some examples of kinetic schemes whose diffusion limit is il in s exponential fitting this paper is concerned with diffusive approximations of some numerical schemes for several linear or weakly nonlinear kinetic models which are motivated by wide range applications including radiative transfer or neutron transport run and tumble models of chemotaxis dynamics and vlasov fokker planck plasma modeling the well balanced method applied to such kinetic equations leads to time marching schemes involving a scattering s matrix itself derived from a normal modes decomposition of the stationary solution one common feature these models share is the type of diffusive approximation their macroscopic densities solve drift diffusion systems for which a distinguished numerical scheme is il in scharfetter gummel s exponential fitting discretization we prove that these well balanced schemes relax within a parabolic rescaling towards such type of discretization by means of an appropriate decomposition of the s matrix hence are asymptotic preserving

eigenvalues of elliptic boundary value problems with an indefinite weight function on etudie les problemes aux valeurs propres elliptiques autoadjoints au r x u dans un ensemble ouvert r k a est un operateur positif d ordre 2m et le poids r est une fonction qui change de signe dans et peut etre discontinue on determine le comportement asymptotique des valeurs propres

limit sets of automatic sequences abstract we show that for every k automatic sequence there exists a natural number p 0 such that the sequences of the form kpn j n 0 with j 0 p 1 are scaling sequences for f moreover we demonstrate that every limit set is the union of certain basic limit sets

the similarity problem for j nonnegative sturm liouville operators abstract sufficient conditions for the similarity of the operator a 1 r x d 2 d x 2 q x with an indefinite weight r x sgn x r x are obtained these conditions are formulated in terms of titchmarsh weyl m coefficients sufficient conditions for the regularity of the critical points 0 and of j nonnegative sturm liouville operators are also obtained this result is exploited to prove the regularity of 0 for various classes of sturm liouville operators this implies the similarity of the considered operators to self adjoint ones in particular in the case r x sgn x and q l 1 r 1 x d x we prove that a is similar to a self adjoint operator if and only if a is j nonnegative the latter condition on q is sharp i e we construct q 1 l 1 r 1 x d x such that a is j nonnegative with the singular critical point 0 hence a is not similar to a self adjoint operator for periodic and infinite zone potentials we show that j positivity is sufficient for the similarity of a to a self adjoint operator in the case q 0 we prove the regularity of the critical point 0 for a wide class of weights r this yields new results for forward backward diffusion equations

automatic sets and delone sets automatic sets are characterized by having a finite number of decimations they are equivalently generated by fixed points of certain substitution systems or by certain finite automata as examples two dimensional versions of the thue morse baum sweet rudin shapiro and paperfolding sequences are presented we give a necessary and sufficient condition for an automatic set to be a delone set in the result is then extended to automatic sets that are defined as fixed points of certain substitutions the morphology of automatic sets is discussed by means of examples

inverse spectral problems for differential pencils on the half line with turning points the inverse spectral problem of recovering pencils of second order differential operators on the half line with turning points is studied we establish properties of the spectral characteristics give a formulation of the inverse problem prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem

on a backward forward parabolic equation and its regularization abstract the steady state diffusion of heat in a fluid which flows rapidly but not unidirectionally in a semi infinite channel leads to a singular perturbation of a backwards forwards heat equation with neumann conditions on the channel walls the temperature is prescribed on the whole channel entrance for the perturbed problems but only at the points where the fluid flow is inward for the limiting problem conditions are obtained for uniqueness and the additional arbitrary constants needed to obtain a well posed problem when uniqueness does not hold are characterized conditions for the convergence of the solution of the perturbed problem to that of the limiting problem are also found

boundary and initial boundary value problems for separable backward forward parabolic problems a backward forward parabolic equation is considered in a strip which is infinite in the timelike direction with boundary conditions on the sides of the strip the unique bounded solution of the problem is given explicitly by separation of variables in addition a similar problem in a semi infinite strip is treated with initial data at the end of the strip it is shown that the solution can be represented arbitrarily closely in the maximum norm by a sum of functions obtain by separation of variables

galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics error estimates for galerkin proper orthogonal decomposition pod methods for nonlinear parabolic systems arising in fluid dynamics are proved for the time integration the backward euler scheme is considered the asymptotic estimates involve the singular values of the pod snapshot set and the grid structure of the time discretization as well as the snapshot locations

mixed finite element formulation and error estimates based on proper orthogonal decomposition for the nonstationary navier stokes equations in this paper proper orthogonal decomposition pod is used for model reduction of mixed finite element mfe for the nonstationary navier stokes equations and error estimates between a reference solution and the pod solution of reduced mfe formulation are derived the basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual mfe method for the nonstationary navier stokes equations or from physics system trajectories by drawing samples of experiments and interpolation or data assimilation and then the basis functions of the usual mfe method are substituted with the pod basis functions reconstructed by the elements of the ensemble to derive the pod reduced mfe formulation for the nonstationary navier stokes equations it is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between pod solution of reduced mfe formulation and the reference solution is consistent with theoretical results moreover it is also shown that this result validates the feasibility and efficiency of the pod method

