optimality conditions with state constraints for semilinear systems determined by operator valued measures abstract in this paper we develop the necessary conditions of optimality for a class of distributed parameter systems partial differential equations determined by operator valued measures and controlled by vector measures based on some recent results on existence of optimal controls from the space of vector measures we develop necessary conditions of optimality for a class of control problems the main results are the necessary conditions of optimality for problems without state constraints and those with state constraints also a conceptual algorithm along with a brief discussion of its convergence is presented

existence of solutions of nonlinear stochastic integrodifferential inclusions in a hilbert space in this paper the existence of mild solutions of nonlinear stochastic integrodifferentialinclusions in a hilbert space is studied the results are obtained by using a fixed point theorem for condensing map due to martelli the result is illustrated with an example

optimal a priori error estimates for the hp version of the local discontinuous galerkin method for convection diffusion problems we study the convergence properties of the hp version of the local discontinuous galerkin finite element method for convection diffusion problems we consider a model problem in a one dimensional space domain we allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh width h in the polynomial degree p and in the regularity of the exact solution we identify a special numerical flux for which the estimates are optimal in both h and p the theoretical results are confirmed in a series of numerical examples

periodic duffing delay equations with state dependent impulses conditions are obtained for the duffing equation with delay and state dependent impulses to admit a periodic solution

error estimates for finite element methods for scalar conservation laws in this paper new a posteriors error estimates for the shock capturing streamline diffusion scsd method and the shock capturing discontinuous galerkin scdg method for scalar conservation laws are obtained these estimates are then used to prove that the scsd method and the scdg method converge to the entropy solution with a rate of at least h 1 8 and h 1 8 respectively in the l infty l 1 norm the triangulations are made of general acute simplices and the approximate solution is taken to be piecewise a polynomial of degree k the result is independent of the dimension of the space

superconvergence of the local discontinuous galerkin method for the linearized korteweg de vries equation we study the superconvergence property of the local discontinuous galerkin ldg method for solving the linearized korteweg de vries kdv equation we prove that if the piecewise p k polynomials with k 1 are used the ldg solution converges to a particular projection of the exact solution with the order k 3 2 when the upwind flux is used for the convection term and the alternating flux is used for the dispersive term numerical examples are provided at the end to support the theoretical results

measure solutions for evolution equations with discontinuous vector fields and their optimal control abstract in this paper we present a brief survey of some recent developments of impulsive dynamic systems in infinite dimensional banach spaces regular as well as measure solutions are discussed including some comments on optimal control

hyperbolic polynomial diffeomorphisms of c2 ii hubbard trees this paper is a sequel to part i y ishii hyperbolic polynomial diffeomorphisms of i a non planar map adv math 218 2 2008 417 464 in the current article we construct an object analogous to a hubbard tree consisting of a pair of trees decorated with loops and a pair of maps between them for a hyperbolic polynomial diffeomorphism f of key notions in the construction are the pinching disks and the pinching locus which determine how local dynamical pieces are glued together to obtain a global picture it is proved that the shift map on the orbit space of a hubbard tree is topologically conjugate to f on its julia set several examples of hubbard trees are also given

vector measure for explicit nonlinear impulsive system of glycerol bioconversion in fed batch cultures and its parameter identification in this paper the nonlinear dynamical system of fed batch fermentation is investigated in the process of bio dissimilation of glycerol to 1 3 propanediol by klebsiella pneumoniae taking into account the abrupt changes of concentrations of glycerol and the priority of the value of state to the jump in fed batch culture we propose a explicit nonlinear impulsive dynamical system of glycerol bioconversion to 1 3 propanediol driven by a vector valued measure in fed batch fermentation which is fit for the factual fermentation better than continuous models in being we study the existence and uniqueness of solutions to the impulsive system because of errors between experimental results and numerical simulation results of the proposed impulsive system we establish the parameter identification model for the impulsive system and discuss the identifiability of the parameters and continuous dependence on parameters finally in order to find the optimal parameters of the impulsive system a hybrid particle swarm optimization algorithm is constructed numerical results of simulations show that the optimization algorithm is valid

