asymptotic stability of impulsive stochastic partial differential equations with infinite delays in this paper we study the existence and asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with infinite delay by employing a fixed point approach sufficient conditions are derived for achieving the required result these conditions do not require the monotone decreasing behaviour of the delays

fixed points and stability of neutral stochastic delay differential equations in this paper we consider a linear scalar neutral stochastic differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically mean square stable by means of fixed point theory these conditions do not require the boundedness of delays nor do they ask for a fixed sign on the coefficient functions an asymptotic mean square stability theorem with a necessary and sufficient condition is proved some well known results are improved and generalized

unimodular congruence of the laplacian matrix of a graph abstract let g be a graph with vertices 1 2 n associated with g there is an integral quadratic form q x on the n tuple of indeterminates x x 1 x n given by q x x i x j 2 where the sum is taken over all edges i j of g in this paper we prove that the quadratic forms q 1 q 2 associated with graphs g 1 g 2 are congruent by a unimodular matrix if and only if g 1 and g 2 are cycle isomorphic

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

duality and the signed laplacian matrix of a graph abstract we give a necessary and sufficient condition for a bijection between the edge sets of two graphs to be a dual bijection the condition involves unimodular congruence of augmented signed laplacian matrices for the two graphs

differential stackel matrices we show that additive separation of variables for linear homogeneous equations of all orders is characterized by differential stackel matrices generalizations of the classical stackel matrices used for multiplicative separation of second order schrodinger equations and additive separation of hamilton jacobi equations we work out the principal properties of these matrices and demonstrate that even for second order laplace equations additive separation may occur when multiplicative separation does not

conditional lie backlund symmetries and solutions to n 1 dimensional nonlinear diffusion equations this paper discusses a class of n 1 dimensional nonlinear diffusion equations with source term which arises in nonlinear shear flows of non newtonian fluids it is shown that some radially symmetric equations admit certain types of conditional lie backlund symmetries as a result exact solutions to the resulting equations are obtained those solutions extend the known ones such as instantaneous source solutions of the porous medium equation with absorption term the phenomena of extinction and blow up and behavior to many of the solutions are described

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

wave equation on spherically symmetric lorentzian metrics wave equation on a general spherically symmetric spacetime metric is constructed noether symmetries of the equation in terms of explicit functions of and are derived subject to certain differential constraints by restricting the metric to flat friedman case the noether symmetries of the wave equation are presented invertible transformations are constructed from a specific subalgebra of these noether symmetries to convert the wave equation with variable coefficients to the one with constant coefficients

functional separation of variables for laplace equations in two dimensions the authors say that a solution psi of a partial differential equation in two real variables x1 x2 is functionally separable in these variables if psi x1 x2 phi a x1 b x2 for single variable functions phi a b such that phi a b not 0 they classify all possibilities for regular functional separation in local coordinates for equations of the form delta 2 psi f psi x1 x2 where delta 2 is the laplace beltrami operator on a two dimensional riemannian or pseudo riemannian space if the dependence of f on x1 x2 is non trivial then separation can occur for conformal cartesian coordinates on any space if f g psi then for orthogonal coordinates they find that true functional separation i e separation other than additive or multiplicative occurs precisely for cartesian coordinates in the euclidean and pseudo euclidean planes the sine gordon equation provides an example of this separation for many of these cases the separated solutions a b can be expressed in terms of elliptic functions for non orthogonal coordinates and f g psi true functional separation occurs precisely for cartesian coordinates in the pseudo euclidean plane and for a coordinate system on the hyperboloid of one sheet a pseudo riemannian space of constant curvature

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

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

exact polynomial solutions to some nonlinear diffusion equations abstract a class of non group invariant exact solutions to a number of nonlinear diffusion equations is constructed expressed in the appropriate variables these solutions are polynomials in the spatial variable the method is mainly applicable to equations which may be written in an appropriate quadratically nonlinear form a number of equivalence transformations between various nonlinear diffusion equations are also derived

symmetry classifications and reductions of some classes of 2 1 nonlinear heat equation abstract the 2 1 nonlinear heat equation u t f u u x x u y y 0 is considered a symmetry classification of the equation using lie group method is presented and reduction to the first or second order ordinary differential equations is provided

