time dependent differential inclusions cocycle attractors and fuzzy differential equations traditional formulations of fuzzy differential equations do not reproduce the rich and varied behavior of crisp differential equations des a recent interpretation in terms of differential inclusions expressed level setwise overcomes this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies attractors of fuzzy des in terms of cocycles and encompasses both the time dependent and autonomous cases

a hybrid method for hydrodynamic kinetic flow part ii coupling of hydrodynamic and kinetic models in this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic kinetic problems in which the knudsen number may span from equilibrium to highly rarefied regimes our approach is characterized by using the full boltzmann equation for the kinetic regime the compressible euler equations for equilibrium with a buffer zone in which the bgk es equation is used to represent the transition between fully kinetic to equilibrium flows in this fashion the boltzmann solver is used only when the collision integral is non stiff and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities thus in principle the same mesh size and time steps can be used in the whole computation moreover the time step is limited only by convective terms since the boltzmann solver is applied only in wholly kinetic regimes we use the reduced noise dsmc scheme we have proposed in part i of the present work this ensures a smooth exchange of information across the different domains with a natural way to construct interface numerical fluxes several tests comparing our hybrid scheme with full boltzmann dsmc computations show the good agreement between the two solutions on a wide range of knudsen numbers

approximating the solution of nonlinear hammerstein fuzzy integral equations abstract in this paper we prove the convergence of the method of successive approximations used to approximate the solution of nonlinear hammerstein fuzzy integral equations and define the notion of numerical stability of the algorithm with respect to the choice of the first iteration this numerical stability is proved and finally some numerical experiments confirm the theoretical results and illustrate the accuracy of the method

fixed point theorems for set valued maps and existence principles for integral inclusions new fixed point theorems of monch type are presented for set valued maps these theorems are then used to establish general existence principles for hammerstein integral inclusions in banach spaces

open fuzzy cubature rule with application to nonlinear fuzzy volterra integral equations in two dimensions abstract in this paper we propose an open fuzzy cubature formula and as application we construct an iterative numerical method for solving nonlinear fuzzy volterra integral equations in two dimensions the convergence of the method is proved by providing the error estimate and it is tested on two numerical experiments

stationary solutions for the ellipsoidal bgk model in a slab abstract we address the boundary value problem for the ellipsoidal bgk model of the boltzmann equation posed in a bounded interval the existence of a unique mild solution is established under the assumption that the inflow boundary data does not concentrate too much around the zero velocity and the gas is sufficiently rarefied

two point boundary value problems of undamped uncertain dynamical systems in this paper we study two point boundary value problems of a class of uncertain dynamical systems i e undamped uncertain dynamical systems we introduce the concept of large solutions and small solutions for two point boundary value problems of undamped uncertain dynamical systems and under some conditions establish the existence and uniqueness of large solutions and small solutions and settle the relationship between large solutions and small solutions

ellipsoidal bgk model for polyatomic molecules near maxwellians a dichotomy in the dissipation estimate abstract we consider the global existence and asymptotic behavior of classical solutions to the ellipsoidal bgk model for polyatomic molecules when the initial data starts sufficiently close to a global polyatomic maxwellian we observe that the linearized relaxation operator is decomposed into a truly polyatomic part and an essentially monatomic part leading to a dichotomy in the dissipative property in the sense that the degeneracy of the dissipation shows an abrupt jump as the relaxation parameter reaches zero accordingly we employ two different sets of micro macro system to derive the full coercivity and close the energy estimate

on the relation between classical and quantum thermodynamic entropy we discuss the unresolved problem of proving rigorously that in the classical limit 0 the quantum thermodynamic entropy functional tends to the classical entropy functional we state rather restrictive conditions that define the general problem of finding a complete classical phase space representation of quantum kinematics whether the problem admits of solutions remains an unresolved question we discuss a physically interesting attempt to relate the structure in the classical limit 0 of the well known blokhintzev wigner and wehrl phase space functions to the spectral expansion of the quantum state operator