superconvergence and a posteriori error estimates of the dg method for scalar hyperbolic problems on cartesian grids in this paper we analyze the discontinuous galerkin dg finite element method for the steady two dimensional transport reaction equation on cartesian grids we prove the l2 stability and optimal l2 error estimates for the dg scheme we identify a special numerical flux for which the l2 norm of the solution is of order p 1 when tensor product polynomials of degree at most p are used we further prove superconvergence towards a particular projection of the directional derivative the order of superconvergence is proved to be p 1 2 we also provide a very simple derivative recovery formula which is o h p 1 superconvergent approximation to the directional derivative moreover we establish an o h 2 p 1 global superconvergence for the solution flux at the outflow boundary of the domain these results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element finally we prove that the proposed a posteriori dg error estimates converge to the true errors in the l2 norm at o h p 1 rate and that the global effectivity index converges to unity at o h rate our results are valid without the flow condition restrictions we perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal

stability analysis and a priori error estimates of the third order explicit runge kutta discontinuous galerkin method for scalar conservation laws in this paper we present an analysis of the runge kutta discontinuous galerkin method for solving scalar conservation laws where the time discretization is the third order explicit total variation diminishing runge kutta method we use an energy technique to prove the mathrm l 2 norm stability for scalar linear conservation laws and to obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws quasi optimal order is obtained for general numerical fluxes and optimal order is given for upwind fluxes the theoretical results are obtained for piecewise polynomials with any degree k geq1 under the standard temporal spatial cfl condition tau leq gamma h where h and tau are the element length and time step respectively and the positive constant gamma is independent of h and tau

discontinuous galerkin error estimation for hyperbolic problems on unstructured triangular meshes we extend the error analysis of adjerid and baccouch 1 2 for the discontinuous galerkin discretization error to variable coefficient linear hyperbolic problems as well as nonlinear hyperbolic problems on unstructured meshes we further extend this analysis to transient hyperbolic problems and prove that the local superconvergence results presented in 1 still hold for both steady and transient variable coefficient linear and nonlinear problems this local error analysis allows us to construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on each element of general unstructured meshes we illustrate the superconvergence and the efficiency of our a posteriori error estimates by showing computational results for several linear and nonlinear numerical examples

stability and boundedness criteria of nonlinear impulsive systems employing perturbing lyapunov functions this paper develops a new comparison principle for nonlinear impulsive differential systems then the stability practical stability and boundedness of impulsive differential systems are proved by using the method of perturbing lyapunov functions the notion of perturbing lyapunov functions enables us to discuss stability properties of impulsive systems under much weaker assumptions the reported novel results complement the existing results it may provide a greater prospect for solving problems which exhibit impulsive effects

asymptotically exact a posteriori ldg error estimates for one dimensional transient convection diffusion problems in this paper new a posteriori error estimates for the local discontinuous galerkin ldg formulation applied to transient convection diffusion problems in one space dimension are presented and analyzed these error estimates are computationally simple and are computed by solving a local steady problem with no boundary conditions on each element we first show that the leading error term on each element for the solution is proportional to a p 1 degree right radau polynomial while the leading error term for the solution s derivative is proportional to a p 1 degree left radau polynomial when polynomials of degree at most p are used these results are used to prove that for smooth solutions these error estimates at a fixed time converge to the true spatial errors in the l 2 norm under mesh refinement more precisely we prove that our ldg error estimates converge to the true spatial errors at o h p 5 4 rate finally we prove that the global effectivity indices in the l 2 norm converge to unity at o h 1 2 rate our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates

a note on the convergence of the discontinuous galerkin method for a scalar hyperbolic equation a numerical example is presented which shows that the known l 2 error estimate left u u h right leqq ch k 1 2 left u right k 1 for the discontinuous galerkin method cannot be improved within the class of quasi uniform meshes not even for smooth exact solutions u

diffusion equation for multivalued stochastic differential equations abstract deterministic oscillations with bilinear hysteresis are governed by a multivalued differential equation of the type k b g where k is maximal monotonic and b is lipschitzian an existence and uniqueness result is proven for corresponding stochastic equation the diffusion equation satisfied by the laws of t is established in the particular case k 0 this equation is equivalent to the fokker planck equation

existence of solutions to differential inclusions and to time optimal control problems in the autonomous case we prove existence of solutions to upper semicontinuous differential inclusions and to time optimal control problems under conditions that are strictly weaker than the usual assumption of convexity

a posteriori error estimation for discontinuous galerkin solutions of hyperbolic problems abstract we analyze the spatial discretization errors associated with solutions of one dimensional hyperbolic conservation laws by discontinuous galerkin methods dgms in space we show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a radau polynomial of degree p 1 on each element we also prove that the local and global discretization errors are o x2 p 1 and o x2p 1 at the downwind point of each element this strong superconvergence enables us to show that local and global discretization errors converge as o xp 2 at the remaining roots of radau polynomial of degree p 1 on each element convergence of local and global discretization errors to the radau polynomial of degree p 1 also holds for smooth solutions as p these results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth

an optimal order error estimate for the discontinuous galerkin method in this paper a new approach is developed for analyzing the discontinuous galerkin method for hyperbolic equations for a model problem in r2 the method is shown to converge at a rate 0 hn l when applied with nth degree polynomial approxi mations over a semiuniform triangulation assuming sufficient regularity in the solution

optimal control for a class of strongly nonlinear impulsive equations in banach spaces abstract in this paper we consider the optimal control problem of system governed by a class of strongly nonlinear impulsive evolution equations based on the existence of strongly nonlinear impulsive evolution equations which contain nonlinear monotone operators and nonmonotone perturbations we prove the existence of optimal controls for the general lagrange problem the necessary conditions for optimality are also presented

on discontinuous continuity equation and impulsive ensemble control abstract the note is devoted to a class of singular ensemble control problems we consider the continuity equation driven by a control affine vector field subject to the constraint on the l 1 norm of the input function since no geometric constraints on control are imposed solutions to the continuity equation can potentially tend somehow to discontinuous measure valued functions we elaborate a representation of such discontinuous generalized solutions through an ordinary control equation being a time reparameterization of the characteristic ordinary differential equation ode of the original continuity equation by virtue of standard results from impulsive control theory of odes we perform a constructive formula for generalized solutions in terms of a measure differential equation finally we state an optimal impulsive control problem for the singular continuity equation and prove the existence of its solution the obtained results can be useful for modeling and analysis of impulsive control problems with uncertain initial data

infinitely many periodic attractors for holomorphic maps of 2 variables an important development in the study of discrete dynamical systems was newhouse s use of persistent homoclinic tangencies to show that a large set of c2 diffeomorphisms of compact surfaces have infinitely many coexisting periodic attractors or sinks 6 where large refers to a residual subset of an open set of diffeomorphisms in the present paper we obtain this result for various spaces of holomorphic maps of two variables newhouse later extended his result to show that such residual sets exist near any surface diffeomorphism which has a homoclinic tangency 7 more recently palis and viana extended this latter result to higher dimensions when the stable manifold has codimension one 10 and romero obtained an analogous result for higher codimension stable manifolds using saddles in place of sinks 12 in each case however the construction reduces to the study of intersecting cantor sets in the line under an appropriate projection the stable and unstable manifolds of a basic set are mapped to cantor sets in the line and a tangency between these manifolds corresponds to a point of intersection of the cantor sets the generic unfolding of tangencies then gives rise to periodic attractors or saddles this reduction to linear cantor sets depends heavily on the fact that there is only one expanding eigenvalue even romero s result for higher codimension stable manifolds involves a reduction to this case in the holomorphic setting eigenvalues come in conjugate pairs from a real point of view so this reduction is not possible instead stable and unstable manifolds are riemann surfaces and after extending the stable and unstable manifolds of a basic set to foliations these foliations will be tangent in a real 2 dimensional disk and the stable and unstable manifolds correspond to cantor sets in this disk hence persistent tangencies between basic sets correspond to the stable intersection of two cantor sets in the plane

hyperbolic polynomial diffeomorphisms of c2 iii iterated monodromy groups abstract this paper is a sequel to part i 13 and part ii 14 15 in the current article we relate several combinatorial descriptions of the julia sets for hyperbolic polynomial diffeomorphisms of c 2 quotients of solenoids 3 automata 22 and hubbard trees 14 15 the notion of iterated monodromy groups are defined for such diffeomorphisms and are used to construct automata from hubbard trees

hyperbolic polynomial diffeomorphisms of c2 i a non planar map abstract in this paper we develop a general framework for verifying hyperbolicity of holomorphic dynamical systems in c 2 our framework in particular enables us to construct the first example of a hyperbolic henon map of c 2 which is non planar i e which is not topologically conjugate on its julia set to a small perturbation of any expanding polynomial in one variable the key ideas in its proof are the poincare box which is a building block to apply our criterion for hyperbolicity an operation called fusion to merge two polynomials in one variable to obtain essentially two dimensional dynamics and rigorous computation by using interval arithmetic some conjectures and problems are also presented

