initial boundary value problem of a parabolic hyperbolic system arising from tumor angiogenesis abstract in this paper we consider the half space problem for a parabolic hyperbolic system arising from tumor angiogenesis under a mixed type boundary condition we construct the green s function for half space problem by relating it with fundamental solution for an initial value problem after differential equation method for the boundary operator and proper estimates for additional exponential factors generated by the asymmetry of characteristics one reduces the green s function for initial boundary value problem into fundamental solution for cauchy problem which could be estimated by singularity removal long wave short wave decomposition and weighted energy method outside cone we finally obtain the pointwise estimate for green s function which results in the pointwise convergence rate for nonlinear problem the green s function constructed is precise enough to avoid a priori estimate for derivatives by energy method the half space problem studied here is a main ingredient for future study of shock profile with presence of boundary

asymptotic behavior toward nonlinear waves for radially symmetric solutions of the multi dimensional burgers equation abstract the present paper is concerned with the asymptotic behaviors of radially symmetric solutions for the multi dimensional burgers equation on the exterior domain in r n n 3 where the boundary and far field conditions are prescribed we show that in some case where the corresponding 1 d riemann problem for the non viscous part admits a shock wave the solution tends toward a linear superposition of stationary and rarefaction waves as time goes to infinity and also show the decay rate estimates furthermore we improve the results on the asymptotic stability of the stationary waves which are treated in the previous papers 2 3 finally for the case of n 3 we give the complete classification of the asymptotic behaviors which includes even a linear superposition of stationary and viscous shock waves

approximate symmetries and approximate solutions for a multidimensional landau ginzburg equation the authors give the approximate symmetries for the multidimensional landau ginzburg equation delta 2u delta x2i delta u delta x4 a1 a2u in un where n in r and 0 in 1 they also construct approximate solutions for this nonlinear equation using the approximate symmetries

transfer matrix study of the adsorption of a flexible self avoiding polymer chain in two dimensions the authors model the polymer as a self avoiding walk on a semi infinite square lattice with an energy contribution in from each step along the lattice boundary the critical adsorption temperature ta and the ordinary and special surface exponents of the walk above and at the adsorption temperature respectively are estimated with transfer matrix finite size scaling methods for the adsorption temperature they obtain exp in kta 2 041 or 0 002 and for the crossover and susceptibility exponents of the special transition phi sp 0 501 or 0 003 and gamma 1sp 1 454 or 0 004

finite element galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems abstract this paper establishes some refined estimates for the approximation of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems by finite element or more generally galerkin methods suppose lambda is an eigenvalue of multiplicity q of a selfajoint problem and let m lambda denote the space of eigenvectors corresponding to lambda keywords hilbert space

study of the θ point by enumeration of self avoiding walks on the triangular lattice the author reports series expansion analyses of self avoiding walks with nearest neighbour bond interactions the estimates 2 nu t 1 07 or 0 05 and phi 0 64 or 0 05 for the correlation and crossover exponents at the theta point were obtained by examining the number of walks and the end to end distance data up to 16 steps on the triangular lattice

symmetry and transformation properties of iterative ordinary differential equation abstract symmetries of linear iterative equations and new conditions on the infinitesimals are obtained regarding the expressions of the solutions in terms of the parameters of the source equation an ansatz is made on the original parameters we have also obtained an expression for the source parameters of the transformed equation under equivalence transformations we conducted this work with a special emphasis on second third and fourth order equations although some of our results are valid for equations of a general order

on the construction of approximate solutions for a multidimensional nonlinear heat equation we study three methods based on continuous symmetries to find approximate solutions for the multidimensional nonlinear heat equation delta u delta x0 delta u aun epsilon f u where a and n are arbitrary real constants f is a smooth function and 0 epsilon 1

on a variational approximation method for a class of elliptic eigenvalue problems in composite structures we consider a second order elliptic eigenvalue problem on a convex polygonal domain divided in m nonoverlapping subdomains the conormal derivative of the unknown function is continuous on the interfaces while the function itself is discontinuous we present a general finite element method to obtain a numerical solution of the eigenvalue problem starting from a nonstandard formally equivalent variational formulation in an abstract setting in product hilbert spaces we use standard lagrange finite element spaces on the subdomains moreover the bilinear forms are approximated by suitable numerical quadrature formulas we obtain error estimates for both the eigenfunctions and the eigenvalues allowing for the case of multiple exact eigenvalues by a pure variational method

crossover exponent for piecewise directed walk adsorption on sierpinski fractals we study the problem of critical adsorption of piecewise directed random walks on a boundary of fractal lattices that belong to the sierpinski gasket family by applying the exact real space renormalization group method we calculate the crossover exponent associated with the number of adsorbed steps for the complete fractal family we demonstrate that our results are very close to the results obtained for ordinary self avoiding walk and discuss the asymptotic behaviour of at the fractal to euclidean lattice crossover

estimation of the effect of numerical integration in finite element eigenvalue approximation finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration the effect of this numerical integration on the eigenvalue and eigenfunction error is estimated specifically for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well known estimates for approximations without quadrature provided the quadrature rules have appropriate degrees of precision