one sided fuzzy numbers and applications to integral equations from epidemiology in this paper we present the algebraic and topological structure of one sided fuzzy numbers and introduce the notion of side preserving fuzzy number valued function we define the dual decomposition of a commutative monoid and prove that the sets of left sided fuzzy numbers and right sided fuzzy numbers realize a dual decomposition of the additive monoid of all fuzzy numbers we prove the usefulness of right sided fuzzy numbers in epidemiology presenting an application of these fuzzy numbers to a mathematical model for the spread of infectious diseases with a rate of contact that varies seasonally

numerical comparison between the boltzmann and es bgk models for rarefied gases abstract rarefied gas flows obey the boltzmann equation but numerical simulations of this equation are not always possible so that simpler models have been introduced the es bgk equation is one of these models it gives the correct transport coefficients for the navier stokes approximation so that boltzmann or es bgk simulations are expected to give the same results for dense gases but in the case of a rarefied flow complete numerical comparisons are needed in this paper we present numerical comparisons between the two models in transitional regimes where the es bgk model is expected to be useful for reentry flows around a compression ramp and a plate we also emphasize that the es bgk model gives flow predictions closer to the boltzmann result than the simpler bgk model

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

an implicit monte carlo method for rarefied gas dynamics for the space homogeneous boltzmann equation we formulate a hybrid monte carlo method that is robust in the fluid dynamic limit this method is based on an analytic representation of the solution over a single time step and involves implicit time differencing derived from a suitable power series expansion of the solution a generalized wild expansion a class of implicit yet explicitly implementable numerical schemes is obtained by substituting a maxwellian distribution in place of the high order terms in the expansion the numerical solution is represented as a convex combination of a non equilibrium particle distribution and a maxwellian the hybrid distribution is then evolved by monte carlo using the implicit formulation for the time evolution computational simulations of spatially homogeneous problems by our method are presented here for the kac model and for the variable hard sphere model including maxwell molecules comparison to exact solutions and to direct simulation monte carlo dsmc computations shows the robustness and the efficiency of the new method

hukuhara difference of z numbers abstract real world problems in decision analysis economics optimization and other fields are characterized by fuzzy and partially reliable information in this regard prof zadeh suggested the concept of a z number as an ordered pair z a b of fuzzy numbers a and b the first of which describes a linguistic value and the second one is a fuzzy value of probability measure of the first one playing a role of reliability of information construction of mathematical models to handle z number based information is a new challenge which requires development of z number valued analysis including z number valued equations z number valued derivative and integral and other concepts one of the basic concepts in this realm is an inverse of addition operation in the existing literature devoted to set valued arithmetic and fuzzy arithmetic this operation is referred to as hukuhara difference for interval numbers and fuzzy numbers the existence conditions of hukuhara difference are derived unfortunately the concept of hukuhara difference for z numbers is not developed in this work for the first time we suggest a fundamental approach for development of the concept of hukuhara difference of z numbers examples are provided in the paper to show validity of the suggested approach

towards the accuracy of iterative numerical methods for fuzzy hammerstein fredholm integral equations abstract an iterative numerical method based on the three point quadrature formula is applied to solve nonlinear fuzzy hammerstein fredholm integral equations of the second kind and provides more accurate results than the trapezoidal quadrature iterative method also we obtain the error estimation of the iterative method in terms of the uniform and partial modulus of continuity moreover we investigate the concept of numerical stability of the iterative method with respect to the choice of the initial approximation illustrative examples are included to demonstrate the accuracy and efficiency of the iterative method

microscopically implicit macroscopically explicit schemes for the bgk equation in this work a new class of numerical methods for the bgk model of kinetic equations is introduced the schemes proposed are implicit with respect to the distribution function while the macroscopic moments are evolved explicitly in this fashion the stability condition on the time step coincides with a macroscopic cfl evaluated using estimated values for the macroscopic velocity and sound speed thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of energetic particles either with the technique proposed here the updating of the distribution function requires the solution of a linear system of equations even though the bgk model is highly non linear thus the proposed schemes are particularly effective for high or moderate mach numbers where the macroscopic cfl condition is comparable to accuracy requirements we show results for schemes of order 1 and 2 and the generalization to higher order is sketched

general entropic approximations for canonical systems described by kinetic equations in this paper we extend the general construction of entropic approximation for kinetic operators modelling canonical systems more precisely this paper aims at pursuing to thermalized systems the works of levermore schneider and junk on moments problems relying on entropy minimization in order to construct bgk approximations and moments based equations

