on the stochastic porous medium equation abstract in this paper we discuss an initial boundary value problem and a cauchy problem for the stochastic porous medium equation our basic estimates are based on the known results due to math sbornik 67 1965 609 642 arch rational mech anal 25 1967 64 80 by the procedure developed in trans amer math soc 354 2001 1117 1135 we obtain solutions over the given probability space rather than martingale solutions we will also establish the existence of invariant measures when the space domain is bounded

yosida approximations for multivalued stochastic partial differential equations driven by levy noise on a gelfand triple abstract we prove the existence and uniqueness of solutions for a class of multivalued stochastic partial differential equations with maximal monotone drift on banach space driven by multiplicative levy noise we also establish the strong convergence result for solutions of the approximating equations where the maximal monotone drift operator is replaced by its yosida approximation as an application the existence and uniqueness of solutions for multivalued stochastic porous medium equations is obtained

a measurement theory view on the granularity of partitions measurement of granularity is one of the foundational issues in granular computing this paper investigates a class of measures of granularity of partitions the granularity of a set is defined by a strictly monotonic increasing transformation of the cardinality of the set the granularity of a partition is defined as the expected granularity of all blocks of the partition with respect to the probability distribution defined by the partition many existing measures of granularity are instances of the proposed class new measures of granularity of partitions are also introduced

asymptotic couplings by reflection and applications for nonlinear monotone spdes abstract asymptotic couplings by reflection are constructed for a class of nonlinear monotone spdes stochastic partial differential equations as applications gradient holder estimates as well as exponential convergence are derived for the associated markov semigroup the main results are illustrated by stochastic generalized porous media equations stochastic p laplacian equations and stochastic generalized fast diffusion equations we emphasize that the gradient estimate is studied for the first time for these equations

random attractors for singular stochastic evolution equations abstract the existence of random attractors for singular stochastic evolution equations see perturbed by general additive noise is proven the drift is only assumed to satisfy the standard assumptions of the variational approach to see with compact embeddings in the gelfand triple and singular coercivity for ergodic monotone contractive random dynamical systems it is proven that the random attractor consists of a single random point in case of real linear multiplicative noise finite time extinction is obtained applications include stochastic generalized fast diffusion equations and stochastic generalized singular p laplace equations perturbed by levy noise with jump measure having finite first and second moments

chebyshev chebyshev spectral collocation method for solving the generalized regularized long wave grlw equation in this research a chebyshev chebyshev spectral collocation method based on kronecker and hadamard products is proposed for solving the generalized regularized long wave grlw equation chebyshev gauss lobatto collocation points are used in both time and space directions three invariants of motion mass momentum and energy are evaluated to determine the conservation properties of the grlw equation the single solitary wave and the interaction of two and three solitary waves are presented to validate the efficiency and the accuracy of the proposed scheme

novel implicit explicit local conservative schemes for the regularized long wave equation and convergence analysis abstract two implicit and two explicit schemes preserving the local momentum conservation laws exactly on any time space region are proposed for the regularized long wave equation with appropriate boundary conditions the schemes will be energy and mass preserving globally combining with the momentum conservation laws we obtain the priori estimates of the numerical solution and the error estimates in l norm for the proposed implicit schemes numerical experiments show the excellent performance of the proposed schemes and the numerical results coincide with the theoretical ones

stochastic evolution of 2d crystals we study a crystalline version of the modified stefan problem in the plane the feature of our approach is that we consider a model with stochastic perturbations and assume the interfacial curve to be a polygon the existence of solution to our stochastic system is established galerkin s method is one of the main tools used in the proof of our assertion

weak solutions to the stochastic porous media equation via kolmogorov equations the degenerate case a stochastic version of the porous medium equation with coloured noise is studied the corresponding kolmogorov equation is solved in the space l 2 h where is an infinitesimally excessive measure then a weak solution is constructed

