algebraic k theory of group rings and the cyclotomic trace map we prove that the farrell jones assembly map for connective alge braic k theory is rationally injective under mild homological niteness condi tions on the group and assuming that a weak version of the leopoldt schneider conjecture holds for cyclotomic elds this generalizes a result of bokstedt hsiang and madsen and leads to a concrete description of a large direct summand of kn z g z q in terms of group homology in many cases the number theoretic conjectures are true so we obtain rational injectivity results about assembly maps in particular for whitehead groups under niteness assumptions on the group only the proof uses the cyclotomic trace map to topological cyclic homology bokstedt hsiang madsen s functor c and new general injectivity results about the assembly maps for topological hochschild homology and c

analytical and numerical studying of the perturbed korteweg de vries equation the perturbed korteweg de vries equation is considered this equation is used for the description of one dimensional viscous gas dynamics nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves the integrability of this equation is investigated using the painleve approach the condition on parameters for the integrability of the perturbed korteweg de vries equation equation is established new classical and nonclassical symmetries admitted by this equation are found all corresponding symmetry reductions are obtained new exact solutions of these reductions are constructed they are expressed via trigonometric and airy functions stability of the exact solutions of the perturbed korteweg de vries equation is investigated numerically

similarity reduction nonlocal and master symmetries of sixth order korteweg devries equation a systematic investigation to derive the lie point symmetries nonlocal and master symmetries of sixth order korteweg de vries equation kdv6 is presented using the obtained point symmetries similarity reductions are derived and constructed their particular solutions wherever possible it is shown that kdv6 admits infinitely many nonlocal and master symmetries the existence of infinitely many master symmetries ensures that kdv6 is completely integrable

new classes of solutions for the schwarzian korteweg de vries equation in 2 1 dimensions we have obtained new classes of solutions for the 2 1 dimensional schwarzian korteweg de vries equation by considering several types of reductions of a system equivalent to this equation the first analysis is done by studying the nonclassical reductions of the system further reductions are attained by means of other types of symmetry reductions or by ansatz based reductions most of the new classes of solutions depend on jacobian elliptic functions and solutions of a riemann wave equation including the cnoidal waves solutions the new classes of solutions can display several types of coherent structures and can exhibit the overturning or intertwining phenomena according to the suitable selection of the functions these solutions depend on

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

iterations of the non classical symmetries method and conditional lie backlund symmetries iterations of the non classical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the original equation invariant solutions of these heir equations supply new solutions of the original equation we show that particular cases of such invariant solutions correspond to zhdanov s conditional lie backlund symmetries

nonclassical analysis and painleve property for the kupershmidt equations by means of the nonclassical lie approach and the direct method all the similarity reductions of the kupershmidt equations are found two reduction equations which can also be obtained by the classical lie approach are the painleve type ii and iv respectively the other three reduction equations which cannot be obtained by the classical lie approach are only linear or riccati type

an optimal galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere the kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically conducting liquid one of its main uses is as a proof of concept tool to test hypotheses about self exciting dynamo action indeed it provided the very first quantitative evidence for the possibility of the geodynamo despite its utility due to the requirement of resolving fine structures historically numerical work has proven difficult and reported solutions were often plagued by poor convergence in this paper we demonstrate the numerical superiority of a galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere after adopting a poloidal toroidal decomposition and expanding in spherical harmonics we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials each basis function is constructed from a terse sum of one sided jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior but is everywhere infinitely differentiable including at the origin this galerkin method exhibits more rapid convergence for a given problem size than any other scheme hitherto reported as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature in the axisymmetric flows we consider in this paper at a magnetic reynolds number of o 100 a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions alternatively 4 significant figures requires 20 radial functions the terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics we exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of bachtiar et al 4 thereby verifying a counter example to the zel dovich anti dynamo theorem in a spherical geometry

on the conditional symmetries of levi and winternitz it is pointed out that the conditional symmetries of levi and winternitz 1989 and the non classical method first introduced by bluman and cole 1989 have a natural group theoretical setting this corresponds to classifying solutions to a pdes according to their full symmetry and not just according to their symmetries which are also symmetries of the pde