measure valued solutions to conservation laws motivated by traffic modelling motivated by a traffic model we study a conservation law whose solutions naturally need to be bounded radon measures we investigate the relations between this equation and finite systems of conservation laws that converge to it

cauchy problem for the ellipsoidal bgk model of the boltzmann equation ellipsoidal bgk model es bgk is a generalized version of the original bgk model designed to reproduce the physically correct prandtl number in the navier stokes limit in this paper we study the cauchy problem for the es bgk model under the condition of finite initial mass energy and entropy equivalence type estimates for the temperature tensor are crucially used

an asymptotic preserving automatic domain decomposition method for the vlasov poisson bgk system with applications to plasmas in this work we present an efficient strategy to deal with plasma physics simulations in which localized departures from thermodynamical equilibrium are present the method is based on the introduction of intermediate regions which allows smooth transitions between kinetic and fluid zones in this paper we extend domain decomposition techniques obtained through dynamic coupling and buffer zones to the study of plasmas and moreover we combine them with asymptotic preserving and asymptotically accurate strategies for the time integration we use a hybrid scheme in which both kinetic and fluid descriptions are considered and coupled together while the kinetic model is solved by asymptotic preserving and accurate methods in order to guarantee high efficiency and accuracy in all regimes the numerical scheme is validated and its performances are analyzed by numerical simulations

a multiscale kinetic fluid solver with dynamic localization of kinetic effects this paper collects the efforts done in our previous works 7 9 10 to build a robust multiscale kinetic fluid solver our scope is to efficiently solve fluid dynamic problems which present non equilibrium localized regions that can move merge appear or disappear in time the main ingredients of the present work are the followings ones a fluid model is solved in the whole domain together with a localized kinetic upscaling term that corrects the fluid model wherever it is necessary this multiscale description of the flow is obtained by using a micro macro decomposition of the distribution function 9 the dynamic transition between fluid and kinetic descriptions is obtained by using a time and space dependent transition function to efficiently define the breakdown conditions of fluid models we propose a new criterion based on the distribution function itself several numerical examples are presented to validate the method and measure its computational efficiency

an h adaptive mesh method for boltzmann bgk hydrodynamics coupling we introduce a coupled method for hydrodynamic and kinetic equations on 2 dimensional h adaptive meshes we adopt the euler equations with a fast kinetic solver in the region near thermodynamical equilibrium while use the boltzmann bgk equation in kinetic regions where fluids are far from equilibrium a buffer zone is created around the kinetic regions on which a gradually varying numerical flux is adopted based on the property of a continuously discretized cut off function which describes how the flux varies the coupling will be conservative in order for the conservative 2 dimensional specularly reflective boundary condition to be implemented conveniently the discrete maxwellian is approximated by a high order continuous formula with improved accuracy on a disc instead of on a square domain the h adaptive method can work smoothly with a time split numerical scheme through h adaptation the cell number is greatly reduced this method is particularly suitable for problems with hydrodynamics breakdown on only a small part of the whole domain so that the total efficiency of the algorithm can be greatly improved three numerical examples are presented to validate the proposed method and demonstrate its efficiency

two point boundary value problems of uncertain dynamical systems abstract recent research shows that in many cases there are no solutions periodic solutions etc to fuzzy two point boundary value problems in the sense of h derivatives to overcome the disadvantage of h derivatives in studying fuzzy two point boundary value problems in this paper we study fuzzy two point boundary value problems in the sense of differential inclusions i e two point boundary value problems of uncertain dynamical systems we obtain the existence and uniqueness of solutions and the existence of periodic solutions to two point boundary value problems of uncertain dynamical systems after introducing the concept of big solutions to two point boundary value problems of undamped uncertain dynamical systems we establish the existence and uniqueness of big solutions discuss the relationship between solutions and big solutions and further get a conclusion concerning the solutions of two point boundary value problems of uncertain dynamical systems

classical solutions for the ellipsoidal bgk model with fixed collision frequency abstract we establish the existence of global in time smooth solutions for the ellipsoidal bgk model which is a variant of the bgk model for the boltzmann equation designed to yield the correct prandtl number in the hydrodynamic approximation at the navier stokes level for this we carefully design a function space which captures the growth of the solution in a weighted sobolev norm and show that the ellipsoidal relaxation operator is lipschitz continuous in the induced metric this approach is restricted to the case when the collision frequency does not depend on the macroscopic field but no smallness on the initial data is required