a reduced high order compact finite difference scheme based on proper orthogonal decomposition technique for kdv equation abstract in this paper a reduced implicit sixth order compact finite difference cfd6 scheme which combines proper orthogonal decomposition pod technique and high order compact finite difference scheme is presented for numerical solution of the korteweg de vries kdv equation high order compact finite difference scheme is applied to attain high accuracy for kdv equation and the pod technique is used to improve the computational efficiency of the high order compact finite difference scheme this method is validated by considering the simulation of five examples and the numerical results demonstrate that the reduced sixth order compact finite difference r cfd6 scheme can largely improve the computational efficiency without a significant loss in accuracy for solving kdv equation

inverse spectral problems for left definite sturm liouville equations with indefinite weight abstract inverse sturm liouville problems with indefinite weight are considered theorems analogous to those of both hochstadt and gelfand levitan are proved as a sample result under suitable symmetry conditions we show that half of one spectrum suffices to determine the potential function uniquely

a reduced finite volume element formulation and numerical simulations based on pod for parabolic problems a proper orthogonal decomposition pod method is applied to a usual finite volume element fve formulation for parabolic equations such that it is reduced to a pod fve formulation with lower dimensions and high enough accuracy the error estimates between the reduced pod fve solution and the usual fve solution are analyzed it is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions moreover it is also shown that the reduced pod fve formulation based on pod method is both feasible and highly efficient

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

elliptic eigenvalue problems with eigenparameter dependent boundary conditions abstract let a be a uniformly elliptic second order linear operator on a smooth bounded domain r n we study the eigenvalue problem au u subject to boundary conditions b0u b1u on where bj are linear boundary operators the problem is recast in the form a u u in a hilbert or krein space and results are given on the location and type of the spectrum full and half range completeness and regularity of critical points

indefinite sturm liouville problems and half range completeness where the function h changes sign on i such problems occur in certain physical models particularly in transport theory and statistical physics see 2 9 11 131 and references therein as for the standard eigenvalue problem it is true under quite general conditions and not difficult to prove that there is a set of eigenfunctions un which is complete in an appropriate hilbert space of functions this is known as full range completeness for the physical problems the question of interest is generally that of half range completeness to formulate this we set

abstract time dependent transport equations abstract we consider the abstract time dependent linear transport equation as an initial boundary value evolution problem in the banach spaces lp 1 p

ray solution of a singularly perturbed elliptic pde with applications to communications networks we analyze a second order linear elliptic pde with mixed boundary conditions this problem arose as a limiting case of a markov modulated queueing model for data handling switches in communications networks we use singular perturbation methods to analyze the problem in particular we use the ray method to solve the pde in the limit where convection dominates diffusion we show that there are both interior and boundary caustics as well as a cusp point where two caustics meet an internal layer boundary layers and a corner layer our analysis leads to approximate formulas for the queue length or buffer content distribution at the switch

half range solutions of indefinite sturm liouville problems abstract for diffusion equations of indefinite sturm liouville type we develop two equivalent methods of constructing explicit representations for the solutions the first method is based on an eigenfunction expansion whereas the second uses a wiener hopf type integral equation and factorization some illustrative examples are worked out

probability representations of a class of two way diffusions there has been little progress in the analysis of two way diffusion in the last few decades due to the difficulties brought by the interface section similar to a free boundary condition in this paper however the equivalent probability model is considered and the interface section is precisely described by an integral equation the solution of two way diffusion is then expressed in an integral form with the integrand being the solution of a classical first passage time model and the solution of a one dimensional integral equation which is relatively easier to solve the exact expression of the two way diffusion enables us to find the explicit solution of the model with infinite horizontal boundaries and without drifting

stability of stationary transport equations with accretive collision operators abstract in this paper we consider transport equations with accretive collision operators we characterize when the equation has a unique solution and show that in this case the solution is stable under small perturbations of the collision operator and the initial value in one case in which there is more than one solution we show how to make a special selection of a solution which is then stable again under small perturbations of both the collision operator and the initial value the results obtained here parallel those obtained earlier for the case where the collision operator is positive semidefinite

the kinetic boundary layer for the klein kramers equation a new numerical approach we explore a numerical technique for determining the structure of the kinetic boundary layer of the klein kramers equation for noninteracting brownian particles in a fluid near a wall that absorbs the brownian particles the equation is of interest in the theory of diffusion controlled reactions and of the coagulation of colloidal suspensions by numerical simulation of the langevin equation equivalent to the klein kramers equation we amass statistics of the velocities at the first return to the wall and of the return times for particles injected into the fluid at the wall with given velocities the data can be used to construct the solutions of the standard problems at an absorbing wall the milne and the albedo problem we confirm and extend earlier results by burschka and titulaer obtained by a variational method vexed by the slow convergence of the underlying eigenfunction expansion we briefly discuss some further boundary layer problems that can be attacked by exploiting the results reported here

