the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

group classification of variable coefficient generalized kawahara equations an exhaustive group classification of variable coefficient generalized kawahara equations is carried out as a result we derive new variable coefficient nonlinear models admitting lie symmetry extensions all inequivalent lie reductions of these equations to ordinary differential equations are performed we also present some examples on the construction of exact and numerical solutions

nonlinear equations invariant under the poincare similitude and conformal groups in two dimensional space time all realizations of the lie algebras p 1 1 sim 1 1 and conf 1 1 are classified under the action of the group of local diffeomorphisms of r3 the result is used to obtain all second order scalar differential equations invariant under the corresponding poincare similitude and conformal groups the invariant equations are in general nonlinear and the requirement of linearity turns out to be very restrictive group invariant solutions of some of the conformally invariant equations are obtained either by quadratures or by a linearizing transformation

contractions of low dimensional lie algebras theoretical background of continuous contractions of finite dimensional lie algebras is rigorously formulated and developed in particular known necessary criteria of contractions are collected and new criteria are proposed a number of requisite invariant and semi invariant quantities are calculated for wide classes of lie algebras including all low dimensional lie algebras an algorithm that allows one to handle one parametric contractions is presented and applied to low dimensional lie algebras as a result all one parametric continuous contractions for both the complex and real lie algebras of dimensions not greater than 4 are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases levels and colevels of low dimensional lie algebras are discussed in detail properties of multiparametric and repeated contractions are also investigated

formal linearization of nonlinear massive representations of the connected poincare group let u1 be an arbitrary finite direct sum of unitary irreducible representations each of positive mass 2 of the connected poincare group p0 r4 sl 2 c it is proved that each nonlinear representation of p0 with linear part u1 on the space of differentiable vectors of u1 is formally linearizable further it is remarked that commonly used nonlinear massless representations of r2 so0 1 1 are nonlinearizable and that the corresponding evolution equations do not have covariant wave operators

semi implicit time discretization schemes for the bidomain model the bidomain model is a system of partial differential equations used to model the propagation of electrical potential waves in the myocardium it is composed of coupled parabolic and elliptic partial differential equations as well as at least one ordinary differential equation to model the ion activity through the cardiac cell membranes the purpose of this paper is to propose and analyze several implicit semi implicit and explicit time stepping methods to solve that model in particular to avoid the expensive resolution of a nonlinear system through the newton raphson method we identify necessary stability conditions on the time step delta t for the proposed methods through a theoretical analysis based on energy estimates we next compare the methods for one and two dimensional test cases in terms of both stability and accuracy of the numerical solutions the theoretical stability conditions are seen to be consistent with those observed in practice our analysis allows us to recommend using either the crank nicolson adams bashforth method or the second order semi implicit backward differention method these semi implicit methods produce a good numerical solution unlike the explicit methods their stability does not depend on the spatial grid size and unlike the implicit methods they do not require the resolution of a system of nonlinear equations

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

on covariant realizations of the euclid group we classify realizations of the lie algebras of the rotation o 3 and euclid e 3 groups within the class of first order differential operators in arbitrary finite dimensions it is established that there are only two distinct realizations of the lie algebra of the group o 3 which are inequivalent within the action of a diffeomorphism group using this result we describe a special subclass of realizations of the euclid algebra which are called covariant ones by analogy to similar objects considered in classical representation theory furthermore we give an exhaustive description of realizations of the lie algebra of the group o 4 and construct covariant realizations of the lie algebra of the generalized euclid group e 4

realizations of real low dimensional lie algebras using a new powerful technique based on the notion of megaideal we construct a complete set of inequivalent realizations of real lie algebras of dimension no greater than four in vector fields on a space of an arbitrary finite number of variables our classification amends and essentially generalizes earlier works on the subject

a scalable newton krylov schwarz method for the bidomain reaction diffusion system a novel two level newton krylov schwarz nks solver is constructed and analyzed for implicit time discretizations of the bidomain reaction diffusion system in three dimensions this multiscale system describes the bioelectrical activity of the heart by coupling two degenerate parabolic equations with several ordinary differential equations at each point in space together with a finite element discretization in space the proposed nks bidomain solver employs an outer inexact newton iteration to solve the nonlinear finite element system originating at each time step of the decoupled implicit discretization the jacobian update during the newton iteration is solved by a krylov method employing a two level overlapping schwarz preconditioner a convergence rate estimate is proved for the resulting preconditioned operator showing that its condition number is independent of the number of subdomains scalability and bounded by the ratio of the subdomains characteristic size and the overlap size this theoretical result is confirmed by several parallel simulations employing up to more than 2000 processors for scaled and standard speedup tests in three dimensions the results show the scalability of the proposed nks bidomain solver in terms of both nonlinear and linear iterations in both cartesian slabs and ellipsoidal cardiac domains