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

partial symmetry of initial value problems abstract it is generally believed that the symmetry of initial value problems ivp must leave both the partial differential equations pdes and initial conditions invariant in this paper we propose partial symmetry of ivp which needs less restrictive conditions on the pdes and initial conditions and only leaves the ivp invariant on some non empty subset of the whole solution set of the governing pdes thus some symmetries which either only leave the pdes invariant or only leave initial conditions invariant are partial symmetries of ivp then considering from two different starting points we define two types of partial symmetry and give the corresponding algorithms where one starts with the pdes and the other is from the initial conditions applications to boussinesq equation and a class of nonlinear heat equation are performed

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

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

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

partial lie point symmetries of differential equations when we consider a differential equation 0 whose set of solutions is a lie point exact symmetry of this is a lie point invertible transformation t such that t i e such that any solution to 0 is transformed into a generally different solution to the same equation here we define partial symmetries of 0 as lie point invertible transformations t such that there is a non empty subset such that t i e such that there is a subset of solutions to 0 which are transformed into one another we discuss how to determine both partial symmetries and the invariant set and show that our procedure is effective by means of concrete examples we also discuss relations with conditional symmetries and how our discussion applies to the special case of dynamical systems our discussion will focus on continuous lie point partial symmetries but our approach would also be suitable for more general classes of transformations the discussion is indeed extended to partial generalized or lie backlund symmetries along the same lines and in the appendix we will discuss the case of discrete partial symmetries

the homotopy theory of cyclic sets the aim of this note is to show that the homotopy theory of the cyclic sets of connes 3 is equivalent to that of s0 2 spaces i e spaces with a circle action and hence to that of spaces over k z 2

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 symmetry and spectrum of the one dimensional schrodinger equation we develop an algebraic approach to studying the spectral properties of the stationary schrodinger equation in one dimension based on its high order conditional symmetries this approach makes it possible to obtain in explicit form representations of the schrodinger operator by n n matrices for any n n and thus to reduce a spectral problem to a purely algebraic one of finding eigenvalues of constant n n matrices the connection to so called quasiexactly solvable models is discussed it is established in particular that the case when conditional symmetries reduce to high order lie symmetries corresponds to exactly solvable schrodinger equations a symmetry classification of schrodinger equation admitting nontrivial high order lie symmetries is carried out which yields a hierarchy of exactly solvable schrodinger equations exact solutions of these are constructed in explicit form possible applications of the technique developed to multidimensional linear and one dimensional nonlinear schrodinger equations are briefly discussed

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

spectral approximation on the unit ball spectral approximation by polynomials on the unit ball is studied in the frame of the sobolev spaces w s p mathbb b d 1 p infty the main results give sharp estimates on the order of approximation by polynomials in the sobolev spaces and explicit construction of approximating polynomials one major effort lies in understanding the structure of orthogonal polynomials with respect to an inner product of the sobolev space w 2 s mathbb b d as an application a direct and efficient spectral galerkin method based on our orthogonal polynomials is proposed for the second and fourth order elliptic equations on the unit ball its optimal error estimates are explicitly derived for both procedures in the sobolev spaces and finally numerical examples are presented to illustrate the theoretic results

an application of one sided jacobi polynomials for spectral modeling of vector fields in polar coordinates a spectral tau method is proposed for solving vector field equations defined in polar coordinates the method employs one sided jacobi polynomials as radial expansion functions and fourier exponentials as azimuthal expansion functions all the regularity requirements of the vector field at the origin and the physical boundary conditions at a circumferential boundary are exactly satisfied by adjusting the additional tau coefficients of the radial expansion polynomials of the highest order the proposed method is applied to linear and nonlinear dispersive time evolution equations of hyperbolic type describing internal kelvin and poincare waves in a shallow stratified lake on a rotating plane

group foliation and non invariant solutions of the heavenly equation the main physical results of this paper are new exact analytical solutions of the heavenly equation of importance in the general theory of relativity these solutions are not invariant under any subgroup of the symmetry group of the equation the main mathematical result is a new method of obtaining non invariant solutions of partial differential equations with infinite dimensional symmetry groups the method involves the compatibility of the given equations with a differential constraint which is automorphic under a specific symmetry subgroup the latter acting transitively on the submanifold of the common solutions by studying the integrability of the resulting conditions one can provide an explicit foliation of the entire solution manifold of the considered equations