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

on random fuzzy differential equations we consider a cauchy problem in a random fuzzy setting under the condition of lipschitzean right hand side the existence and uniqueness of the solution is proven also the continuous dependence on the right hand side and initial condition is shown some kind of boundedness of the solution is established

convergence of discrete velocity schemes for the boltzmann equation in this paper we prove the convergence of two discrete velocity deterministic schemes for the boltzmann equation namely buet s scheme and a new finite volume scheme that we introduce here we write the discretized equation in the form of a boltzmann continuous equation in order to be in the framework of the diperna lions theory of renormalized solutions in order to prove convergence we have to overcome two difficulties the convergence of the discretized collision kernel is very weak and the lemma on the compactness of velocity averages can be recovered only asymptotically when the parameter of discretization tends to zero

a consistency result for a discrete velocity model of the boltzmann equation in this paper we study the link between a certain class of discrete velocity models dvms and the boltzmann equation those models possess an infinite number of velocities laying on a regular grid of step h in r3 our aim is to prove that it is possible to construct models consistent with the boltzmann equation i e such that the discrete collision term can be seen as an approximation of the collision integral of the boltzmann equation

weighted l bounds and uniqueness for the boltzmann bgk model we prove rather general l bounds for hydrodynamical fields in terms of weighted l norms of the kinetic density we use these estimates to prove l bounds and uniqueness for the solution of the bgk equation which is a simple relaxation model introduced by bhatnagar gross krook to mimic boltzmann flows

a generalization of hukuhara difference and division for interval and fuzzy arithmetic we propose a generalization of the hukuhara difference first the case of compact convex sets is examined then the results are applied to generalize the hukuhara difference of fuzzy numbers using their compact and convex level cuts finally a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown

classical information entropy and phase distributions of optical fields the wehrl phase distribution is defined as a phase density of the wehrl classical information entropy the new measure is applied to describe the quantum phase properties of some optical fields including fock states coherent and squeezed states and superposition of chaotic and coherent fields the wehrl phase distribution is compared with both the conventional wehrl entropy and husimi phase distribution the marginal husimi q function it is shown that the wehrl phase distribution is a good measure of the phase space uncertainty noise phase locking and phase bifurcation effects it is also demonstrated that the wehrl phase distribution properly describes phase randomization processes and thus can be used in a description of the quantum optical phase

periodic problems of first order uncertain dynamical systems it is well known that there are no periodic solutions to fuzzy differential equations in the sense of hukuhara derivatives i e h derivatives so fuzzy differential equations in the sense of h derivatives cannot be used to describe periodic phenomena in the real world to overcome this disadvantage this paper studies periodic problems of first order fuzzy differential equations in the sense of differential inclusions i e periodic problems of first order uncertain dynamical systems and obtains the existence of periodic solutions for first order uncertain dynamical systems

stability and periodicity in fuzzy differential equations formulations of fuzzy differential equations des in terms of the hukuhara derivative do not reproduce the rich and varied behavior of crisp des another interpretation in terms of differential inclusions expressed level setwise overcomes much of this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies studies lyapunov stability of fuzzy des and the periodicity of the fuzzy solution set for both the time dependent and autonomous cases

convergence of a discrete velocity model for the boltzmann bgk equation abstract we prove the convergence of a conservative and entropic discrete velocity model for the bathnagar gross krook bgk equation in this model the approximation of the maxwellian is based on a discrete entropy minimization principle the main difficulty due to its implicit definition is to prove that this approximation is consistent we also demonstrate the existence and uniqueness of a solution to the discrete velocity model by using a fixed point theorem finally the model is written in a continuous equation form and we prove the convergence of its solution toward a solution of the bgk equation