surface critical exponents for models of polymer collapse and adsorption the universality of the theta and theta points the surface critical exponents are estimated both in ordinary and in special regimes from exact series expansions of up to 28 terms for a model of the theta point they are found to be in agreement with those derived for the theta point confirming the conjecture that the theta and theta points lie in the same universality class some essential features of the phase diagram for a self interacting self avoiding polymer in solution are also calculated to a higher degree of accuracy than previously obtained

external finite element approximations of eigenfunctions in the case of multiple eigenvalues abstract the paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains h are not subdomains of the original domain r 2 the considerations are restricted to piecewise linear approximations special attention is devoted to the convergence of approximate eigenfunctions in the case of multiple exact eigenvalues as yet the approximate solutions have been compared with linear combinations of exact eigenfunctions with coefficients depending on the mesh parameter h we avoid this disadvantage

the effect of attractive monomer monomer interactions on adsorption of a polymer chain the authors have used transfer matrix techniques to study a model of polymer adsorption and collapse they consider the conventional model of a self avoiding walk with attractive nearest neighbour interactions on the square lattice and introduce adsorbing boundaries the numerical results are consistent with the existence of a multicritical point where the adsorption and collapse transitions coincide and suggest that the adsorption transition of a collapsed polymer chain in first order there is also clear evidence that the collapse transition lowers the adsorption temperature estimates of the surface scaling dimensions at the ordinary theta transition support the conjecture of seno and stella 1988 that the surface critical behaviour at the theta and theta points belong to different universality classes

finite element approximation of spectral problems with neumann boundary conditions on curved domains this paper deals with the finite element approximation of the spectral problem for the laplace equation with neumann boundary conditions on a curved nonconvex domain convergence and optimal order error estimates are proved for standard piecewise linear continuous elements on a discrete polygonal domain h in tile framework of the abstract spectral approximation theory

finite element approximation for 2nd order elliptic eigenvalue problems with nonlocal boundary or transition conditions in this paper we use two symplectic schemes to simulate the ablowitz ladik model associated to the cubic nonlinear schrodinger equation and we compare them with nonsymplectic methods

coalescence cascades and special function solutions for the continuous and discrete painleve equations we present a systematization of the special function type solutions of continuous and discrete painleve equations our method is to start from and construct its special solutions from the solutions of the hypergeometric equation and by coalescence obtain the lower painleve equations together with their special solutions in the discrete case we study the symmetric one component three point mapping forms of discrete painleve equations starting from d

collapsing and adsorbing polygons the adsorption transition in the phase diagram of a self interacting lattice polygon is examined the polygon has a nearest neighbour contact fugacity and the interaction between the polygon and an impenetrable wall is modelled by a visit fugacity which is conjugate to the number of vertices of the polygon incident with the wall the partition function of this model is z c y z d p v cp c v c y v z c where p c v c is the number of polygons with c nearest neighbour contacts v visits to the wall and n edges and counted up to translations parallel to the wall the limiting free energy of this model is f c y z d limn 1 1 logz c y z and it is known to be a non analytic function of y for each z 1 for all z2 0 1

equivalence transformations and approximate solutions of a nonlinear heat conduction model we propose a method of finding approximate solutions associated with nonlinear partial differential equations involving a small parameter by using equivalence transformations as an application of our method we consider a one dimensional nonlinear heat conduction model and study its approximate solutions

propagation of a stationary shock layer in the presence of a boundary we are interested in the large time behavior of a stationary shock layer in the presence of a boundary effect it is shown that the stationary shock propagates away from the boundary with speed inversely proportional to the time t consequently the location of the wave front is of order log thinspace 1 t we show therefore that the zero viscosity and time asymptotic limits do not commute our analysis involves detailed point wise estimates which allow us to trace the shock

boundary critical behavior of d 2 self avoiding walks on correlated and uncorrelated vacancies in this paper we present exact results for the critical exponents of interacting self avoiding walks with ends at a linear boundary effective interactions are mediated by vacancies correlated and uncorrelated on the dual lattice by choosing different boundary conditions several ordinary and special regimes can be described in terms of clusters geometry and of critical and lowtemperature properties of the mathcal o n 1 model in particular the problem of boundary exponents at the point is fully solved and implications for point universality are discussed the surface crossover exponent at the special transition of noninteracting self avoiding walks is also interpreted in terms of percolation dimensions

on approximate symmetry and approximate solutions of the nonlinear wave equation with a small parameter the concept of approximate symmetry is introduced the authors describe all nonlinearities f u with which the nonlinear wave equation square operator u lambda u3 epsilon f u 0 with a small parameter epsilon is approximately scale and conformally invariant some approximate solutions of wave equations in question are obtained using the approximate symmetry