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

some non self similar solutions to a nonlinear diffusion equation the author generates exact solutions to the nonlinear diffusion equation ut del u 1 2 del u which are not similarity solutions some applications and generalizations are indicated

variable separation approach for a differential difference system special toda equation a bilinear variable separation approach is used to construct some special solutions for a differential difference toda equation the semi discrete form of the continuous formula which describes some types of special solutions for many 2 1 dimensional continuous systems is found for a suitable quantity of the differential difference toda equation thus abundant semi discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions

conditional lie backlund symmetries and invariant subspaces to nonlinear diffusion equations with source abstract this paper considers conditional lie backlund symmetries of the radially symmetric nonlinear diffusion equations with source we obtain a complete list of canonical forms for such equations which admit higher order conditional symmetries as a consequence the solutions of the resulting equations are constructed on the invariant subspaces admitted by the corresponding equations

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

similarity reductions of partial differential equations the author determine the relation between two methods of reduction the similarity reduction method using nonclassical symmetry groups and the direct approach used by clarkson and kruskal 1989 they prove that the solutions which are obtained by similarity in correspondence to nonclassical groups constitute a larger family than the one obtained by the method of clarkson and kruskal the two procedures are equivalent only if the generators zeta x t u tau x t u eta x t u of the nonclassical groups are such that zeta tau is independent of u to explain these results he proves the existence of families of solutions of the burgers equation which are found by means of nonclassical symmetry reduction and which cannot be determined via the general reduction form of clarkson and kruskal

symmetries and integrability of generalized diffusion reaction equations using lie group methods and the painleve test the authors analyse nonlinear diffusion reaction equations ut d u u f u with power law diffusion coefficients d approximately uv and arbitrary nonlinear reaction terms f u which have a wide spectrum of applications in many areas of science the lie group based similarity method leads to a classification of the reaction terms according to its symmetry properties with the help of the adjoint representation the optimal system of similarity reductions is calculated to check the integrability of the partial differential equation the existence of generalized lie backlund symmetries is investigated apart from three known cases no further cases with third order symmetries exist examining the integrability of the second order ordinary differential equations resulting from the reductions only a few parameter combinations can be found for which the painleve property is given however they are able to construct unknown integrals of motion for a much larger range of parameter values from the integrals exact similarity solutions may be derived this is demonstrated by examples corresponding to the important moving wave reduction

a note on 2 isomorphisms and the signed laplacian matrix of a graph abstract let g and h be graphs with a bijection on their edges in a recent paper we observed that the laplacian matrices of signed versions of g and h contain enough information to decide whether or not g and h are dual graphs with respect to in this note we show that g and h are 2 isomorphic if and only if the determinants of the reduced forms of the consistently signed laplacian matrices for g and h are equal thus the laplacian matrices of signed versions of g and h also contain enough information to decide whether or not g and h are 2 isomorphic with respect to

a symmetry classification for a class of 2 1 nonlinear wave equation abstract in this paper a symmetry classification of a 2 1 nonlinear wave equation u t t f u u x x u y y 0 where f u is a smooth function on u using lie group method is given the basic infinitesimal method for calculating symmetry groups is presented and used to determine the general symmetry group of this 2 1 nonlinear wave equation

finite time stability of a class of fuzzy cellular neural networks with multi proportional delays abstract this paper is concerned with finite time stability for a class of fuzzy cellular neural networks with multi proportional delays based on the finite time stability theory we establish a novel result to ensure the finite time stability of the addressed system by employing the differential inequality techniques we also use numerical simulations to demonstrate our theoretical result

separation of variables in the nonlinear wave equation we develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations using it we obtain a number of new two dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions

separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source abstract a solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q u x t for some single variable functions q and in this paper the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source we obtain a complete list of canonical forms for such equations which admit the functionally separable solutions as a result we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source the behavior and blow up properties of some solutions are described

painleve analysis of the generalized burgers huxley equation a complete painleve test 1900 is applied to the generalized burgers huxley equation using the version of the painleve analysis recently developed by weiss tabor and carnevale 1983 for partial nonlinear differential equations in so doing the authors are able to find a complete set of new solutions as well as recovering some previous particular solutions already found by using ad hoc methods which have been recently published

the inverse symmetry problem for a 2d generalized second order evolutionary equation abstract the paper is devoted to the lie symmetries of 2 d nonlinear dynamical systems described by second order partial differential equations by imposing the invariance condition of the equations under the action of the lie symmetry operator we obtained a determining system which could be solved in two directions i to find the symmetries of a concrete equation ii to obtain all the compatible equations with an imposed form of symmetry algebra this paper will pay attention to the indirect problem ii and will use the algorithm for determining the most general 2 d equations which belong following their symmetries to the same class with two interesting and important physical models the nonlinear heat equation and the transfer equation with power law nonlinearities

evolution equations invariant surface conditions and functional separation of variables abstract this paper is devoted to a discussion of the reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables the relationship of these methods with nonclassical and weak point symmetries is stressed applications to diffusion equations with an inhomogeneous reaction term or with saturating dissipation are provided

functionally separable solutions to nonlinear wave equations by group foliation method abstract we apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force u t t a x d u u x x b x q u a x 0 a classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases

homogeneous diffusion in ℝ with power like nonlinear diffusivity we study the nonnegative solutions of the initial value problem ut ur ux p 1ux x u x 0 l1 where p 0 r p 0 the local velocity of propagation of the solutions is identified as v vx vx p 1 where v cu with r p 1 p and c r p r p 1 is the nonlinear potential our main result is the a priori estimate vx vx p 1 x frac 1 2p r t

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

homology and the nonlinear heat diffusion equation the concept of homology siam j appl math 40 1981 pp 191 207 is generalized to an arbitrary nonlinear diffusion equation a large class of self homologous diffusion coefficients is introduced and used to examine possible changes in initial and boundary conditions through homologous transformations generalized homology is shown to be identical to backlund transformations

fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays the fixed point theory is first used to consider the stability for stochastic partial differential equations with delays some conditions for the exponential stability in pth mean as well as in sample path of mild solutions are given these conditions do not require the monotone decreasing behavior of the delays which is necessary in t caraballo k liu exponential stability of mild solutions of stochastic partial differential equations with delays stoch anal appl 17 1999 743 763 ruhollan jahanipur stability of stochastic delay evolution equations with monotone nonlinearity stoch anal appl 21 2003 161 181 even in this special case our results also improve the results in t caraballo k liu exponential stability of mild solutions of stochastic partial differential equations with delays stoch anal appl 17 1999 743 763

nonclassical symmetries of evolutionary partial differential equations and compatibility the determining equations for the nonclassical reductions of a general nth order evolutionary partial differential equations is considered it is shown that requiring compatibility with a first order quasilinear partial differential equation the determining equations are obtained burgers equation and the kdv equation and generalizations serve as examples illustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

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

fixed points and exponential stability for stochastic volterra levin equations in this paper we study a stochastic volterra levin equation by using fixed point theory we give some conditions for ensuring that this equation is exponentially stable in mean square and is also almost surely exponentially stable our result generalizes and improves on the results in 14 1 30