discontinuous galerkin methods for nonlinear scalar hyperbolic conservation laws divided difference estimates and accuracy enhancement in this paper an analysis of the accuracy enhancement for the discontinuous galerkin dg method applied to one dimensional scalar nonlinear hyperbolic conservation laws is carried out this requires analyzing the divided difference of the errors for the dg solution we therefore first prove that the alpha th order 1 le alpha le k 1 1 k 1 divided difference of the dg error in the l 2 l2 norm is of order k frac 3 2 frac alpha 2 k 32 2 when upwind fluxes are used under the condition that f u f u possesses a uniform positive lower bound by the duality argument we then derive superconvergence results of order 2k frac 3 2 frac alpha 2 2k 32 2 in the negative order norm demonstrating that it is possible to extend the smoothness increasing accuracy conserving filter to nonlinear conservation laws to obtain at least frac 3 2 k 1 32k 1 th order superconvergence for post processed solutions as a by product for variable coefficient hyperbolic equations we provide an explicit proof for optimal convergence results of order k 1 k 1 in the l 2 l2 norm for the divided differences of dg errors and thus 2k 1 2k 1 th order superconvergence in negative order norm holds numerical experiments are given that confirm the theoretical results

existence results for impulsive differential inclusions with nonlocal conditions abstract in this paper we shall establish sufficient conditions for the existence of mild solutions for nonlocal impulsive differential inclusions on the basis of the fixed point theorems for multivalued maps and the technique of approximate solutions new results are obtained examples are also provided to illustrate our results

transient l1 error estimates for well balanced schemes on non resonant scalar balance laws abstract the ability of well balanced wb schemes to capture very accurately steady state regimes of non resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by greenberg and leroux 1996 15 see also the anterior wb glimm scheme in e 1992 8 this paper aims at showing by means of rigorous c t 0 l x 1 estimates that these schemes deliver an increased accuracy in transient regimes too namely after explaining that for the vast majority of non resonant scalar balance laws the c t 0 l x 1 error of conventional fractional step tang and teng 1995 45 numerical approximations grows exponentially in time like exp max g t x as a consequence of the use of gronwall s lemma it is shown that wb schemes involving an exact riemann solver suffer from a much smaller error amplification thanks to strict hyperbolicity their error grows at most only linearly in time see also layton 1984 30 numerical results on several test cases of increasing difficulty including the classical leveque yee s benchmark problem leveque and yee 1990 34 in the non stiff case confirm the analysis

superconvergence of discontinuous galerkin and local discontinuous galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension in this paper we study the superconvergence property for the discontinuous galerkin dg and the local discontinuous galerkin ldg methods for solving one dimensional time dependent linear conservation laws and convection diffusion equations we prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection diffusion equations the order of superconvergence for both cases is proved to be k frac 3 2 when piecewise p k polynomials with k geq1 are used the proof is valid for arbitrary nonuniform regular meshes and for piecewise p k polynomials with arbitrary k geq1 improving upon the results in y cheng and c w shu j comput phys 227 2008 pp 9612 9627 y cheng and c w shu computers and structures 87 2009 pp 630 641 in which the proof based on fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise p 1 polynomials

corrigendum to hyperbolic polynomial diffeomorphisms of c2 ii hubbard trees adv math 220 4 2009 985 1022

measure solutions for impulsive evolution equations with measurable vector fields in this paper we present some results on the question of existence of generalized or measure valued solutions for semilinear impulsive evolution equations on banach spaces with the nonlinear parts being merely measurable and bounded on bounded sets this is a far reaching generalization of the previous results of the author and others it may be interesting to consider extension of the results of this paper to cover impulsive differential inclusions and their potential applications to control theory and uncertain dynamics

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

superconvergence of the local discontinuous galerkin method for elliptic problems on cartesian grids in this paper we present a superconvergence result for the local discontinuous galerkin ldg method for a model elliptic problem on cartesian grids we identify a special numerical flux for which the l2 norm of the gradient and the l2 norm of the potential are of orders k 1 2 and k 1 respectively when tensor product polynomials of degree at most k are used for arbitrary meshes this special ldg method gives only the orders of convergence of k and k 1 2 respectively we present a series of numerical examples which establish the sharpness of our theoretical results