a finite element method for differential eigenvalue problems with mixed classical and semi periodic boundary conditions this paper deals with a class of elliptic eigenvalue problems evps for second order differential equations des on a rectangular domain email protected r 2 with periodic or semi periodic boundary conditions bcs on two opposite sides of w on the remaining sides classical robin or dirichlet type bcs are imposed this type of evp is given a suitable variational formulation which allows us to recast the evp in the framework of abstract evps for symmetric bounded coercive bilinear forms in hilbert spaces leading to the existence of exact eigenpairs from the other side the variational evp serves as the starting point for finite element methods without and with numerical quadrature both triangular and rectangular meshes are considered moreover we allow for the case of multiple exact eigenvalues well known error estimates established for finite element approximations of elliptic evps with classical bcs are shown to remain valid for the present type of evp too some attention is also paid to the computational aspects of the resulting algebraic evp finally the analysis is illustrated by a numerical example

asymptotic stability of traveling wave solutions of systems for one dimensional gas motion the asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics 1 the solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet for some 0 ast provided that the initial disturbance is small and of integral zero and in addition decays at an algebraic rate for x 2 the traveling wave solution with nishida and smoller s condition of the system of a viscous heat conductive ideal gas is asymptotically stable provided the initial disturbance is small and of integral zero 3 the traveling wave solution with weak shock profile of the broadwell model system of the boltzmann equation is asymptotically stable provided the initial disturbance is small and its hydrodynamical moments are of integral zero each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one the property of integral zero of the initial disturbance plays a crucial role in this procedure

boundary effect on a stationary viscous shock wave for scalar viscous conservation laws the initial boundary value problem on the negative half line r formula is considered subsequently to t p liu and k nishihara 1997 j differential equations133 296 320 here the flux f is a smooth function satisfying f u 0 and the oleinik shock condition f 0 for u u if u u in this situation the corresponding cauchy problem on the whole line r to has a stationary viscous shock wave x x0 for any fixed x0 our aim in this paper is to show that the solution u x t to behaves as x d t with d t o ln t as t under the suitable smallness conditions when f u2 2 the fact was shown by t p liu and s h yu 1997 arch rational mech anal 139 57 82 based on the hopf cole transformation our proof is based on the weighted energy method

asymptotic stability of traveling waves for scalar viscous conservation laws with non convex nonlinearity the asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t f u x u xx with the initial datau0 which tend to the constant statesu asx stability theorems are obtained in the absence of the convexity off and in the allowance ofs shock speed f u moreover the rate of asymptotics in time is investigated for the casef u s f u if the integral of the initial disturbance over x is small and decays at the algebraic rate as x then the solution approaches the traveling wave at the corresponding rate ast this rate seems to be almost optimal compared with the rate in the casef u2 2 for which an explicit form of the solution exists the rate is also obtained in the casef u s under some additional conditions proofs are given by applying an elementary weighted energy method to the integrated equation of the original one the selection of the weight plays a crucial role in those procedures

finite element approximation of spectral acoustic problems on curved domains this paper deals with the finite element approximation of the displacement formulation of the spectral acoustic problem on a curved non convex two dimensional domain convergence and error estimates are proved for raviart thomas elements on a discrete polygonal domain h inline equation in the framework of the abstract spectral approximation theory similar results have been previously proved only for polygonal domains numerical tests confirming the theoretical results are reported

finite element approximation of the elasticity spectral problem on curved domains we analyze the finite element approximation of the spectral problem for the linear elasticity equation with mixed boundary conditions on a curved non convex domain in the framework of the abstract spectral approximation theory we obtain optimal order error estimates for the approximation of eigenvalues and eigenvectors two kinds of problems are considered the discrete domain does not coincide with the real one and mixed boundary conditions are imposed some numerical results are presented

asymptotic behavior for scalar viscous conservation laws with boundary effect abstract we consider the asymptotic stability of viscous shock wave for scalar viscous conservation lawsut f u x uxxon the half space 0 with boundary valuesu x u u x 0 u our problem is divided into three cases depending on the sign of shock speedsof the shock u u whens 0 the asymptotic state ofubecomes d t whered t depends implicitly on the initial datau x 0 and is related to the boundary layer of the solution at the boundaryx 0 the stability of this state fors 0 is also treated

exact enumeration results for self avoiding walks on the honeycomb lattice attached to a surface we consider self avoiding walks on the honeycomb lattice interacting with a surface with different energies associated between sites in contact with a linear boundary to the left of the origin and those in contact with the right of the boundary we numerically confirm recent exact results for the polymer adsorption transition and corresponding critical exponents with mixed ordinary and special boundary conditions the phase diagram is elucidated with the aid of some rigorous arguments

estimates for the errors in eigenvalue and eigenvector approximation by galerkinmethods with particular attention to the case of multiple eigenvalues refined estimates for the errors in eigenvalue and eigenvector approximation by finite element or more generally galerkin methods as they apply to selfadjoint problems are presented particular attention is given to the case of multiple eigenvalues the results are new in this case the proof is based on a novel approach which yields the known results for simple eigenvalues in a simple way numerical computations are presented and analyzed in light of the theoretical results

bilinear discrete painleve ii and its particular solutions by analogy to the continuous painleve ii equation we present particular solutions of the discrete painleve ii d pii equation these solutions are of a rational and special function airy type our analysis is based on the bilinear formalism that allows us to obtain the tau function for d pii two different forms of bilinear d pii are obtained and we show that they can be related by a simple gauge transformation