convergence of a semi lagrangian scheme for the bgk model of the boltzmann equation recently a new class of semi lagrangian methods for the bgk model of the boltzmann equation has been introduced f filbet and g russo kinet relat models 2 2009 pp 231 250 g russo and p santagati a new class of large time step methods for the bgk models of the boltzmann equation arxiv 1103 5247 p santagati high order semi lagrangian methods for the bgk model of the boltzmann equation ph d thesis university of catania italy 2007 these methods work in a satisfactory way either in a rarefied or a fluid regime moreover because of the semi lagrangian feature the stability property is not restricted by the cfl condition these aspects make them very attractive for practical applications in this paper we prove that the discrete solution of the scheme converges in a weighted l 1 norm to the unique smooth solution by deriving an explicit error estimate

a hybrid method for hydrodynamic kinetic flow part i a particle grid method for reducing stochastic noise in kinetic regimes in this work we present a hybrid particle grid monte carlo method for the boltzmann equation which is characterized by a significant reduction of the stochastic noise in the kinetic regime the hybrid method is based on a first order splitting in time to separate the transport from the relaxation step the transport step is solved by a deterministic scheme while a hybrid dsmc based method is used to solve the collision step such a hybrid scheme is based on splitting the solution in a collisional and a non collisional part at the beginning of the collision step and the dsmc method is used to solve the relaxation step for the collisional part of the solution only this is accomplished by sampling only the fraction of particles candidate for collisions from the collisional part of the solution performing collisions as in a standard dsmc method and then projecting the particles back onto a velocity grid to compute a piecewise constant reconstruction for the collisional part of the solution the latter is added to a piecewise constant reconstruction of the non collisional part of the solution which in fact remains unchanged during the relaxation step numerical results show that the stochastic noise is significantly reduced at large knudsen numbers with respect to the standard dsmc method indeed in this algorithm the particle scheme is applied only on the collisional part of the solution so only this fraction of the solution is affected by stochastic fluctuations but since the collisional part of the solution reduces as the knudsen number increases stochastic noise reduces as well at large knudsen numbers

investigations of information quantifiers for the tavis cummings model abstract in this article a system of two two level atoms interacting with a single mode quantized electromagnetic field in a lossless resonant cavity via a multi photon transition is considered the quantum fisher information negativity classical fisher information and reduced von neumann entropy for the two atoms are investigated we found that the number of photon transitions plays an important role in the dynamics of different information quantifiers in the cases of two symmetric and two asymmetric atoms our results show that there is a close relationship between the different quantifiers also the quantum and classical fisher information can be useful for studying the properties of quantum states which are important in quantum optics and information

convergence of a weighted particle method for solving the boltzmann b g k equation we consider in this paper the numerical solution of the b g k model for the boltzmann equation the numerical method used here was introduced by mas gallic transport theory statist phys 16 1987 pp 885 887 using the results of perthame and pulvirenti arch rational mech anal 125 1993 pp 289 295 we first derive a uniqueness result for the b g k model and also regularity properties of the solution which are necessary for our analysis we then prove following mas gallic and poupaud transport theory statist phys 17 1988 pp 311 345 the convergence of the numerical approximation for the b g k model finally some numerical experiments are presented

wehrl information entropy and phase distributions of schrodinger cat and cat like states the wehrl information entropy and its phase density the so called wehrl phase distribution are applied to describe schrcat and cat like kitten states the advantages of the wehrl phase distribution over the wehrl entropy in a description of the superposition principle are presented the entropic measures are compared with a conventional phase distribution from the husimi q function compact form formulae for the entropic measures are found for superpositions of well separated states examples of schr odinger cats including even odd and yurke stoler coherent states as well as the cat like states generated in the kerr medium are analysed in detail it is shown that in contrast to the wehrl entropy the wehrl phase distribution properly distinguishes between different superpositions of unequally weighted states with respect to their number and phase space configuration

smooth solutions to the bgk equation and the es bgk equation with infinite energy abstract the bgk model and the es bgk model of the boltzmann equation are of great importance in the kinetic theory of rarefied gases for the cauchy problems with general initial data smooth solutions were obtained only in bounded domains for both models in this paper we establish the existence and uniqueness of smooth solutions in the whole space more important some smooth solutions can permit infiniteness of energy