superconvergence and a posteriori error estimates for the ldg method for convection diffusion problems in one space dimension in this paper we investigate the superconvergence properties of the local discontinuous galerkin ldg method applied to transient convection diffusion problems in one space dimension we show that the leading terms of the local discretization errors for the p degree ldg solution and its spatial derivative are proportional to p 1 degree right and left radau polynomials respectively thus the discretization errors for the p degree ldg solution and its spatial derivative are o h p 2 superconvergent at the roots of p 1 degree right and left radau polynomials respectively the superconvergence results are used to construct asymptotically correct a posteriori error estimates these error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element numerical computations suggest that these a posteriori ldg error estimates for the solution and its spatial derivative at a fixed time t converge to the true errors at o h p 3 and o h p 2 rates respectively we also show that the global effectivity indices for the solution and its derivative in the l 2 norm converge to unity at o h 2 and o h rates respectively finally we show that the ldg method combined with the a posteriori error estimation procedure yields both accurate error estimates and o h p 2 superconvergent solutions our proofs are valid for arbitrary regular meshes and for p p polynomials with p 1 and for periodic dirichlet and mixed dirichlet neumann boundary conditions several numerical simulations are performed to validate the theory

stability and boundedness of nonlinear impulsive systems in terms of two measures via perturbing lyapunov functions this paper develops the concepts of stability practical stability and boundedness in terms of two measures for nonlinear impulsive differential systems using the method of perturbing lyapunov functions the notion of perturbing lyapunov functions enables us to discuss stability properties of solutions of nonlinear impulsive differential systems in terms of two measures under much weaker assumptions the novel results offer a way to unify a variety of stability results found in the relative literature

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

evolution equations determined by operator valued measures and optimal control abstract in this paper we consider a general class of systems determined by operator valued measures briefly we review some recent results on optimal control with controls being vector measures then we consider problems of structural controls where the operator valued measures are considered as the controls we prove existence of optimal controls for linear and semilinear problems the basic results are illustrated by several examples from systems governed by partial differential equations of hyperbolic as well as parabolic types containing coefficients which are vector measures

discontinuous galerkin error estimation for linear symmetric hyperbolic systems in this manuscript we present an error analysis for the discontinuous galerkin discretization error of multi dimensional first order linear symmetric hyperbolic systems of partial differential equations we perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of legendre polynomials of degree p and p 1p 1 we apply these asymptotic results and solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error we present computational results for several linear hyperbolic systems in one two and three dimensions

superconvergence and time evolution of discontinuous galerkin finite element solutions in this paper we study the convergence and time evolution of the error between the discontinuous galerkin dg finite element solution and the exact solution for conservation laws when upwind fluxes are used we prove that if we apply piecewise linear polynomials to a linear scalar equation the dg solution will be superconvergent towards a particular projection of the exact solution thus the error of the dg scheme will not grow for fine grids over a long time period we give numerical examples of p k polynomials with 1

periodic lienard type delay equations with state dependent impulses abstract conditions are obtained for lienard type equations with delay and state dependent impulses to admit an absolutely continuous periodic solution with first derivative of bounded variation and consequently with lebesgue integrable second derivative the results are applied to josephson s equation and the nonconservative forced pendulum equation

periodic solutions of semilinear impulsive periodic system on banach space abstract in this paper a class of semilinear impulsive periodic systems on banach space is considered using impulsive periodic evolution operator the t 0 periodic p c mild solution is introduced and suitable poincare operator is constructed showing the compactness of the poincare operator and using a new generalized gronwall inequality with mixed type integral operators we utilize the leray schauder fixed point theorem to prove the existence of the t 0 periodic p c mild solutions our method is much different from methods of other papers an example illustrates the applicability of our results

superconvergence of discontinuous galerkin solutions for a nonlinear scalar hyperbolic problem in this paper we study the superconvergence of the discontinuous galerkin solutions for nonlinear hyperbolic partial differential equations on the first inflow element we prove that the p degree discontinuous finite element solution converges at radau points with an o h p 2 rate we further show that the solution flux converges on average at o h 2p 2 o n element outflow boundary when no reaction terms are present for reaction convection problems we establish an o h min 2p 2 p 4 superconvergence rate of the flux on element outflow boundary globally we prove that the flux converges at o h 2p 1 on average at the outflow of smooth solution regions for nonlinear conservation laws numerical computations indicate that our results extend to nonrectangular meshes and nonuniform polynomial degrees we further include a numerical example which shows that discontinuous solutions are superconvergent to the unique entropy solution away from shock discontinuities 2005 elsevier b v all rights reserved msc 65m60 65m25 65m15