measurement of general granules abstract the measurement of granules is often considered the most foundational problem in granular computing and it has been studied in many fields but in relative isolation this paper attempts to explore the measurement for general granules whether they are granules based on equivalence or general relations whether they are macrocosmic or microcosmic granules and whether they are granules on a finite or an infinite set for a finite set we discuss the measurement of granules corresponding to equivalence relations and introduce definitions of granularity fineness and granular entropy in order to distinguish those granules with the same granularity fineness and granular entropy we present conditional granularity conditional fineness and conditional granular entropy and obtain the necessary and sufficient condition which can judge one granule as coarser or finer than another while the conditional granularity is a subsethood measure which satisfies monotonic properties what s more we introduce relation matrix to measure the granules and it not only can be easily generalized to measure the granules on general relations but also connects granular computing to graph theory for an infinite set we can define different granularities in different questions but we suggest the basic idea of granularity that is the granularity of a granule is the sum of the relative information contents of its atomic granules

stochastic porous media equations and self organized criticality the existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and wiener forcing term is proven the finite time extinction of solutions with high probability is also proven in 1 d the results are relevant for self organized criticality behavior of stochastic nonlinear diffusion equations with critical states

relational granulation method based on quotient space theory for maximum flow problem abstract granular computing grc is a problem solving concept deeply rooted in human thinking grc as a new and rapidly growing paradigm of information processing has attracted the attention of many researchers and practitioners grc is related to granulation i e a process of drawing a set of objects or points together based on their indiscernibility similarity proximity or functionality in general two types of granulation processes exist functional granulation and relational granulation if the process is based entirely on the attributes of the objects it is known as functional granulation whereas if the granulation process is based on the relationship between objects it is known as relational granulation this paper proposes a novel method called the maximum flow based on quotient space theory m f q s t for solving the maximum flow problems based on quotient space theory for relational granulation using the method m f q s t substructures are first detected and the community is described as a substructure next the relational granulation criterion is discussed in detail the substructure that satisfies the relational granulation criterion is regarded as a coarse grained node subsequently the construction of a quotient network that is coarser than the original network is described finally the maximum flow algorithm is used to compute the maximum flow on the quotient network as the approximated maximum flow on the original network within a much shorter period of time experimental results demonstrate that the novel method m f q s t reduces the cumulative running time after simplifying the network structure with a low error rate the size of the quotient network is significantly reduced and the node and edge scales are reduced to 20 59 and 21 62 on average respectively

solitary waves of the regularized long wave equation abstract a finite element solution of the regularised long wave equation based on galerkin s method using cubic splines as element shape functions is set up a linear stability analysis shows the scheme to be unconditionally stable test problems including the migration and interaction of solitary waves are used to validate the method which is found to be accurate and efficient the three invariants of the motion are evaluated to determine the conservation properties of the algorithm the temporal evolution of a maxwellian initial pulse is then studied

a new explicit multisymplectic scheme for the regularized long wave equation in this paper we derive a new ten point multisymplectic scheme for the regularized long wave equation from its bridges multisymplectic form the new scheme is an explicit scheme in the sense that it does not need iteration we discuss some properties of the new scheme the performance and the efficiency of the new scheme are illustrated by solving several test examples the obtained results are presented and compared with previous methods numerical results indicate that the multisymplectic scheme cannot only obtain satisfied solutions for the regularized long wave equation but also keep three invariants of motion which are evaluated to determine the conservation properties of the algorithm very well

a variational approach to stochastic nonlinear parabolic problems abstract this work is concerned with an optimal control approach to stochastic nonlinear parabolic diffusion equations with monotonically increasing nonlinearity this approach leads to sharper existence and uniqueness results under minimal growth conditions on nonlinear diffusion coefficients

a collocation method with cubic b splines for solving the mrlw equation the modified regularized long wave mrlw equation is solved numerically by collocation method using cubic b splines finite element a linear stability analysis of the scheme is shown to be marginally stable three invariants of motion are evaluated to determine the conservation properties of the algorithm also the numerical scheme leads to accurate and efficient results moreover interaction of two and three solitary waves are studied through computer simulation and the development of the maxwellian initial condition into solitary waves is also shown

toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic there are three basic concepts that underlie human cognition granulation organization and causation informally granulation involves decomposition of whole into parts organization involves integration of parts into whole and causation involves association of causes with effects granulation of an object a leads to a collection of granules of a with a granule being a clump of points objects drawn together by indistinguishability similarity proximity or functionality for example the granules of a human head are the forehead nose cheeks ears eyes etc in general granulation is hierarchical in nature a familiar example is the granulation of time into years months days hours minutes etc modes of information granulation ig in which the granules are crisp c granular play important roles in a wide variety of methods approaches and techniques crisp ig however does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy f granular the granules of a human head for example are fuzzy in the sense that the boundaries between cheeks nose forehead ears etc are not sharply defined furthermore the attributes of fuzzy granules e g length of nose are fuzzy as are their values long short very long etc the fuzziness of granules their attributes and their values is characteristic of ways in which humans granulate and manipulate information

application of the variational iteration method to the regularized long wave equation this paper applies the variational iteration method to the regularized long wave equation in this method a generalized lagrange multiplier is introduced to construct a correction functional for the discussed problem the multiplier can be identified optimally via variational theory three invariants of the motion are evaluated to determine the conservation properties of the system

some linearly and non linearly implicit schemes for the numerical solutions of the regularized long wave equation in this paper we present some linearly implicit and non linearly implicit schemes for investigating the numerical solutions of the regularized long wave equation numerical experiments indicate that all the present schemes can give satisfactory numerical solutions meanwhile numerical results also confirm the excellent conservations of three invariants in long time computations

solving the generalized regularized long wave equation on the basis of a reproducing kernel space on the basis of a reproducing kernel space an iterative algorithm for solving the generalized regularized long wave equation is presented the analytical solution in the reproducing kernel space is shown in a series form and the approximate solution u n is constructed by truncating the series to n terms the convergence of u n to the analytical solution is also proved results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such evolution equations

stochastic generalized porous media and fast diffusion equations abstract we present a generalization of krylov rozovskii s result on the existence and uniqueness of solutions to monotone stochastic differential equations as an application the stochastic generalized porous media and fast diffusion equations are studied for finite reference measures where the drift term is given by a negative definite operator acting on a time dependent function which belongs to a large class of functions comparable with the so called n functions in the theory of orlicz spaces

uncertainly measures of rough set prediction abstract the main statistics used in rough set data analysis the approximation quality is of limited value when there is a choice of competing models for predicting a decision variable in keeping within the rough set philosophy of non invasive data analysis we present three model selection criteria using information theoretic entropy in the spirit of the minimum description length principle our main procedure is based on the principle of indifference combined with the maximum entropy principle thus keeping external model assumptions to a minimum the applicability of the proposed method is demonstrated by a comparison of its error rates with results of c4 5 using 14 published data sets

stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise in this work we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation rm d x t rm div left frac nabla x t nabla x t right rm d t x t rm d w t rm in 0 infty times mathcal o where mathcal o is a bounded and open domain in mathbb r n n geqq 1 and w t is a wiener process of the form w t sum infty k 1 mu k e k beta k t e k in c 2 overline mathcal o cap h 1 0 mathcal o and beta k k in mathbb n are independent brownian motions this is a stochastic diffusion equation with a highly singular diffusivity term one main result established here is that for all initial conditions in l 2 mathcal o it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality thus one obtains a stochastic version of the minimal total variation flow the new approach developed here also allows us to prove the finite time extinction of solutions in dimensions 1 leqq n leqq3 which is another main result of this work