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

exponential stability of a class of nonlinear systems via fixed point theory abstract in this paper the problem of globally stochastically exponential stability in the p th moment for a class of t s fuzzy stochastic impulsive genetic regulatory networks with random discrete delays distributed delays and parameter uncertainties is discussed by utilizing the theory of stochastic analysis semigroup theory and fixed point theory a novel sufficient condition to guarantee the globally stochastically exponential stability in the p th moment of the considered genetic regulatory networks is derived eventually an example is given to demonstrate that the obtained result is effective

dual graphs and knot invariants abstract let g be a signed plane graph and g d its signed dual graph methods from knot theory are used to show that the signed laplacian matrices l g and l g d are goeritz congruent there exists diagonal 0 1 matrices 1 and 2 and a unimodular matrix u such that 1 l g u 2 l g d u t for laplacian matrices of an unsigned plane graph and its dual l g is goeritz congruent to l g d

on spectral properties of signed laplacians with connections to eventual positivity signed graphs have appeared in a broad variety of applications ranging from social networks to biological networks from distributed control and computation to power systems in this paper we investigate spectral properties of signed laplacians for undirected signed graphs we find conditions on the negative weights under which a signed laplacian is positive semidefinite via the kron reduction and multiport network theory for signed laplacians that are indefinite we characterize their inertias with the same framework furthermore we build connections between signed laplacians generalized m matrices and eventually exponentially positive matrices

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

iterating the nonclassical symmetries method we show that iterations of the nonclassical symmetries method for any evolution equation give raise to new nonlinear equations which inherit the symmetry algebra of the given equation invariant solutions of these heir equations supply new solutions of the original equation the same method generates heir equations for hyperbolic and elliptic equations some examples are given

derivative dependent functional separable solutions for the kdv type equations abstract a complete list of canonical forms for a special kdv type of equations which admit derivative dependent functional separable solutions is obtained by using the generalized conditional symmetry approach some exact solutions to the resulting equations are explicitly obtained

a note on unimodular congruence of graphs abstract let q q g be an oriented vertex edge incidence matrix for the graph g then k g q t q is an edge version of the kirchhoff or laplacian matrix the purpose of this note is to examine k g under integer unimodular congruence

some non local transformations between nonlinear diffusion equations generalized backlund transformations are applied to derive links between a large number of different types of nonlinear diffusion equations including many which are of physical significance some new exactly linearizable forms are determined

stability analysis of uncertain genetic sum regulatory networks this paper addresses the problem of establishing robust stability of uncertain genetic networks with sum regulatory functions specifically we first consider uncertain genetic networks where the regulation occurs at the transcriptional level and we derive a sufficient condition for robust stability by introducing a bounding set of the uncertain nonlinearity we hence show that this condition can be formulated as a convex optimization through polynomial lyapunov functions and polynomial descriptions of the bounding set by exploiting the square matricial representation smr of polynomials which allows to establish whether a polynomial is a sum of squares sos via a linear matrix inequality lmi then we propose a method for computing a family of bounding sets by means of convex optimizations it is worthwhile to remark that these results are derived in spite of the fact that the variable equilibrium point cannot be computed as being the solution of a system of parameter dependent nonlinear equations and is hence unknown lastly the proposed approach is extended to models where the regulation occurs at different levels and both mrna and protein dynamics are nonlinear

nonclassical symmetries and backlund transformations abstract in this paper we show how to obtain well known backlund transformations of several equations by applying the nonclassical symmetries method in particular we derive the backlund transformation between the modified korteweg de vries and the korteweg de vries equations the backlund transformation between the calogero and the third order linear equations the auto backlund transformations of the potential korteweg de vries the potential modified korteweg de vries equation and the burgers equations as a consequence the n solitons solution of the korteweg de vries equation can be obtained by the nonclassical symmetries method

new explicit solutions of quasilinear heat equations with general first order sign invariants abstract for the one dimensional quasilinear heat equations ut u uxx f u with smooth functions u 0 and f u we derive conditions under which a general quadratic hamilton jacobi operator h u ut h2 u ux 2 h1 u ux h0 u with the functions h2 h1 h0 to be determined preserves both the signs 0 and 0 on the solutions u x t h u is then called a sign invariant of the given parabolic equation several types of different sign invariants are established which are governed by a linear second order ordinary differential equations new explicit solutions via finite dimensional dynamical systems corresponding to zero preserving of the first order operator are constructed