a mixed basis spectral projection method a new grid less no collocation spectral projection method is presented the unsteady navier stokes equations are approximated according to the variational framework of guermond and quartapelle which accommodates two vector spaces for the velocity fields obtained in the two half steps of the fractional step method but retains only one in the final solution algorithm two different bases built on legendre polynomials are used for the velocity and pressure to solve the corresponding helmholtz and poisson equations by direct spectral elliptic solvers interpolations pn and pn 2 are employed for velocity and pressure to satisfy the lbb stability requirement and a gauss legendre quadrature formula with 3 2n integration points is used to prevent aliasing error in the pseudospectral evaluation of the nonlinear terms a bdf second order time stepping is implemented to provide accurate numerical results about the stability of the singular driven cavity problem

differential constraints and exact solutions of nonlinear diffusion equations the differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations certain linear determining equations with parameters are used to find such differential constraints they generalize the determining equations used in the search for classical lie symmetries

on an extension of the notion of reedy category we extend the classical notion of a reedy category so as to allow non trivial automorphisms our extension includes many important examples occurring in topology such as segal s category or the total category of a crossed simplicial group such as connes cyclic category for any generalized reedy category mathbb r and any cofibrantly generated model category mathcal e the functor category mathcal e mathbb r is shown to carry a canonical model structure of reedy type

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

the approximation of vector functions and their derivatives on the sphere a vector basis is defined for vector functions that are given on the sphere it is shown that the basis is orthogonal that it is complete and that the convergence properties of the spectral representation are determined by the smoothness of the vector function in cartesian coordinates the method for analyzing a vector function in terms of this basis uses the methods for analyzing scalar functions in terms of surface harmonics a vector function that is smooth in cartesian coordinates and nonzero at the pole will be discontinuous in spherical coordinates this is a result of the discontinuous spherical coordinate system rather than the vector function itself this in turn induces singularities in certain surface derivatives on the sphere even though the function is bounded and differentiable in cartesian coordinates many of the individual terms in a partial differential equation on the sphere are unbounded at the poles however the cancellation between these terms is sufficient to yield a bounded resu

conditional symmetries and riemann invariants for hyperbolic systems of pdes this paper contains an analysis of rank k solutions in terms of riemann invariants obtained from interrelations between two concepts that of the symmetry reduction method and of the generalized method of characteristics for first order quasilinear hyperbolic systems of pdes in many dimensions a variant of the conditional symmetry method for obtaining this type of solutions is proposed a lie module of vector fields which are symmetries of an overdetermined system defined by the initial system of equations and certain first order differential constraints is constructed it is shown that this overdetermined system admits rank k solutions expressible in terms of riemann invariants finally examples of applications of the proposed approach to the fluid dynamics equations in k 1 dimensions are discussed in detail several new soliton like solutions among them kinks bumps and multiple wave solutions have been obtained these solutions remain bounded even when the riemann invariants admit a gradient catastrophe some physical interpretation of these results is discussed

new similarity reductions of the steady state boundary layer equations new similarity reductions and exact solutions of the steady state boundary layer equations which are of considerable importance in physics are presented besides similarity reductions obtained by applying the non classical group method due to bluman and cole some similarity solutions which figure neither among classical nor among non classical reductions are determined using an extension of the non classical method

a reduction procedure for generalized riemann problems with application to nonlinear transmission lines generalized simple wave solutions to quasilinear hyperbolic nonhomogeneous systems of pdes are obtained through the differential constraint method these solutions prove to be flexible enough to solve generalized riemann problems where discontinuous initial data are involved within such a theoretical framework the governing model of nonlinear transmission lines is investigated throughout

new conditional symmetries and exact solutions of reaction diffusion convection equations with exponential nonlinearities abstract a complete description of q conditional symmetries for a class of reaction diffusion convection equations with exponential diffusivities is derived it is shown that all the known results for reaction diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa the symmetries obtained for constructing exact solutions of the relevant equations are successfully applied the exact solutions are compared with those found earlier by means of different approaches an application of the solutions for solving a boundary value problem arising in population dynamics is presented

weak transversality and partially invariant solutions new exact solutions are obtained for several nonlinear physical equations namely the navier stokes and euler systems an isentropic compressible fluid system and a vector nonlinear schrodinger equation the solution method makes use of the symmetry group of the system in situations when the standard lie method of symmetry reduction is not applicable