nonlinear impulsive systems on infinite dimensional spaces abstract in this paper we consider two different classes of nonlinear impulsive systems one driven purely by dirac measures at a fixed set of points and the second driven by signed measures the later class is easily extended to systems driven by general vector measures the principal nonlinear operator is monotone hemicontinuous and coercive with respect to certain triple of banach spaces called gelfand triple the other nonlinear operators are more regular non monotone continuous operators with respect to suitable banach spaces we present here a new result on compact embedding of the space of vector valued functions of bounded variation and then use this result to prove two new results on existence and regularity properties of solutions for impulsive systems described above the new embedding result covers the well known embedding result due to aubin

on a cell entropy inequality for discontinuous galerkin methods we prove a cell entropy inequality for a class of high order dis continuous galerkin finite element methods approximating conservation laws which implies convergence for the one dimensional scalar convex case

error estimates to smooth solutions of runge kutta discontinuous galerkin methods for scalar conservation laws in this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for runge kutta discontinuous galerkin rkdg methods where the time discretization is the second order explicit total variation diminishing tvd runge kutta method error estimates for the mathbb p 1 piecewise linear elements are obtained under the usual cfl condition tau leq gamma h for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions where h and tau are the maximum element lengths and time steps respectively and the positive constant gamma is independent of h and tau however error estimates for higher order mathbb p k k geq 2 elements need a more restrictive time step tau leq gamma h 4 3 we remark that this stronger condition is indeed necessary as the method is linearly unstable under the usual cfl condition tau leq gamma h for the mathbb p k elements of degree k geq 2 error estimates of o h k 1 2 tau 2 are obtained for general monotone numerical fluxes and optimal error estimates of o h k 1 tau 2 are obtained for upwind numerical fluxes

optimality condition of the nonlinear impulsive system in fed batch fermentation abstract based on the nonlinear impulsive system of fed batch fermentation which is used in the process of bio dissimilation of glycerol to 1 3 propanediol by klebsiella pneumoniae this paper is concerned with the optimal control of the volumes of infused glycerol at impulsive moments the optimal control model is developed the authors study the properties of both the nonlinear impulsive system and the solution of it and ascertain the existence of the optimal control then the optimality condition is obtained which can be used as a necessary condition that the optimal control should satisfy

superconvergence of discontinuous galerkin method for scalar nonlinear hyperbolic equations in this paper we study the superconvergence behavior of the semi discrete discontinuous galerkin dg method for scalar nonlinear hyperbolic equations in one spatial dimension superconvergence results for problems with fixed and alternating wind directions are established on the one hand we prove that if the wind direction is fixed i e the derivative of the flux function is bounded away from zero both the cell average error and numerical flux error at cell interfaces converge at a rate of 2k 1 when upwind fluxes and piecewise polynomials of degree k are used moreover we also prove that the function value approximation of the dg solution is superconvergent at interior right radau points and the derivative value approximation is superconvergent at interior left radau points with an order of k 2 and k 1 respectively as a byproduct we show a k 2 th order superconvergence of the dg solution towards the gauss radau projection of the exact solution on the other hand superconvergence resu

a class of differential equations with impulses at variable times on banach spaces abstract in this paper two typical classes of semilinear evolution equation with impulses at variable times on infinite dimensional spaces are considered introducing the reasonable pc mild solutions we discuss the local existence maximal existence interval global existence of pc mild solution for two models in addition to the pulse phenomena an example is given for demonstration

solution bundle for a class of impulsive differential inclusions on banach spaces in this paper a class of impulsive differential inclusions is investigated the existence of solution bundle is proved and we also construct a nonlinear semigroup of operators on cb e closed bounded subset of e to describe the set of attainable states

superconvergence of the local discontinuous galerkin method for the sine gordon equation in one space dimension abstract in this paper we present superconvergence results for the local discontinuous galerkin ldg method for the sine gordon nonlinear hyperbolic equation in one space dimension we identify a special numerical flux and a suitable projection of the initial conditions for the ldg scheme for which the l 2 norm of the ldg solution and its spatial derivative are of order p 1 when piecewise polynomials of degree at most p are used our numerical experiments demonstrate optimal order of convergence we further prove superconvergence toward particular projections of the exact solutions more precisely we prove that the ldg solution and its spatial derivative are o h p 3 2 super close to particular projections of the exact solutions while computational results show higher o h p 2 convergence rate our analysis is valid for arbitrary regular meshes and for p p polynomials with arbitrary p 1 numerical experiments validating these theoretical results are presented

