polynomial functions some new solutions are given for equations v 4av3 3bv2 2cv d v exp v v sin v v cosh v and v sinh v

orthogonal and non orthogonal separation of variables in the wave equation utt uxx v x u 0utt uxx v x u 0 we develop a direct approach to the separation of variables in partial differential equations within the framework of this approach the problem of the separation of variables in the wave equation with time independent potential reduces to solving an over determined system of nonlinear differential equations we have succeeded in constructing its general solution and as a result all potentials v x permitting variable separation have been found for each of them we have constructed all inequivalent coordinate systems providing separability of the equation under study it should be noted that the above approach yields both orthogonal and non orthogonal systems of coordinates

a precise definition of reduction of partial differential equations we give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry classical and non classical reductions of partial differential equations using the introduced definition of reduction of differential equations we establish equivalence of the non classical conditional symmetry and direct ansatz approaches to reduction of partial differential equations as an illustration we give an example of non classical reduction of the nonlinear wave equation in 1 3 dimensions the conditional symmetry approach when applied to the equation in question yields a number of non lie reductions which are far reaching generalizations of the well known symmetry reductions of the nonlinear wave equations

some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables abstract several inequalities for differentiable convex wright convex and quasi convex mapping are obtained respectively that are connected with the celebrated hermite hadamard integral inequality also some error estimates for weighted trapezoid formula and higher moments of random variables are given

strong convergence of a new composite iterative method for equilibrium problems and fixed point problems in this paper we propose a new composite iterative method for finding a common point of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of nonexpansive mappings in a hilbert space it is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of nonexpansive mappings our results improve and extend the corresponding ones announced by many others

extended symmetry analysis of generalized burgers equations using enhanced classification techniques we carry out the extended symmetry analysis of the class of generalized burgers equations of the form ut uux f t x uxx 0 this enhances all the previous results on symmetries of these equations and includes the description of admissible transformations lie symmetries lie and nonclassical reductions hidden symmetries conservation laws potential admissible transformations and potential symmetries the study is based on the fact that the class is normalized and its equivalence group is finite dimensional

a new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings abstract in this paper we introduce a new iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for inverse strongly monotone mappings we show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions this main theorem extends a recent result of yao and yao y yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings applied mathematics and computation 186 2 2007 1551 1558

auxiliary principle and iterative algorithm for a new system of generalized mixed equilibrium problems in banach spaces abstract in this paper we study a new system of generalized mixed equilibrium problems involving skew symmetric bifunctions sgmep in reflexive banach spaces a system of auxiliary mixed equilibrium problems samep for solving the sgmep is introduced and the existence and uniqueness of the solutions of the samep is first proved next by using the auxiliary principle technique a new iterative algorithm to compute the approximate solutions of the sgmep is suggested and analyzed finally the strong convergence of the iterative sequences generated by the algorithm is also proved under quite mild conditions these results improve unify and generalize some known results in recent literature

the symmetry and some exact solutions of the nonlinear many dimensional liouville d alembert and eikonal equations multiparametrical exact solutions of the many dimensional nonlinear d alembert liouville sine gordon and eikonal equations are obtained the maximally extensive local invariance groups of the equations are determined and invariants of the extended poincare group are found

refinements of hadamard type inequalities for quasi convex functions with applications to trapezoidal formula and to special means in this paper some inequalities of hadamard s type for quasi convex functions are given some error estimates for the trapezoidal formula are obtained applications to some special means are considered

singular reduction modules of differential equations the notion of singular reduction modules i e of singular modules of nonclassical conditional symmetry of differential equations is introduced it is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in depth prior study of the associated singular modules of vector fields the form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations as examples singular reduction modules of evolution equations and second order quasi linear equations are studied reductions of differential equations to algebraic equations and to first order ordinary differential equations are considered in detail within the framework proposed and are related to previous no go results on nonclassical symmetries

on a new method for constructing exact solutions of the nonlinear differential equations of mathematical physics a simple new method for constructing solutions of multidimensional nonlinear wave equations is proposed

a new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization abstract in this paper we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for inverse strongly monotone mappings in a hilbert space under suitable conditions some strong convergence theorems for approximating a common element of the above three sets are obtained as applications at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings the results presented in the paper extend and improve some recent results of yao and yao y y yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings appl math comput 186 2 2007 1551 1558 plubtieng and punpaeng s plubtieng r punpaeng a new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings appl math comput 2007 doi 10 1016 j amc 2007 07 075 s takahashi and w takahashi s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 2006 506 515 su shang and qin y f su m j shang x l qin an iterative method of solution for equilibrium and optimization problems nonlinear anal 2007 doi 10 1016 j na 2007 08 045 and chang cho and kim s s chang y j cho j k kim approximation methods of solutions for equilibrium problem in hilbert spaces dynam systems appl in print

hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings abstract recently takahashi and takahashi s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 2006 doi 10 1016 j jmaa 2006 08 036 suggested and analyzed an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space in this paper we introduce a new iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinitely many nonexpansive mappings in a hilbert space then we prove a strong convergence theorem which is the improvements and extension of takahashi and takahashi s 2006 corresponding result using this theorem we obtain two corollaries which improve and extend their corresponding results