lie symmetries of a generalised non linear schrodinger equation ii exact solutions for pt i see ibid vol 21 p 1493 1988 the authors obtain group invariant solutions of the non linear equation i psi t delta psi a0 psi a1 psi mod psi mod 2 a2 psi mod psi mod 4 for which the symmetry group was previously shown to be the extended galilei group for a1a2 not 0 and the extended galilei similitude group for a1 0 or a2 0 they use the symmetry subgroups to reduce the equation to ordinary differential equations which are solved whenever possible with the help of a singularity analysis solutions are obtained in terms of elementary functions jacobi elliptic functions and painleve transcendents

multilevel additive schwarz preconditioners for the bidomain reaction diffusion system multilevel additive schwarz methods are analyzed and studied numerically for the anisotropic cardiac bidomain model in three dimensions this is the most complete model to date of the bioelectrical activity of the heart tissue consisting of a degenerate parabolic system of nonlinear reaction diffusion equations coupled with a stiff system of several ordinary differential equations describing the ionic currents through the cellular membrane due to the presence of very different scales in both space and time the numerical discretization of this system by finite elements in space and semi implicit methods in time produces very ill conditioned linear systems that must be solved at each time step the proposed multilevel algorithm employs a hierarchy of nested meshes with overlapping schwarz preconditioners on each level and is fully additive hence parallel within and among levels convergence estimates are proved for the resulting multilevel algorithm showing that its convergence rate is independent of the number of subdomains scalability of the mesh sizes of each level and of the number of levels optimality several parallel tests on a linux cluster confirm the scalability and optimality of the method as well as its parallel efficiency on both cartesian and deformed domains in three dimensions

a hybrid multilevel schwarz method for the bidomain model abstract a hybrid multilevel schwarz method is studied numerically for the anisotropic bidomain model in both two and three dimensions this multiscale system models the electrical activity of the heart and it consists of two degenerate parabolic non linear reaction diffusion equations coupled with a stiff system of ordinary differential equations the numerical discretization of the whole system by finite elements in space and semi implicit methods in time generates ill conditioned linear systems that must be solved at each time step the multilevel algorithm studied employs a hierarchy of nested meshes with overlapping schwarz preconditioners on each level and is additive within the levels and multiplicative among the levels we perform several parallel tests on two linux clusters showing that the convergence of the method is independent of the number of subdomains scalability the discretization parameters and the number of levels optimality moreover the comparison with the traditional block jacobi ilu parallel preconditioner and the v cycle multigrid parallel preconditioner shows that on a whole heart cycle simulation the proposed method attains the best performances in terms of cpu times

a model based block triangular preconditioner for the bidomain system in electrocardiology we introduce a preconditioner for the solution of the bidomain system governing the propagation of action potentials in the myocardial tissue the bidomain model is a degenerate parabolic set of nonlinear reaction diffusion equations the nonlinear term describes the ion flux at the cellular level the degenerate nature of the problem results in a severe ill conditioning of its discretization our preconditioning strategy is based on a suitable adaptation of the monodomain model a simplified version of the bidomain one which is by far simpler to solve nevertheless is unable to capture significant features of the action potential propagation the monodomain preconditioner application to a non symmetric formulation of the bidomain system results at the algebraic level in a lower block triangular preconditioner we prove optimality of the preconditioner with respect to the mesh size and corroborate our theoretical results with 3d numerical simulations both on idealized and real ventricle geometries

realizations of galilei algebras all inequivalent realizations of the galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by shirokov the varieties of the deformed galilei algebras are discussed and families of one parametric deformations are presented in explicit form it is also shown that a number of well known and physically interesting equations and systems are invariant with respect to the considered galilei algebras or their deformations

improved formulations of qcs6 hybrid element for enhancing computational efficiency this article discusses two alternative methods of formulating the recently proposed quadrilateral plane element qcs6 which is based on a generalized functional with multiple stress fields the first method is derived by an extended hu washizu principle with the presence of incompatible modes and enforcement of energy orthogonality while the second one is established on a non conventional discrete local stress smoothing both methods can be applied to other structural element formulations with enhanced efficiency

the convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations this paper reviews the convergence properties of linear stationary iterative processes for solving large unstructured systems of linear equations special attention is given to the case where the system to be solved is singular and possibly inconsistent the review starts by investigating the iteration bf x k 1 h bf x k bf f k 0 1 2 cdots when the system i h bf x bf f is inconsistent in this case the sequence left bf x k right always diverges nevertheless when h is convergent there exists a vector bf q in text null i h such that the sequence hat bf x k bf x k k bf q converges the limit of this sequence hat bf x solves a certain least squares problem and the sequence of residual vectors bf r k i h bf x k bf f converges to i h hat bf x bf f furthermore if h is similar to a real diagonal matrix then the corresponding norms of bf r k and hat bf x k hat bf x decr

efficient algebraic solution of reaction diffusion systems for the cardiac excitation process in this paper we deal with the problem of solving the large algebraic linear system arising in the numerical solution of a reaction diffusion r d system associated with myocardial excitation process modeling we show that an ad hoc preconditioning technique can be devised so as to efficiently and simultaneously handle the differential equations of the r d system with no additional memory requirements two different formulations are commonly considered for the theoretical and numerical analyses respectively we observe that the formulation employed for the theoretical analysis of the problem actually yields the best numerical performance when compared with the usual numerical scheme