extensions of amenable groups by recurrent groupoids we show that the amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital schreier graphs this applies to a wide class of groups the amenability of which was an open problem as well as unifies many known examples to one general proof in particular this includes grigorchuk s group basilica group group associated to fibonacci tiling the topological full groups of cantor minimal systems groups acting on rooted trees by bounded automorphisms groups generated by finite automata of linear activity growth and groups naturally appearing in holomorphic dynamics

optimal convergence of the original dg method for the transport reaction equation on special meshes we show that the approximation given by the original discontinuous galerkin method for the transport reaction equation in d space dimensions is optimal provided the meshes are suitably chosen the l 2 norm of the error is of order k 1 when the method uses polynomials of degree k these meshes are not necessarily conforming and do not satisfy any uniformity condition they are required only to be made of simplexes each of which has a unique outflow face we also find a new element by element postprocessing of the derivative in the direction of the flow which superconverges with order k 1

hp version discontinuous galerkin methods for hyperbolic conservation laws a parallel adaptive strategy this paper describes a parallel algorithm based on discontinuous hp finite element approximations of linear scalar hyperbolic conservation laws the paper focuses on the development of an effective parallel adaptive strategy for such problems numerical experiments suggest that these techniques are highly parallelizable and exponentially convergent thereby yielding efficiency many times superior to conventional schemes for hyperbolic problems

existence of optimal controls for a general class of impulsive systems on banach spaces this paper presents some results on the question of existence of optimal controls for a large class of semilinear impulsive systems in infinite dimensional spaces with admissible controls from the space of vector measures this also includes as a special case the class of purely impulsive controls two physical examples are presented for illustration

adaptive finite element methods for parabolic problems ii optimal error estimates in l l 2 and l l optimal error estimates are derived for a complete discretization of linear parabolic problems using space time finite elements the discretization is done first in time using the discontinuous galerkin method and then in space using the standard galerkin method the underlying partitions in time and space need not be quasi uniform and the partition in space may be changed from time step to time step the error bounds show in particular that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same given level i e without error accumulation effects the derivation of the estimates is based on the orthogonality of the galerkin procedure and the use of strong stability estimates the particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration

second order nonlinear impulsive integro differential equations of mixed type with time varying generating operators and optimal controls on banach spaces in this paper a class of second order nonlinear impulsive integro differential equations of mixed type whose principle is time varying generating operators with unbounded perturbation on banach spaces is considered discussing the perturbation of time varying operator matrix and constructing corresponding the evolution system generated by operator matrix we introduce the reasonable mild solution of second order nonlinear impulsive integro differential equations of mixed type and prove the existence of mild solutions the existence of optimal controls for a lagrange problem of systems governed by the second order nonlinear impulsive integro equations of mixed type is also presented an example is given for demonstration

superconvergence of discontinuous galerkin methods for scalar nonlinear conservation laws in one space dimension in this paper an analysis of the superconvergence property of the semidiscrete discontinuous galerkin dg method applied to one dimensional time dependent nonlinear scalar conservation laws is carried out we prove that the error between the dg solution and a particular projection of the exact solution achieves left k frac32 right th order superconvergence when upwind fluxes are used the results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree k k ge 1 under the condition that f u possesses a uniform positive lower bound numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions indicating that the restriction on f u is artificial

numerical resolution of discontinuous galerkin methods for time dependent wave equations abstract the discontinuous galerkin dg method is known to provide good wave resolution properties especially for long time simulation in this paper using fourier analysis we provide a quantitative error analysis for the semi discrete dg method applied to time dependent linear convection equations with periodic boundary conditions we apply the same technique to show that the error is of order k 2 superconvergent at radau points on each element and of order 2 k 1 superconvergent at the downwind point of each element when using piecewise polynomials of degree k an analysis of the fully discretized approximation is also provided we compute the number of points per wavelength required to obtain a fixed error for several fully discrete schemes numerical results are provided to verify our error analysis

optimal feedback control for a class of strongly nonlinear impulsive evolution equations in this paper we consider the optimal feedback control problems of a system governed by strongly nonlinear impulsive differential equations which contains monotone operators and nonlinear nonmonotone perturbations based on the existence of feasible pairs an existence result of optimal control pairs for the lagrange problem is presented

