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

on the group classification of variable coefficient nonlinear diffusion convection equations we consider the variable coefficient diffusion convection equation of the form f x ut g x d u ux x h x k u ux which has considerable interest in mathematical physics biology and chemistry we present a complete group classification for this class of equations also we derive equivalence transformations between equations that admit lie symmetries furthermore we obtain mappings that connect variable and constant coefficient equations exact solutions of special forms of this equations are constructed using lie symmetries and equivalence transformations

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

conditional lie backlund symmetries of hamilton jacobi equations abstract the conditional lie backlund symmetry method as a generalization of the conditional symmetry and lie backlund symmetry methods is developed to study hamilton jacobi equations it is shown that two hamilton jacobi equations admit conditional lie backlund symmetries of second order for certain smooth functions as a result a complete description of the structure of solutions to the resulting equations associated with the conditional lie backlund symmetries is performed

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

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

potential symmetries of a porous medium equation potential symmetries which are not local symmetries are carried out for the porous medium equation where when it can be written in a conserved form these symmetries are realized as local symmetries of a related auxiliary system and lead to the construction of corresponding invariant solutions as well as to the linearization of the equation by non invertible mappings

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

classical point symmetries of a porous medium equation the lie group formalism is applied to deduce symmetries of the porous medium equation we study those spatial forms that admit the classical symmetry group the reduction obtained from the optimal system of subalgebras are derived some new exact solutions can be obtained

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

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

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

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

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

classification and approximate solutions to perturbed diffusion convection equations abstract approximate symmetries of the perturbed nonlinear diffusion convection equations are completely classified by the method originated with fushchich and shtelen moreover for some interesting cases symmetry reductions and approximate solutions are discussed in detail

fast slow diffusion and collapsing sandpiles we regard the limit as p of the flow governed by the p laplacian as providing a simplistic model for the collapse of an initially unstable sandpile upon rescaling to stretch out the initial layer we obtain some simple dynamics and provide fairly explicit solutions in certain cases involving nonlocal geometric curvature motion in particular we note that such models entail instantaneous mass transfer governed by monge kantorovich theory

weighted weierstrass theorem with first derivatives abstract we characterize the set of functions which can be approximated by continuous functions with the norm f l w for every weight w this fact allows to determine the closure of the space of polynomials in l w for every weight w with compact support we characterize as well the set of functions which can be approximated by smooth functions with the norm f w 1 w 0 w 1 f l w 0 f l w 1 for a wide range of even non bounded weights w j s we allow a great deal of independence among the weights w j s

conditional lie backlund symmetries and functionally generalized separable solutions to the generalized porous medium equations with source abstract the functionally generalized separable solutions of the generalized porous medium equations with power law and exponential diffusivity are studied by using the conditional lie backlund symmetry method the variant forms of the considered equations which admit the linear conditional lie backlund symmetries are identified a number of examples are considered and some exact solutions defined on the polynomial trigonometric and exponential invariant subspaces determined by the linear conditional lie backlund symmetries are constructed

a stochastic model for growing sandpiles and its continuum limit we study a stochastic model for the sandpile growth the height of the sandpile above every site of a d dimensional lattice is denoted by an integer the height changes randomly and after a suitable scaling of the time and space we show that the rescaled height converges to a deterministic function the limit is governed by a nonlinear evolution equation

sobolev orthogonal polynomial in the complex plane sobolev orthogonal polynomials with respect to measures supported on compact subsets of the complex plane are considered for a wide class of such sobolev orthogonal polynomials it is proved that their zeros are contained in a compact subset of the complex plane and their asymptotic zero distribution is studied we also find the nth root asymptotic behavior of the corresponding sequence of sobolev orthogonal polynomials

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

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

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

extended rotation and scaling groups for nonlinear evolution equations a 1 1 dimensional nonlinear evolution equation is invariant under the rotation group if it is invariant under the infinitesimal generator v x u u x then the solution satisfies the condition ux x u for equations that do not admit the rotation group we provide an extension of the rotation group the corresponding exact solution can be constructed via the invariant set r0 u ux xf u of a contact first order differential structure where f is a smooth function to be determined the time evolution on r0 is shown to be governed by a first order dynamical system we introduce an extension of the scaling groups characterized by an invariant set tilde s 0 that depends on two constants and n 1 when 0 it reduces to the invariant set s0 introduced by galaktionov we also introduce a generalization of both the scaling and rotation groups which is described by an invariant set e0 with parameters a and b when a 0 or b 0 it respectively reduces to r0 or s0 these approaches are used to obtain exact solutions and reductions of dynamical systems of nonlinear evolution equations

analytic aspects of sobolev orthogonal polynomials revisited this paper surveys some recent achievements in the analytic theory of polynomials orthogonal with respect to inner products involving derivatives asymptotic properties zero location approximation and moment theory are some of the topic considered

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

separation of variables and exact solutions to nonlinear diffusion equations with x dependent convection and absorption abstract this paper considers nonlinear diffusion equations with x dependent convection and source terms u t d u u x x q x u u x p x u the functional separation of variables of the equations is studied by using the generalized conditional symmetry approach we formulate conditions for such equations which admit the functionally separable solutions as a consequence some exact solutions to the resulting equations are constructed finally we consider a special case for the equations which admit the functionally separable solutions when the convection and source terms are independent of x

on nonlinear diffusion equations with x dependent convection and absorption abstract for the 1 1 dimensional nonlinear diffusion equations with x dependent convection and source terms u t d u u x x q x u u x p x u we obtain conditions under which the equations admit the second order generalized conditional symmetries x u u xx h u u x 2 g x u u x f x u and the first order sign invariants j x u u t a u u x 2 b x u u x c x u on the solutions u x t several different generalized conditional symmetries and first order sign invariants for equations in which the diffusion term offers different possibilities power law exponential mullin fujita are presented exact solutions to the resulting equations corresponding to the generalized conditional symmetries and the first order sign invariants are constructed

nonclassical symmetry solutions for reaction diffusion equations with explicit spatial dependence abstract nonclassical symmetry methods are used to study the linear diffusion equation with a nonlinear source term which includes explicit spatial dependence mathematical forms for the spatial dependence are found which enable strictly nonclassical symmetries to be admitted when the nonlinearity is cubic a number of new exact solutions are constructed and an application of one of these solutions to diploid population genetics is discussed

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

symmetries and solutions to the thin film equations we study symmetries and solutions of the generalized fourth order nonlinear partial differential equations which arise from studies of thin liquid films it is shown that the equations admit extended scaling and rotation symmetries and a class of generalized conditional symmetries for certain coefficient functions exact solutions associated to the symmetries are obtained

discrete collapsing sandpile model abstract our main goal is to introduce and study a discrete model for the collapsing of a pile of cubes this is a typical example of self organized critical phenomena exhibited by a critical slope we prove the existence and uniqueness of the solution for the model then by using dual arguments we study the numerical computation of the solution and we present some numerical simulations

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

fast slow diffusion and growing sandpiles abstract we study a coupled system of ode introduced by the first author insiam j appl math 22 1972 437 458 for the heights of growing interacting sand cones we show that these ode correspond to the evolution inl2generated by the subdifferential of the convex functional which vanishes on functions whose gradient has length less than or equal to one and is infinity otherwise additionally we explain how the ode arise from evolutions governed by thep laplacian in the infinitely fast infinitely slow diffusion limit asp

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