rarefied flow computations using nonlinear model boltzmann equations high resolution finite difference schemes for solving the nonlinear model boltzmann equations are presented for the computations of rarefied gas flows the discrete ordinate method is first applied to remove the velocity space dependency of the distribution function which renders the model boltzmann equation in phase space to a set of hyperbolic conservation laws with source terms in physical space then a high order essentially nonoscillatory method due to harten et al j comput phys 71 231 1987 is adapted and extended to solve them explicit methods using operator splitting and implicit methods using the lower upper factorization are described to treat multidimensional problems the methods are tested for both steady and unsteady rarefied gas flows to illustrate its potential use the computed results using model boltzmann equations are found to compare well both with those using the direct simulation monte carlo results in the transitional regime flows and those with the continuum navier stokes calculations in near continuum regime flows

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

a note on two point boundary value problems associated with non linear fuzzy differential equations in recent papers lakshmikantham et al and o regan et al assert that two point boundary value problem for a fuzzy differential equation is equivalent to an integral equation and the proofs of the main results in the above cited papers use this equivalence we prove by a counterexample that by using h differentiability or hukuhara differentiability this assertion does not hold moreover it is proved that a large class of two point boundary value problems have no solutions at all under h differentiability or hukuhara differentiability

existence uniqueness and positivity of solutions for bgk models for mixtures abstract we consider kinetic models for a multi component gas mixture without chemical reactions in the literature one can find two types of bgk models in order to describe gas mixtures one type has a sum of bgk type interaction terms in the relaxation operator for example the model described by klingenberg pirner and puppo 20 which contains well known models of physicists and engineers for example hamel 16 and gross and krook 15 as special cases the other type contains only one collision term on the right hand side for example the well known model of andries aoki and perthame 1 for each of these two models 20 and 1 we prove existence uniqueness and positivity of solutions in the first part of the paper in the second part we use the first model 20 in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by dellacherie 11

theory and applications of fuzzy volterra integral equations formulations of fuzzy integral equations in terms of the aumann integral do not reflect the behavior of corresponding crisp models consequently they are ill adapted to describe physical phenomena even when vagueness and uncertainty are present a similar situation for fuzzy odes has been obviated by interpretation in terms of families of differential inclusions the paper extends this formalism to fuzzy integral equations and shows that the resulting solution sets and attainability sets are fuzzy and far better descriptions of uncertain models involving integral equations the investigation is restricted to volterra type equations with mildly restrictive conditions but the methods are capable of extensive generalization to other types and more general assumptions the results are illustrated by integral equations relating to control models with fuzzy uncertainties

global existence to the bgk model of boltzmann equation abstract we present an existence and a stability proof for solutions to the bgk model of boltzmann equation f v x f f m f t 0 x r n v r n m f p 2 t n 2 exp v u 2 2t p pu p u 2 pt 1 x r n 1 v v 2 f t x v dv it relies on the strong compactness of u t and on a new estimate on the third moment of v dv we also prove the entropy relation for 1

existence and comparison theorems to volterra fuzzy integral equation in e n d abstract kaleva 1987 studied the measurability and integrability for the fuzzy set valued mappings of a real variable whose values are in the fuzzy number space e n d in this paper by using the embedding theorem of kaleva 1990 on e n d the darbo fixed points theorem in the convex cone c i e n where i denotes a bounded closed interval is given further the criteria for the existence and comparison of solutions to the volterra fuzzy integral equation in e n d are established under the compactness type conditions

a moving interface method for dynamic kinetic fluid coupling this paper is a continuation of earlier work p degond s jin l mieussens a smooth transition between kinetic and hydrodynamic equations journal of computational physics 209 2005 665 694 in which we presented an automatic domain decomposition method for the solution of gas dynamics problems which require a localized resolution of the kinetic scale the basic idea is to couple the macroscopic hydrodynamics model and the microscopic kinetic model through a buffer zone in which both equations are solved discontinuities or sharp gradients of the solution are responsible for locally strong departures to local equilibrium which require the resolution of the kinetic model the buffer zone is drawn around the kinetic region by introducing a cut off function which takes values between zero and one and which is identically zero in the fluid zone and one in the kinetic zone in the present paper we specifically consider the possibility of moving the kinetic region or creating new kinetic regions by evolving the cut off function with respect to time we present algorithms which perform this task by taking into account indicators which characterize the non equilibrium state of the gas the method is shown to be highly flexible as it relies on the time evolution of the buffer cut off function rather than on the geometric definition of a moving interface which requires remeshing by contrast to many previous methods numerical examples are presented which validate the method and demonstrate its performances