a posteriori discontinuous finite element error estimation for two dimensional hyperbolic problems abstract we analyze the discontinuous finite element errors associated with p degree solutions for two dimensional first order hyperbolic problems we show that the error on each element can be split into a dominant and less dominant component and that the leading part is o hp 1 and is spanned by two p 1 degree radau polynomials in the x and y directions respectively we show that the p degree discontinuous finite element solution is superconvergent at radau points obtained as a tensor product of the roots of p 1 degree radau polynomial for a linear model problem the p degree discontinuous galerkin solution flux exhibits a strong o h2p 2 local superconvergence on average at the element outflow boundary we further establish an o h2p 1 global superconvergence for the solution flux at the outflow boundary of the domain these results are used to construct simple efficient and asymptotically correct a posteriori finite element error estimates for multi dimensional first order hyperbolic problems in regions where solutions are smooth

parabolic systems determined by strongly nonlinear operator valued measures abstract in this paper we consider a class of distributed parameter systems partial differential equations governed by strongly nonlinear operator valued measures we prove existence uniqueness and regularity properties of weak solutions this extends our recent results on semilinear and strongly nonlinear parabolic problems similar extensions to hyperbolic problems and stochastic systems remain open

a discontinuous galerkin method for higher order ordinary differential equations in this paper we propose a new discontinuous finite element method to solve initial value problems for ordinary differential equations and prove that the finite element solution exhibits an optimal o tp 1 convergence rate in the l2 norm we further show that the p degree discontinuous solution of differential equation of order m and its first m 1 derivatives are o t2p 2 m superconvergent at the end of each step we also establish that the p degree discontinuous solution is o tp 2 superconvergent at the roots of p 1 m degree jacobi polynomial on each step finally we present several computational examples to validate our theory and construct asymptotically correct a posteriori error estimates

perturbation of the mapping and solvability theorems in banach space abstract here we consider perturbations of continuous mappings on banach spaces and investigate their images under various conditions consequently we study the solvability of some classes of equations and inclusions for these we start by the investigation of local properties of the considered mapping and local comparisons of this mapping with certain smooth mappings moreover we study different mixed problems

hp version discontinuous galerkin methods for hyperbolic conservation laws abstract the development of hp version discontinuous galerkin methods for hyperbolic conservation laws is presented in this work a priori error estimates are derived for a model class of linear hyperbolic conservation laws these estimates are obtained using a new mesh dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation the results generalize and extend previous results on mesh dependent norms to hp version discontinuous galerkin methods a posteriori error estimates which provide bounds on the actual error are also developed in this work numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution

a remark on existence of solutions of infinite dimensional noncompact optimal control problems without compactness of the semigroup optimal controls for semilinear infinite dimensional control problems may not exist even under the usual convexity lower semicontinuity assumptions this is shown with two counterexamples and two fixes for nonexistence are discussed

the optimal error estimate and superconvergence of the local discontinuous galerkin methods for one dimensional linear fifth order time dependent equations in this paper we investigate the optimal error estimate and the superconvergence of linear fifth order time dependent equations we prove that the local discontinuous galerkin ldg solution is k 1 th order convergent when the piecewise p k space is used also the numerical solution is k 3 2 th order superconvergent to a particular projection of the exact solution the numerical experiences indicate that the order of the superconvergence is k 2 which implies the result obtained in this paper is suboptimal

a high order discontinuous galerkin method for ito stochastic ordinary differential equations in this paper we develop a high order discontinuous galerkin dg method for strong solution of ito stochastic ordinary differential equations sdes driven by one dimensional wiener processes motivated by the dg method for deterministic ordinary differential equations odes we first construct an approximate deterministic ode with a random coefficient on each element using the well known wong zakai approximation theorem since the resulting ode converges to the solution of the corresponding stratonovich sde we apply a transformation to the drift term to obtain a deterministic ode which converges to the solution of the original sde the corrected equation is then discretized using the standard dg method for deterministic odes we prove that the proposed stochastic dg sdg method is equivalent to an implicit stochastic runge kutta method then we study the numerical stability of the sdg scheme applied to linear sdes with an additive noise term the method is shown to be numerically stable in the mean sense and also a stable as a result it is suitable for solving stiff sdes moreover the method is proved to be convergent in the mean square sense numerical evidence demonstrates that our proposed dg scheme for sdes with additive noise has a strong convergence order of 2 p 1 when p degree piecewise polynomials are used when applied to sdes with multiplicative noise the sdg method is strongly convergent with order p several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method in particular we demonstrate that our proposed scheme is suitable for stiff stochastic differential systems