half range completeness for the fokker planck equation half range completeness theorems are proved for eigenfunctions associated to the one dimensional fokker planck equation in a semi infinite medium existence and uniqueness results for perfectly absorbing partially absorbing and purely specularly reflecting boundary conditions are deduced for the stationary and time dependent problems similar results are obtained for a slab geometry

half range completeness for the fokker planck equation with an external force this paper deals with a generalization of a classical result obtained by r beals and v protopopescu for the fokker planck equations to the case in which a constant external force is present

proper orthogonal decomposition based reduced order stabilized mixed finite volume element extrapolating model for the nonstationary incompressible boussinesq equations abstract in this study we employ a proper orthogonal decomposition pod method to establish a pod based reduced order stabilized mixed finite volume element smfve extrapolating model with very few degrees of freedom for the nonstationary incompressible boussinesq equations we provide error estimates of the pod based reduced order smfve solutions and the algorithm implementation for the pod based reduced order smfve extrapolating model in addition we present some numerical experiments that verify the validity and reliability of the pod based reduced order smfve extrapolating model

the similarity problem for indefinite sturm liouville operators and the help inequality abstract we study two problems the first one is the similarity problem for the indefinite sturm liouville operator a sgn x d w d x d r d x acting in l w 2 b b it is assumed that w r l loc 1 b b are even and positive a e on b b the second object is the so called help inequality 0 b 1 r f d x 2 k 2 0 b f 2 w d x 0 b 1 w 1 r f 2 w d x where the coefficients w r l loc 1 0 b are positive a e on 0 b both problems are well understood when the corresponding sturm liouville differential expression is regular the main objective of the present paper is to give criteria for both the validity of the help inequality and the similarity to a self adjoint operator in the singular case namely we establish new criteria formulated in terms of the behavior of the corresponding weyl titchmarsh m functions at 0 and at as a biproduct of this result we show that both problems are closely connected namely the operator a is similar to a self adjoint one precisely if the help inequality with w r and r w is valid next we characterize the behavior of m functions in terms of coefficients and then these results enable us to reformulate the obtained criteria in terms of coefficients finally we apply these results for the study of the two way diffusion equation also known as the time independent fokker planck equation

convergence of finite volume schemes for poisson s equation on nonuniform meshes an error analysis of a family of cell center finite volume schemes for poisson s equation on cartesian product nonuniform meshes in two dimensions is presented optimal order error estimates are derived in the discrete h 1 norm under minimum smoothness requirements on the exact solution and without any additional assumption on the regularity of the mesh on quasi uniform meshes analogous estimates are obtained in the maximum norm

a krein space approach to symmetric ordinary differential operators with an indefinite weight function on considere les proprietes spectrales du probleme differentiel l f 1 n p 0 f n n 1 n 1 p 1 f n 1 n 1 p n f rf sur un intervalle fini ou infini a b avec des coefficients reels localement sommables 1 p 0 p 1 p n r sous l hypothese que p 0 0 et que la fonction poids r change de signe sur a b

abstract kinetic equations relevant to supercritical media abstract the abstract hilbert space equation t x a x x r is studied with a partial range boundary condition q 0 ran q here t is bounded injective and self adjoint a is fredholm and self adjoint with finite dimensional negative part and q is the orthogonal projection onto the maximal t positive t invariant subspace this models half space stationary transport problems in supercritical media a complete existence and uniqueness theory is developed

galerkin proper orthogonal decomposition methods for parabolic problems in this work error estimates for galerkin proper orthogonal decomposition pod methods for linear and certain non linear parabolic systems are proved the resulting error bounds depend on the number of pod basis functions and on the time discretization numerical examples are included

full and partial range eigenfunction expansions for sturm liouville problems with indefinite weights this article is concerned with eigenvalue problems of the form au tu in a hilbert space h where ais a selfadjoint positive operator generated by a second order sturm liouville differential expression and t a selfadjoint indefinite multiplicative operator which is one to one emphasis is on the full range and partial range expansionproperties of the eigenfunctions

an analogue of cobham s theorem for fractals we introduce the notion of k self similarity for compact subsets of n and show that it is a natural analogue of the notion of k automatic subsets of integers we show that various well known fractals such as the triadic cantor set the sierpinski carpet or the menger sponge turn out to be k self similar for some integers k we then prove an analogue of cobham s theorem for compact sets of that are self similar with respect to two multiplicatively independent bases k and l namely we show that x is both a k and an l self similar compact subset of if and only if it is a finite union of closed intervals with rational endpoints