numerical passage from kinetic to fluid equations a numerical way to pass from the bhatnagar gross krook model of the boltzmann equation to compressible euler equations is presented in order to do so a stable discretization of the kinetic equati

cauchy problem for the ellipsoidal bgk model for polyatomic particles abstract we establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal bgk model which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

large time step shock capturing techniques for scalar conservation laws for a scalar conservation law u t f u x with f of constant sign the first order upwind difference scheme is a special case of godonov s method the method is equivalent to solving a sequence of riemann problems at each step and averaging the resulting solution over each cell in order to obtain the numerical solution at the next time level the difference scheme is stable and the solutions to the associated sequence of riemann problems do not interact provided the courant number nu is less than 1 by allowing and explicitly handling such interactions it is possible to obtain a generalized method which is stable for nu much larger than 1 in many cases the resulting solution is considerably more accurate than solutions obtained by other numerical methods in particular shocks can be correctly computed with virtually no smearing the generalized method is rather unorthodox and still has some problems associated with it nonetheless preliminary results are quite encouraging

a coupled ordinates method for solution acceleration of rarefied gas dynamics simulations non equilibrium rarefied flows are frequently encountered in a wide range of applications including atmospheric re entry vehicles vacuum technology and microscale devices rarefied flows at the microscale can be effectively modeled using the ellipsoidal statistical bhatnagar gross krook esbgk form of the boltzmann kinetic equation numerical solutions of these equations are often based on the finite volume method fvm in physical space and the discrete ordinates method in velocity space however existing solvers use a sequential solution procedure wherein the velocity distribution functions are implicitly coupled in physical space but are solved sequentially in velocity space this leads to explicit coupling of the distribution function values in velocity space and slows down convergence in systems with low knudsen numbers furthermore this also makes it difficult to solve multiscale problems or problems in which there is a large range of knudsen numbers in this paper we extend the coupled ordinates method comet previously developed to study participating radiative heat transfer to solve the esbgk equations in this method at each cell in the physical domain distribution function values for all velocity ordinates are solved simultaneously this coupled solution is used as a relaxation sweep in a geometric multigrid method in the spatial domain enhancements to comet to account for the non linearity of the esbgk equations as well as the coupled implementation of boundary conditions are presented the methodology works well with arbitrary convex polyhedral meshes and is shown to give significantly faster solutions than the conventional sequential solution procedure acceleration factors of 5 9 are obtained for low to moderate knudsen numbers on single processor platforms

a note on fuzzy integral equations in this paper existence theorems for certain volterra integral equations and fredholm integral equation for the fuzzy set valued mappings are obtained

a possibilistic argument for irreversibility abstract our aim is to explicate the idea of irreversibility by a mathematical representation of irreversible processes occurring in possibilistic dynamical systems with this in mind solutions of the cauchy problem in the space of fuzzy points are examined to meet the correspondence argument solutions of the cauchy problem in the space of concentrated points are examined too

numerical methods for solution of volterra and fredholm integral equations for functions with values in l spaces we consider volterra and fredholm integral equations for functions with values in l spaces this includes corresponding problems for set valued functions fuzzy valued functions and many others we prove theorems of existence and uniqueness of the solution for such equations and suggest some algorithms for finding approximate solutions we get initial results in the approximation of functions with values in l spaces by piecewise linear functions and we also get the error estimates of trapezoidal quadrature formulas we apply these results for analysis of convergence of suggested algorithms