c integrable nonlinear partial differentiation equations i a technique to generate c integrable nonlinear partial differentiation equations pdes i e nonlinear pdes linearizable by an appropriate change of variables is reported and several examples of such pdes are exhibited

the modulation of weakly non linear dispersive waves near the marginal state of instability the modulation of a one dimensional weakly non linear purely dispersive quasi monochromatic wave the carrier is usually governed by the non linear schrodinger ns equation the critical wavenumber for which the carrier is marginally modulationally unstable is determined by the condition that the product of the coefficients of the non linear and dispersive terms in the ns equation is zero however near this marginal state the assumptions that lead to the ns equation are invalid and a modified form of the ns equation that involves higher order non linearities is appropriate this modified ns equation is here derived formally for a general system involving a single dependent variable and a revised form of the instability criterion is obtained the results are illustrated by considering a particular system described by a generalised korteweg de vries equation

on integrable doebner goldin equations we integrate sub families of a family of nonlinear schrodinger equations proposed by doebner and goldin in the 1 1 dimensional case which have exceptional lie symmetries since the method of integration involves non local transformations of dependent and independent variables general solutions obtained include implicitly determined functions by properly specifying one of the arbitrary functions contained in these solutions we obtain broad classes of explicit square integrable solutions the physical significance and some analytical properties of the solutions obtained are briefly discussed

a quadrilateral mixed finite element with two enhanced strain modes summary an improved plane strain stress element is derived using a hu washizu variational formulation with bilinear displacement interpolation seven strain and stress terms and two enhanced strain modes the number of unknowns of the four node element is increased from eight to ten degrees of freedom for linear and non linear applications the two unknowns associated with the enhanced strain terms can be eliminated by static condensation so that eight displacement degrees of freedom remain for the proposed element which is denoted by qe2 the excellent performance of the proposed element is demonstrated using several linear and non linear examples

block preconditioners for coupled physics problems finite element discretizations of multiphysics problems frequently give rise to block structured linear algebra problems that require effective preconditioners we build two classes of preconditioners in the spirit of well known block factorizations m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 i c f ipsen siam j sci comput 23 2001 pp 1050 1051 and apply these to the diffusive portion of the bidomain equations and the benard convection problem an abstract generalized eigenvalue problem allows us to give application specific bounds for the real parts of eigenvalues for these two problems this analysis is accompanied by numerical calculations with several interesting features one of our preconditioners for the bidomain equations converges in five iterations for a range of problem sizes for benard convection we observe mesh independent convergence with reasonable robustness with respect to physical parameters and offer some preliminary para

evolution equations invariant under two dimensional space time schrodinger group the most general second order evolution equation t f x t x x xx xx 0 invariant under the galilei galilei similitude and schrodinger groups in two dimensions is constructed a preliminary step is a classification of all possible realizations of the corresponding lie algebras of vector fields in r2 c parametrized by x t and applications of this study include the investigation of nonlinear alternatives to quantum mechanics and nonrelativistic classical field theories among the schrodinger invariant equations in particular are found integrable equations linearizable by contact transformations

on new galilei invariant equations in two dimensional spacetime we study realizations of galilei groups acting as transformation lie groups in the space of two independent variables and one dependent variable classification of realizations of the lie algebras and within the class of lie vector fields is carried out utilizing the classification results we have constructed the full sets of second order scalar differential equations in two dimensional spacetime invariant under the lie algebras and

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

a splitting preconditioner for a block two by two linear system with applications to the bidomain equations we construct an alternating splitting iteration scheme for solving and preconditioning a block two by two linear system arising from numerical discretizations of the bidomain equations the convergence theory of this class of splitting iteration methods is established and some useful properties of the preconditioned matrix are analyzed the potential of this approach is illustrated by numerical experiments

partially invariant solutions of a class of nonlinear schrodinger equations partially invariant solutions of a general class of nonlinear schrodinger equations involving four arbitrary functions of the modulus rho of the solution and its derivative rho x are obtained the modulus rho xi is assumed to depend on a symmetry variable xi whereas the phase omega x t depends on both independent variables both rho and omega are obtained explicitly as are the conditions on the coefficients in the equation necessary for such solutions to exist

relations between incompatible displacement model and hybrid stress model an examination of the variational formulations confirms the similarity between the incompatible displacement model and the assumed stress hybrid model that was pointed out by irons in 1972 but the basic differences between the two are also identified for 8 node solid elements the assumed stress terms obtained through a rational procedure also agree with those deduced by irons purely from his physical insight

computation of invariants of lie algebras by means of moving frames a new purely algebraic algorithm is presented for computation of invariants generalized casimir operators of lie algebras it uses the cartan method of moving frames and the knowledge of the group of inner automorphisms of each lie algebra the algorithm is applied in particular to computation of invariants of real low dimensional lie algebras a number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature bases of invariants of the real six dimensional solvable lie algebras with four dimensional nilradicals are newly calculated and listed in a table