analysis of the cell centred finite volume method for the diffusion equation abstract this paper presents a rigorous theoretical analysis of the cell centred finite volume method for poisson s equation we review the traditional taylor series expansion technique which suggests that the cell centred method is inconsistent on nonuniform grids which is not confirmed by numerical experiments we then present an analysis of the method which confirms that the solution error does indeed reduce as the cell size is reduced this is supported by numerical calculations

the ring of k regular sequences ii in this paper we continue our study of k regular sequences begun in 1992 we prove some new results give many new examples from the literature and state some open problems

automaticity of double sequences generated by one dimensional linear cellular automata abstract we give a complete answer to the question whether a double sequence that is generated by a one dimensional linear cellular automaton and whose states are integers modulo m is k automatic or not

half range analysis of a counter current separator abstract chemical separations are usually accomplished in counter current reactors because in these reactors the chemical equilibrium continually shifts toward increasing the purity of the product stream as it flows through the reactor however as a result of the counter current design these devices are often governed by forward backward two way diffusion equations in this article we analyze the mathematical model of a single interface counter current separator introduced by fitt ockendon and shillor for relatively long reactors for such reactors the mathematical model which is a forward backward diffusion system can be solved by eigenfunction expansions we construct these expansions using complex variable methods developed for kinetic theory from this analysis we obtain explicit formulas for the power of the reactor both in terms of the average purity of the output stream and in terms of the equivalent number of theoretical plates

on the accuracy of the finite volume element method for diffusion equations on composite grids this paper develops the finite volume element fve method which is similar to the so called control volume finite element method but tailored to composite grid applications fve is described for the general two dimensional diffusion equation in divergence form o h 3 2 discretization error estimates are then developed for the case of triangular elements these results are improved to o h 2 using a special modification involving rectilinear elements at the grid interface

partial range completeness and existence of solutions to two way diffusion equations separating variables to solve a two way diffusion equation leads to a nonstandard eigenvalue problem in one variable it is shown that the eigenfunctions having negative eigenvalues are complete on the part of the domain where initial conditions are imposed while those with positive eigenvalues are complete where final conditions are imposed the corresponding exponentially growing and exponentially decaying solutions may be used to expand arbitrary solutions on semi infinite intervals in the time variable a natural iterative procedure for obtaining solutions on finite intervals is shown to converge in some cases a linearly growing solution must also be taken into account

estimates on the non real eigenvalues of regular indefinite sturm liouville problems regular sturm liouville problems with indefinite weight functions may possess finitely many non real eigenvalues in this paper we prove explicit bounds on the real and imaginary parts of these eigenvalues in terms of the coefficients of the differential expression

overlapping schwarz waveform relaxation method for the solution of the forward backward heat equation in this article we analyzed the convergence of the schwarz waveform relaxation method for solving the forward backward heat equation numerical results are presented for a specific type of model problem

a reduced order crank nicolson finite volume element formulation based on pod method for parabolic equations in this paper a proper orthogonal decomposition pod method is used to deal with a classical crank nicolson finite volume element cnfve method for two dimensional parabolic equations a reduced order cnfve formulation with lower dimensions and sufficiently high accuracy based on pod technique is established the error estimates between reduced order cnfve solutions based on the pod method and classical cnfve solutions are provided and the extrapolation algorithm for solving reduced order cnfve formulation is implemented some numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions moreover it is shown that the reduced order cnfve formulation based on pod method is feasible and efficient for solving two dimensional parabolic equations

a reduced order dg formulation based on pod method for the time domain maxwell s equations in dispersive media abstract in this work a proper orthogonal decomposition pod method is applied to time domain maxwell s equations coupled to a drude dispersion model which are discretized in space by a discontinuous galerkin dg method an auxiliary differential equation ade method is used to represent the constitutive relation for the dispersive medium a pod dgtd formulation with lower dimension and sufficiently high accuracy is established together with the description of the pod reduced order basis its construction from a snapshot set and its application to the solution of the time domain maxwell s equations the overall goal is to reduce the computational time while maintaining an acceptable level of accuracy in order to obtain an efficient time domain solver to be used as a starting point for an optimization strategy we provide results from numerical experiments for two dimensional problems that illustrate the capabilities of the proposed pod dgtd formulation and assess its efficiency

spectral properties of singular sturm liouville operators with indefinite weight sgn x we consider a singular sturm liouville expression with the indefinite weight sgnx to this expression there is naturally a self adjoint operator in some krein space associated we characterize the local definitizability of this operator in a neighbourhood of 1 moreover in this situation the point 1 is a regular critical point we construct an operator a sgnx id 2 dx 2 q with non real spectrum accumulating to a real point the obtained results are applied to several classes of sturm liouville operators