numerical simulation of flows from free molecular regime to continuum regime by a dvm with streaming and collision processes a discrete velocity method dvm with streaming and collision processes is presented in this work for simulation of flows from free molecular regime to continuum regime the present scheme can be considered as a semi lagrangian like scheme at first we follow the conventional dvm to discretize the phase velocity space by a number of discrete velocities then for each discrete velocity the kinetic equation with bgk shakhov model is integrated in space and time within one time step as a result a simple algebraic formulation can be obtained and its solution can be marched in time by the streaming and collision processes however differently from the conventional semi lagrangian scheme the present scheme uses the muscl approach with van albada limiter in the process of reconstructing the distribution function at the surrounding points of the cell center and the transport distance is controlled in order to avoid extrapolation this makes the present scheme be capable of simulating the hypersonic rarefied flows in addition as compared to the unified gas kinetic scheme ugks the present scheme is simpler and easier for implementation thus the computational efficiency can be improved accordingly to validate the proposed numerical scheme test examples from free molecular regime to continuum regime are simulated numerical results showed that the present scheme can predict the flow properties accurately even for hypersonic rarefied flows

linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems abstract we are interested in high order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems first we consider optimal control problems for ordinary differential equations and show loss of accuracy for adams moulton and adams bashforth methods whereas bdf methods preserve high order accuracy subsequently we extend these results to semi lagrangian discretizations of hyperbolic relaxation systems computational results illustrate theoretical findings

asymptotic preserving implicit explicit runge kutta methods for nonlinear kinetic equations we discuss implicit explicit imex runge kutta methods which are particularly adapted to stiff kinetic equations of boltzmann type we consider both the case of easy invertible collision operators and the challenging case of boltzmann collision operators we give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate their monotonicity properties are also studied in the case of the boltzmann operator the methods are based on the introduction of a penalization technique for the collision integral this reformulation of the collision operator permits us to construct penalized imex schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator finally we show some numerical results which confirm the theoretical analysis

discrete velocity models and numerical schemes for the boltzmann bgk equation in plane and axisymmetric geometries we present new numerical models for computing transitional or rarefied gas flows as described by the boltzmann bgk and bgk es equations we first propose a new discrete velocity model based on the entropy minimization principle this model satisfies the conservation laws and the entropy dissipation moreover the problem of conservation and entropy for axisymmetric flows is investigated we find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator then we propose some models that satisfy these constraints owing to these properties we obtain numerical schemes that are economic in terms of discretization and robust in particular we develop a linearized implicit scheme for computing stationary solutions of the discrete velocity bgk and bgk es models this scheme is the basis of a code which can compute high altitude hypersonic flows in 2d plane and axisymmetric geometries our results are analyzed and compared to other methods

boltzmann type schemes for gas dynamics and the entropy property let h be a convex function a boltzmann equation is built following the b g k model for which h is an entropy in the fluid limit the compressible euler equations are obtained with a convex entropy naturally associated to h the motivation is to define numerical boltzmann schemes with finite speed of propagation corresponding to an equilibrium density function with bounded support the classical scheme obtained with h f f log f is based on the equilibrium density function exp v 2 2 which has infinite support

accuracy analysis of high order lattice boltzmann models for rarefied gas flows in this work we have theoretically analyzed and numerically evaluated the accuracy of high order lattice boltzmann lb models for capturing non equilibrium effects in rarefied gas flows in the incompressible limit the lb equation is shown to be able to reduce to the linearized bhatnagar gross krook bgk equation therefore when the same gauss hermite quadrature is used lb method closely resembles the discrete velocity method dvm in addition the order of hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the knudsen number to the bgk equation for incompressible flows meanwhile we have numerically evaluated the lb models for a standing shear wave problem which is designed specifically for assessing model accuracy by excluding the influence of gas molecule surface interactions at wall boundaries the numerical simulation results confirm that the high order terms in the discrete equilibrium distribution function play a negligible role in capturing non equilibrium effect for low speed flows by contrast appropriate gauss hermite quadrature has the most significant effect on whether lb models can describe the essential flow physics of rarefied gas accurately our simulation results where the effect of wall gas interactions is excluded can lead to conclusion on the lb modeling capability that the models with higher order quadratures provide more accurate results for the same order gauss hermite quadrature the exact abscissae will also modestly influence numerical accuracy using the same gauss hermite quadrature the numerical results of both lb and dvm methods are in excellent agreement for flows across a broad range of the knudsen numbers which confirms that the lb simulation is similar to the dvm process therefore lb method can offer flexible models suitable for simulating continuum flows at the navier stokes level and rarefied gas flows at the linearized boltzmann model equation level