a meshfree method for the numerical solution of the rlw equation in this paper we present a meshfree technique for the numerical solution of the regularized long wave rlw equation this approach is based on a global collocation method using the radial basis functions rbfs different kinds of rbfs are used for this purpose accuracy of the new method is tested in terms of l 2 and l error norms in case of non availability of the exact solution performance of the new method is compared with existing methods stability analysis of the method is established propagation of single and double solitary waves wave undulation and conservation properties of mass energy and momentum of the rlw equation are discussed

self organization and anomalous diffusion abstract the self organizing process is investigated in the continuum limit of the cellular automaton model introduced by bak tang and wiesenfeld an anomalous diffusion equation is proposed for the description of this process and an analytical method for the solutionis presented in one dimension based on an adiabatic approximation

a computational method for the regularized long wave rlw equation a numerical solution of the regularized long wave rlw equation based on collocation method using cubic b spline finite elements is used to simulate the migration and interaction of solitary waves interaction of solitary waves with different amplitudes are shown the three invariants of the motion are evaluated to determine the conservation properties of the system the temporal evaluation of a maxwellian initial pulse is then studied and then we prove that the number of solitons which are generated from maxwellian initial condition are determined

a conservative weighted finite difference scheme for regularized long wave equation in this paper a new finite difference method for initial boundary value problem of regularized long wave rlw equation is considered two energy conservative finite difference schemes are proposed and the prior estimate of the solution is obtained existence and uniqueness of numerical solution are derived it is proved that the finite difference schemes are convergent and stable numerical experiment results show that the theory is accurate and the schemes are efficient and reliable

finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise abstract in this paper one surveys and sharpens some recent results obtained by the authors on the finite time extinction property of solutions to stochastic diffusion equations of the form d x x m d t x d w and d x sgn x d t x d w where 0 m 1 0 these equations arise as models for nonlinear diffusion processes in porous media plasma and self organized criticality under stochastic gaussian perturbation these equations can be also viewed as control systems governed by fast diffusion porous media equations with a stochastic feedback controller x d w

stochastic porous media equations and self organized criticality convergence to the critical state in all dimensions if x x t is the solution to the stochastic porous media equation in mathcal o subset mathbf r d 1 le d le 3 modelling the self organized criticality barbu et al in commun math phys 285 901 923 2009 and x c is the critical state then it is proved that int infty 0m mathcal o setminus mathcal o t 0 dt infty mathbb p hbox a s and lim t to infty int mathcal o x t x c d xi ell infty mathbb p hbox a s here m is the lebesgue measure and mathcal o t c is the critical region xi in mathcal o x t xi x c xi and x c x 0 a e xi in mathcal o if the stochastic gaussian perturbation has only finitely many modes but is still function valued lim t to infty int k x t x c d xi 0 exponentially fast for all compact k subset mathcal o with probability one if the noise is sufficiently strong we also recover that in the deterministic case l 0

existence of strong solutions for stochastic porous media equation under general monotonicity conditions this paper addresses existence and uniqueness of strong solutions to stochastic porous media equations dx x dt b x dw t in bounded domains of r d with dirichlet boundary conditions here is a maximal monotone graph in r r possibly multivalued with

the conservative and fourth order compact finite difference schemes for regularized long wave equation abstract in this paper two conservative and fourth order compact finite difference schemes are proposed and analyzed for solving the regularized long wave rlw equation the first compact finite difference scheme is two level and nonlinear implicit the second scheme is three level and linearized implicit conservations of the discrete mass and energy and unique solvability of the numerical solutions are proved convergence and unconditional stability are also derived without any restrictions on the grid ratios by using discrete energy method the optimal error estimates in norm and l are of fourth order and second order accuracy for the spatial and temporal step sizes respectively numerical examples are presented to simulate the collision of different solitary waves and support the theoretical analysis

a 2n order compact finite difference method for solving the generalized regularized long wave grlw equation the generalized regularized long wave grlw equation is solved by fully different numerical scheme the equation is discretized in space by 2n order compact finite difference method and in time by a backward finite difference method at the inner and the boundary nodes the first and the second order derivatives with 2n order of accuracy are obtained to determine the conservation properties of the grlw equation three invariants of motion are evaluated the single solitary wave and the interaction of two and three solitary waves are presented to validate the efficiency and the accuracy of the proposed scheme