approximate similarity reduction for singularly perturbed boussinesq equation via symmetry perturbation and direct method we investigate the singularly perturbed boussinesq equation in terms of the approximate symmetry perturbation method and the approximate direct method the similarity reduction solutions and similarity reduction equations of different orders display formal coincidence for both methods series reduction solutions are consequently derived for the approximate symmetry perturbation method similarity reduction equations of the zero order are equivalent to the painleve iv painleve i and weierstrass elliptic equations for the approximate direct method similarity reduction equations of the zero order are equivalent to the painleve iv painleve ii painleve i or weierstrass elliptic equations the approximate direct method yields more general approximate similarity reductions than the approximate symmetry perturbation method

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

approximate conditional symmetries and approximate solutions of the perturbed fitzhugh nagumo equation we consider a perturbation of the fitzhugh nagumo equation the perturbation is proportional to the electric potential across the cell membrane the purpose of this investigation is to determine the effects of a change in electric potential across the cell membrane exact solutions of the perturbed equation are easily obtained from the well known solutions of the unperturbed fitzhugh nagumo equation the method of approximate conditional symmetries is used to obtain first order approximate solutions of the perturbed fitzhugh nagumo equation the approximate solutions are compared with the exact solutions of the perturbed equation the exact solutions of the perturbed equation do not indicate a change in the wave front connecting one constant state to another there is only a proportional increase or decrease in the constant nonzero state the approximate solutions do show a change in the shape of the wave front connecting two constant states as well as a proportional increase or decrease in the constant nonzero state

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

navier stokes spectral solver in a sphere the paper presents the first implementation of a primitive variable spectral method for calculating viscous flows inside a sphere a variational formulation of the navier stokes equations is adopted using a fractional step time discretization with the classical second order backward difference scheme combined with explicit extrapolation of the nonlinear term the resulting scalar and vector elliptic equations are solved by means of the direct spectral solvers developed recently by the authors the spectral matrices for radial operators are characterized by a minimal sparsity diagonal stiffness and tridiagonal mass matrix closed form expressions of their nonzero elements are provided here for the first time showing that the condition number of the relevant matrices grows as the second power of the truncation order a new spectral elliptic solver for the velocity unknown in spherical coordinates is also described that includes implicitly the coriolis force in a rotating frame but requires a minimal coupling between the modal velocity components in the fourier space the numerical tests confirm that the proposed method achieves spectral accuracy and ensures infinite differentiability to all orders of the numerical solution by construction these results indicate that the new primitive variable spectral solver is an effective alternative to the spectral method recently proposed by kida and nakayama where the velocity field is represented in terms of poloidal and toroidal functions

all solutions of standard symmetric linear partial differential equations have classical lie symmetry abstract it is proven that every solution of any linear partial differential equation with an independent variable deforming classical lie point symmetry is invariant under some classical lie point symmetry this is true for any number of independent variables and for equations of any order higher than one although this result makes use of the infinite dimensional component of the lie symmetry algebra due to linear superposition it is shown that new similarity solutions previously thought not to be classical can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra this result applies to all constant coefficient equations and to many variable coefficient equations such as fokker planck equations

the cyclic homology of crossed product algebras in their article 9 on cyclic homology feigin and tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra generalizing burghelea s calculation 4 of the cyclic homology of a group algebra for an analogous spectral sequence for the hochschild homology of a crossed product algebra see brylinski 2 3 in this article we give a new derivation of this spectral sequence and generalize it to negative and periodic cyclic homology hc a and hp a the method of proof is itself of interest since it involves a natural generalization of the notion of a cyclic module in which the condition that the morphism n n is cyclic of order n 1 is relaxed to the condition that it be invertible we call this category the paracyclic category given a paracyclic module p we can define a chain complex c p with differentials b and b which respectively lower and raise degree the condition that the module p is paracyclic translates to the condition on c p that 1 bb bb is invertible our main result is to show that there is an analogue of the eilenberg zilber theorem for bi paracyclic modules it is then easy to obtain a new expression for the cyclic homology of a crossed product algebra which leads immediately to the spectral sequence of feigin and tsygan if m is a module over a commutative ring k we will denote by m k the iterated tensor product defined by m 0 k and m k 1 m k m if m and n are graded modules we will denote by m n their graded tensor product we would like to thank j block j l brylinski and c kassel for a number of interesting discussions on the results presented here

nonclassical symmetry reduction and riemann wave solutions abstract in this paper we employ the nonclassical symmetry method in order to obtain riemann multiple wave solutions of a system of first order quasilinear differential equations we show how to construct a lie module of vector fields which are symmetries of the system supplemented by certain first order differential constraints we demonstrate the usefulness of our approach on several examples