information theoretic measures of uncertainty for rough sets and rough relational databases rough set theory has become well established as a mechanism for uncertainty management in a wide variety of applications including databases this paper addresses the measurement of uncertainty in rough sets and rough relational databases by introducing a measurement based on information theory this rough entropy is discussed as it applies to rough sets in general and in particular to aspects of the rough relational database model

a finite difference scheme for the mrlw and solitary wave interactions the modified regularized long wave mrlw equation is solved numerically using the finite difference method fourier stability analysis of the linearized scheme shows that it is a marginally stable also the local truncation error of the method is investigated three invariants of motion are evaluated to determine the conservation properties of the problem and the numerical scheme leads to accurate and efficient results moreover interaction of two and three solitary waves is shown the development of the maxwellian initial condition into solitary waves is also shown and we show that the number of solitons which are generated from the maxwellian initial condition can be determined numerical results show also that a tail of small amplitude appears after the interactions

numerical schemes for a model for nonlinear dispersive waves abstract a description is given of a number of numerical schemes to solve an evolution equation that arises when modelling the propagation of water waves in a channel the discussion also includes the results of numerical experiments made with each of the schemes it is suggested on the basis of these experiments that one of the schemes may have discrete solitary wave solutions

an entropy based uncertainty measurement approach in neighborhood systems abstract uncertainty measures can provide us with principled methodologies to analyze uncertain data and unveil the substantive characteristics of the data sets accuracy and roughness proposed by pawlak are two main tools to deal with uncertainty measurement issue in rough set theory many uncertainty measure methodologies for discrete valued information systems or discrete valued decision systems have been developed however there are only limited on the uncertainty measurement for neighborhood systems in this paper we address the issues of uncertainty of a neighborhood system and extend the traditional accuracy and roughness measures to deal with neighborhood systems in particular a concept called neighborhood entropy is first introduced to evaluate the uncertainty of a neighborhood information system consequently the entropy based roughness and approximation roughness measures of neighborhood system are presented theoretical analysis indicates that entropy based measures can be used to evaluate the uncertainty in neighborhood systems experiments are conducted on artificial data sets and standard uci data sets to test our proposed methodologies results demonstrate that the entropy based measures are effective and valid for evaluating the uncertainty of neighborhood systems

non monotone stochastic generalized porous media equations abstract by using the nash inequality and a monotonicity approximation argument existence and uniqueness of strong solutions are proved for a class of non monotone stochastic generalized porous media equations moreover we prove for a large class of stochastic pde that the solutions stay in the smaller l 2 space provided the initial value does so that some recent results in the literature are considerably strengthened

entropy measures and granularity measures for set valued information systems set valued information systems are generalized models of single valued information systems in this paper we propose two new relations for set valued information systems based on these two relations the concepts of knowledge information entropy knowledge rough entropy knowledge granulation and knowledge granularity measure are defined in set valued information systems and some properties are investigated moreover relationship between knowledge information entropy and knowledge granulation knowledge rough entropy and knowledge granularity measure are studied it is also shown that knowledge information entropy and knowledge granularity measure can be used to evaluate the certainty degree of knowledge in set valued information systems and knowledge rough entropy and knowledge granulation can be used to evaluate the uncertainty degree of knowledge in set valued information systems these results may supply a further understanding the essence of uncertainty and granularity in set valued information systems

finite time extinction for stochastic sign fast diffusion and self organized criticality we prove finite time extinction for stochastic sign fast diffusion equations driven by linear multiplicative space time noise corresponding to the bak tang wiesenfeld model for self organized criticality this solves a problem posed and left open in several works barbu methods appl sci 36 1726 1733 2013 rockner and wang j lond math soc 2 87 545 560 2013 barbu et al j math anal appl 389 147 164 2012 barbu and rockner comm math phys 311 539 555 2012 barbu et al comm math phys 285 901 923 2009 c r math acad sci paris 347 1 2 81 84 2009 the highly singular degenerate nature of the drift in interplay with the stochastic perturbation causes the need for new methods in the analysis of mass diffusion and several new estimates and techniques are introduced

mixed galerkin finite element methods for modified regularized long wave equation in this paper mixed galerkin finite methods for modified regularized long wave mrlw equation are studied we discrete in space by finite elements and for the time discretization by extrapolated crank nicolson scheme and explicit linear multistep scheme linear stability analysis based on von neumann theory are performed numerical experiments verify the efficiency and accuracy of the proposed numerical schemes

asymptotic behavior of the nonlinear diffusion equation nt n 1nx x the asymptotic behavior of the equation nt ln n xx is studied on the finite interval 0 x 1 with the boundary conditions n 0 t n 1 t n0 and initial data n x 0 n0 we prove that asymptotically ln n x t n0 a exp 2t n0 21 2 sin x and also provide rigorous upper and lower bounds on the asymptotic amplitude a in terms of integrals of nonlinear functions of the initial data the rigorous bounds are compared to values of a obtained from computer experiments the lower bound l 23 2 exp li 1 q where li is the logarithmic integral is euler s constant and q 2 f n x 0 n0 1 sin x dx is found to be the best known estimate of a

numerical implementation for solving the symmetric regularized long wave equation the paper presents a novel finite difference method for the symmetric regularized long wave equation the time discretization is performed by using a four level average difference technique for solving the fluid velocity independently from the density at this stage the numerical solution is easily solved by using the presented method since it does not require an extra effort to deal with a nonlinear term and the density the existence and uniqueness of the numerical solution and the conservation of mass are guaranteed the stability and convergence of the numerical solution with second order accuracy on both space and time are also verified numerical results are carried out to confirm the accuracy of our theoretical results and the efficiency of the scheme to illustrate the effectiveness and the advantage of the proposed method the results at long time behavior are compared with the ones obtained from previously known methods moreover in the computation the present method is applied to the collision of solitons under the effect of variable parameters

the two phase stochastic stefan problem this work is concerned with existence for a stochastic free boundary problem arising in phase transition the stefan two phase problem the existence of an invariant ergodic measure associated with the corresponding transition semigroup p t is also proved together with an integration by parts formula for the generator of p t

interpolating element free galerkin method for the regularized long wave equation and its error analysis abstract combining the simplified interpolating moving least squares imls method with the galerkin weak form an interpolating element free galerkin iefg method is presented for the numerical study of the regularized long wave rlw equation the trial function in the imls method has one less unknown coefficient than in the traditional mls approximation and the shape functions satisfy the interpolating property at nodes then the iefg method can apply the essential boundary conditions naturally and directly and has high computational accuracy and then this paper studies the error estimates of the iefg method for rlw equation the error estimate of the semi discrete iefg method for the rlw equation on the influence domain radius is first studied and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied the theoretical results show that the iefg method for rlw equation has good convergence rate for the purpose of demonstration three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper

a new measure of uncertainty based on knowledge granulation for rough sets in rough set theory accuracy and roughness are used to characterize uncertainty of a set and approximation accuracy is employed to depict accuracy of a rough classification although these measures are effective they have some limitations when the lower upper approximation of a set under one knowledge is equal to that under another knowledge to overcome these limitations we address in this paper the issues of uncertainty of a set in an information system and approximation accuracy of a rough classification in a decision table an axiomatic definition of knowledge granulation for an information system is given under which these three measures are modified theoretical studies and experimental results show that the modified measures are effective and suitable for evaluating the roughness and accuracy of a set in an information system and the approximation accuracy of a rough classification in a decision table respectively and have a much simpler and more comprehensive form than the existing ones