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

efficient spectral galerkin methods iv spherical geometries fast spectral galerkin algorithms are developed for elliptic equations on the sphere the algorithms are based on a double fourier expansion and have quasi optimal optimal up to a logarithmic term computational complexity numerical experiments indicate that they are significantly more efficient and or accurate when compared with the algorithms based on spherical harmonics and on finite difference extensions to problems in spherical layers and to vector equations are also discussed

cyclic symmetries of an quiver representations abstract this note contains a combinatorial construction of symmetries arising in symplectic geometry partially wrapped or infinitesimal fukaya categories algebraic geometry derived categories of singularities and k theory waldhausen s s construction our specific motivation in the spirit of expectations of kontsevich is a combinatorial construction of categorical quantizations of lagrangian skeleta of symplectic manifolds the main result of this paper gives an immediate solution in the one dimensional case of ribbon graphs

nonclassical symmetry reductions for the kadomtsev petviashvili equation abstract in this paper we discuss dimensional reductions of the physically and mathematically significant kadomtsev petviashvili equation in particular we exhibit some new reductions of the kadomtsev petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by clarkson and kruskal j math phys 30 1989 2201 new solutions of the kadomtsev petviashvili equation are obtained in terms of weierstrass elliptic functions solutions of the lame equations and the first second and fourth painleve transcendents a group theoretical explanation of the new solutions is provided in terms of conditional symmetries leaving only specific subsets of solutions of the kadomtsev petviashvili equation invariant

on symmetries of radially symmetric nonlinear diffusion equations in this paper symmetries are sought for radially symmetric nonlinear diffusion equations of the type ut r1 n rn 1f u ur r the nonlinear forms of this equation which admit lie classical symmetries are completely classified and are shown to exist when f un eu also a trivial symmetry exists for arbitrary f lie backlund symmetries are also considered when f un and shown to exist when n 0 linear case and when n 2 and n 1

nonclassical symmetries as special solutions of heir equations in phys d 78 1994 124 we have found that iterations of the nonclassical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the given equation in the present paper we show that special solutions of the right order heir equation correspond to classical and nonclassical symmetries of the original equations an infinite number of nonlinear equations which possess nonclassical symmetries are derived

partially invariant solutions of nonlinear klein gordon and laplace equations the concept of partially invariant solutions is discussed in the framework of the group analysis of systems of partial differential equations using a general and systematic approach based on subgroup classification methods nontrivial partially invariant solutions for specific equations are explicitly obtained belonging to the nlkge and nlle classes in two dimensions it is proven that the obtained partially invariant solutions are distinct from the invariant ones and from solutions obtained by other methods of reducing pdes to odes

constructing conditionally integrable evolution systems in 1 1 dimensions a generalization of invariant modules approach given a generalized lie backlund vector field satisfying certain nondegeneracy assumptions we explicitly describe all 1 1 dimensional evolution systems that admit this vector field as a generalized conditional symmetry the connection with the theory of symmetries of systems of odes and with the theory of invariant modules is discussed

spectral solvers for spherical elliptic problems a new family of direct spectral solvers for the 3d helmholtz equation in a spherical gap and inside a sphere for nonaxisymmetric problems is presented a variational formulation no collocation is adopted based on the fourier expansion and the associated legendre functions to represent the angular dependence over the sphere and using basis functions generated by legendre or jacobi polynomials to represent the radial structure of the solution in the present method boundary conditions on the polar axis and at the sphere center are not required and never mentioned by construction the spectral solution of the vector dirichlet problem is also considered by employing a transformation that uncouples the spherical components of the fourier modes and that is implemented here for the first time the condition numbers of the matrices involved in the scalar solvers are computed and the spectral convergence of all the proposed solution algorithms is verified by numerical tests

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant