effect algebras with state operator state operators on convex effect algebras in particular effect algebras of unital jc algebras jw algebras and convex mv algebras are studied and their relations with conditional expectations in algebraic sense as well as in the sense of probability on mv algebras are shown

state morphism algebras general approach we present a complete description of subdirectly irreducible state bl algebras as well as of subdirectly irreducible state morphism bl algebras in addition we present a general theory of state morphism algebras that is algebras of general type with state morphism which is a fixed idempotent endomorphism we define a diagonal state morphism algebra and we show that every subdirectly irreducible state morphism algebra can be embedded into a diagonal one we describe generators of the varieties of state morphism algebras in particular generators of state morphism bl algebras state morphism mtl algebras state morphism non associative bl algebras and state morphism pseudo mv algebras

states of finite gbl algebras with monoidal sum abstract generalized bl algebras i e divisible residuated lattices provide the semantics for a generalization of basic logic where the axiom of prelinearity does not hold informally gbl algebras generalize heyting algebras in a similar way as mv algebras generalize boolean algebras we introduce the operation of sum in finite gbl algebras and we axiomatize the obtained finite structures called gbl algebras we hence define states of gbl algebras extending mv algebraic states and we prove that they are determined by their restriction on the heyting skeleton extremal states are also characterized in terms of densities concentrated in a unique join prime idempotent

an optimal order error estimate for the mass conservative characteristic finite element scheme abstract the mass conservative characteristic mcc finite element scheme preserves the mass balance which is an important property in the numerical simulation of the convection diffusion problems in this paper an optimal order error estimate is obtained for the model problem in one space dimension the convergence rate is o h 2 t in the l l 2 norm for the linear finite element some numerical examples are given to confirm the conclusion

extensions of states on mv quantales abstract we consider the lattice of the ordered right continuous families of elements of a dual quantale with the index set 0 1 called its rc extension and the lattice of the ordered n 1 tuples with n 1 of elements of this dual quantale its n extension equipped with a dual quantale structure that generalizes the canonical extension proposed by hohle and weber of particular interest is the existence of states on n extensions of finite mv chains we generalize the recent results of weber by incorporating effectible girard quantales which are introduced by the author

finite difference spectral approximations for the fractional cable equation fujian nsf s0750017 national nsf of china 10531080 ministry of education of china 973 high performance scientific computation research program 2005cb321703

lexicographic mv algebras and lexicographic states abstract in this paper we introduce lexicographic mv algebras and prove that they are the counterpart of unital abelian lattice ordered groups defined via lexicographic products the di nola lettieri categorical equivalence between perfect mv algebras and abelian lattice ordered groups is extended to lexicographic mv algebras we also investigate lexicographic states of lexicographic mv algebras these are additive and normalized maps from any lexicographic mv algebra into an ad hoc defined mv subalgebra of a non principal ultraproduct 0 1 of the real unit interval 0 1 for lexicographic states we prove a representation theorem which can be regarded as the measure theoretical analog of the representation theorem for lexicographic mv algebras we also provide necessary and sufficient conditions for a lexicographic state to be faithful

conditional measures and their applications to fuzzy sets abstract given a decomposable measure with respect to a continuous t conorm as introduced by the author in an earlier paper see section 1 we can construct conditional measures as implications these fulfil a generalized product law replacing the product in the classical law by any other strict t norm and turn out to be decomposable with respect to an operation v depending on and the condition set v section 2 more general conditional measures are introduced axiomatically and are shown to be conditional measures with respect to some decomposable measure section 3 bayesian like models are given which are alternatives to that presented by the author in a recent paper section 4 finally some applications to fuzzy sets are sketched section 5

time stepping along characteristics with incomplete iteration for a galerkin approximation of miscible displacement in porous media miscible displacement of one incompressible fluid by another in a porous medium is modeled by a nonlinear coupled system of two partial differential equations the pressure equation is elliptic while the concentration equation is parabolic but normally convection dominated a sequential backward difference time stepping scheme is defined it approximates the pressure by a standard galerkin procedure and the concentration by a combination of a galerkin method and the method of characteristics optimal convergence rates in l 2 and h 1 are demonstrated for this scheme and for a modified version in which the algebraic equations at each time step are solved approximately by a limited number of preconditioned conjugate gradient iterations time stepping along the characteristics of the hyperbolic part of the concentration equation is shown to result in smaller time truncation errors than those of standard methods numerical results published elsewhere have confirmed that larger time steps are appropriate

a new fractional numerical differentiation formula to approximate the caputo fractional derivative and its applications in the present work first a new fractional numerical differentiation formula called the l1 2 formula to approximate the caputo fractional derivative of order 0 1 is developed it is established by means of the quadratic interpolation approximation using three points t j 2 f t j 2 t j 1 f t j 1 and t j f t j for the integrand f t on each small interval t j 1 t j j 2 while the linear interpolation approximation is applied on the first small interval t 0 t 1 as a result the new formula can be formally viewed as a modification of the classical l1 formula which is obtained by the piecewise linear approximation for f t both the computational efficiency and numerical accuracy of the new formula are superior to that of the l1 formula the coefficients and truncation errors of this formula are discussed in detail two test examples show the numerical accuracy of l1 2 formula second by the new formula two improved finite difference schemes with high order accuracy in time for solving the time fractional sub diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed respectively in addition the application of the new formula into solving fractional ordinary differential equations is also presented several numerical examples are computed the comparison with the corresponding results of finite difference methods by the l1 formula demonstrates that the new l1 2 formula is much more effective and more accurate than the l1 formula when solving time fractional differential equations numerically

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

filters in fuzzy class theory a concept of filter is introduced in fuzzy class theory graded properties of filters prime filters and related constructions are investigated in particular the properties of filters relativized with respect to the three basic t norms godel product and lukasiewicz are studied a naturally arising example of a graded filter is constructed and it is shown that the models of filters include well known families of set functions

convergence and stability of compact finite difference method for nonlinear time fractional reaction diffusion equations with delay abstract this paper is concerned with numerical solutions of nonlinear time fractional reaction diffusion equations with time delay a linearized compact finite difference scheme is proposed to solve the equations in terms of a new developed fractional gronwall type inequality convergence and stability of the proposed scheme are obtained numerical experiments are given to illustrate the theoretical results

convergence of finite element solutions of stochastic partial integro differential equations driven by white noise numerical approximation of a stochastic partial integro differential equation driven by a space time white noise is studied by truncating a series representation of the noise with finite element method for spatial discretization and convolution quadrature for time discretization sharp order convergence of the numerical solutions is proved up to a logarithmic factor numerical examples are provided to support the theoretical analysis

galerkin spectral method for nonlinear time fractional cable equation with smooth and nonsmooth solutions abstract in this work we study the numerical solutions of the time fractional cable equations with nonlinear term where the fractional derivatives are described in riemann liouville sense an explicit scheme is constructed based upon finite difference method in time and legendre spectral method in space stability and convergence of scheme are proved rigorously moreover an improved algorithm for the problem with nonsmooth solutions is proposed by adding correction terms to the approximations of first order derivative fractional derivatives and nonlinear term numerical examples are given to support theoretical analysis

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

two fully discrete schemes for fractional diffusion and diffusion wave equations with nonsmooth data we consider initial boundary value problems for the subdiffusion and diffusion wave equations involving a caputo fractional derivative in time we develop two fully discrete schemes based on the piecewise linear galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward euler method second order backward difference method and establish error estimates optimal with respect to the regularity of problem data these two schemes are first and second order accurate in time for both smooth and nonsmooth data extensive numerical experiments for two dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity

state bck algebras and state morphism bck algebras abstract in the paper we define the notion of a state bck algebra and a state morphism bck algebra extending the language of bck algebras by adding a unary operator which models probabilistic reasoning we present a relation between state operators and state morphism operators and measures and states on bck algebras respectively we study subdirectly irreducible state morphism bck algebras we introduce the concept of an adjoint pair in bck algebras and show that there is a one to one correspondence between adjoint pairs and state morphism operators in addition we show the generators of quasivarieties of state morphism bck algebras

an analysis of the rayleigh stokes problem for a generalized second grade fluid we study the rayleigh stokes problem for a generalized second grade fluid which involves a riemann liouville fractional derivative in time and present an analysis of the problem in the continuous space semidiscrete and fully discrete formulations we establish the sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v v including v in l 2 omega v l2 a space semidiscrete galerkin scheme using continuous piecewise linear finite elements is developed and optimal with respect to initial data regularity error estimates for the finite element approximations are derived further two fully discrete schemes based on the backward euler method and second order backward difference method and the related convolution quadrature are developed and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data numerical results for one and two dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method and to verify the convergence theory

measure free conditioning and extensions of additive measures on finite mv algebras using the well known representation of any finite mv algebra as a product of finite mv chains as factors we obtain a representation of its canonical extension as a girard algebra product of the canonical extensions of the mv chain factors based on this representation and using the results from our last paper we characterize the additive measures on any finite mv algebra resp the weakly and the strongly additive measures on its canonical girard algebra extension and that as convex combinations of the corresponding measures on the respective factors after that we apply the results to measure free defined conditional events which for this reason are considered as elements of the canonical extension of a finite mv algebra of events this definition for conditional events is naturally equivalent to the well known definition as lattice intervals of events here the weak additivity is considered as a reasonable notion for a global additivity property of a measure extension these extensions are essentially different from the measure extension which has a form analogous to the classical conditional probability but which is only levelwise additive moreover the measure free approach of this paper is completely different from the classical approach in which conditional measures of events given a fixed condition event are considered based on additive measures states of events without having defined conditional events

error estimates for a semidiscrete finite element method for fractional order parabolic equations we consider the initial boundary value problem for a homogeneous time fractional diffusion equation with an initial condition v x and a homogeneous dirichlet boundary condition in a bounded convex polygonal domain omega we study two semidiscrete approximation schemes i e the galerkin finite element method fem and lumped mass galerkin fem using piecewise linear functions we establish almost optimal with respect to the data regularity error estimates including the cases of smooth and nonsmooth initial data i e v in h 2 omega cap h 1 0 omega and v in l 2 omega for the lumped mass method the optimal l 2 norm error estimate is valid only under an additional assumption on the mesh which in two dimensions is known to be satisfied for symmetric meshes finally we present some numerical results that give insight into the reliability of the theoretical study

representing uncertainty on set valued variables using belief functions a formalism is proposed for representing uncertain information on set valued variables using the formalism of belief functions a set valued variable x on a domain w is a variable taking zero one or several values in w while defining mass functions on the frame 2 2 w is usually not feasible because of the double exponential complexity involved we propose an approach based on a definition of a restricted family of subsets of 2 w that is closed under intersection and has a lattice structure using recent results about belief functions on lattices we show that most notions from dempster shafer theory can be transposed to that particular lattice making it possible to express rich knowledge about x with only limited additional complexity as compared to the single valued case an application to multi label classification in which each learning instance can belong to several classes simultaneously is demonstrated

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

measures states and de finetti maps on pseudo bck algebras in this paper we extend the notions of states and measures presented in dvurecenskij and pulmannova 2000 12 to the case of pseudo bck algebras and study similar properties we prove that under some conditions the notion of a state in the sense of dvurecenskij and pulmannova 2000 12 coincides with the bosbach state and we extend to the case of pseudo bck algebras some results proved by kuhr only for pseudo bck semilattices we characterize extremal states and show that the quotient pseudo bck algebra over the kernel of a measure can be embedded into the negative cone of an archimedean group additionally we introduce a borel state and using results by kuhr and mundici 2007 28 we prove a relationship between de finetti maps bosbach states and borel states

stochastic independence for probability mv algebras we propose a notion of stochastic independence for probability mv algebras addressing an open problem posed by riecan and mundici furthermore we prove a representation theorem for probability mv algebras and we get mv algebraic versions of the holder inequality and the hausdorff moment problem

a unified numerical scheme for the multi term time fractional diffusion and diffusion wave equations with variable coefficients abstract we consider the numerical solutions of the multi term time fractional diffusion and diffusion wave equations with variable coefficients in a bounded domain the time fractional derivatives are described in the caputo sense a unified numerical scheme based on finite difference method in time and legendre spectral method in space is proposed detailed error analysis is given for the fully discrete scheme the convergence rate of the proposed scheme in l 2 norm is o 2 n 1 m where n and m are the time step size polynomial degree and regularity in the space variable of the exact solution respectively numerical examples are presented to illustrate the theoretical results

the modified method of characteristics with adjusted advection the mmoc procedure for approximating the solutions of transport dominated diffusion problems does not automatically preserve integral conservation laws leading to mass balance errors in many kinds of flow problems the variant called the mmocaa discussed herein preserves the conservation law at a minor additional computational cost it is shown that its solution in either galerkin or finite difference form converges at the same rates as were proved earlier by dougl as and russell for the standard mmoc procedure

state operators on generalizations of fuzzy structures we enlarge the language of email protected monoids which are a non commutative generalizations of both mv algebras and bl algebras by adding a unary operation that describes algebraic properties of a state an analog of probability measures the resulting algebras are called email protected monoids and email protected monoids we present basic properties of such algebras we describe subdirectly irreducible algebras some generators of the varieties of state morphism email protected monoids and an interplay between states and state operators

uncertainty measures decomposability and admissibility abstract uncertainty measures are defined in a general manner as morphism between certain posets which permits a unified treatment of different measures for classical and for fuzzy events admissible measures are discussed in a decision theoretical notation making use of quasilinear entropies connections and differences to decomposable measures are illustrated

every state on semisimple mv algebra is integral integral representation theorem will be established for finitely additive probability measures states on semisimple mv algebras this result generalizes the well known theorem of butnariu and klement in case of order continuous states on tribes of fuzzy sets precisely it will be demonstrated that every state on a separating clan of continuous fuzzy sets arises as an integral with respect to a unique borel probability measure the key technique leading to this result exploits the geometrical topological properties of the state space the set of all states on every mv algebra forms a bauer simplex

subdiffusion with time dependent coefficients improved regularity and second order time stepping this article concerns second order time discretization of subdiffusion equations with time dependent diffusion coefficients high order differentiability and regularity estimates are established for subdiffusion equations with time dependent coefficients using these regularity results and a perturbation argument of freezing the diffusion coefficient we prove that the convolution quadrature generated by the second order backward differentiation formula with proper correction at the first time step can achieve second order convergence for both nonsmooth initial data and incompatible source term numerical experiments are consistent with the theoretical results

numerical algorithm for the space time fractional fokker planck system with two internal states the fractional fokker planck system with multiple internal states is derived in xu and deng math model nat phenom mathbf 13 10 2018 where the space derivative is laplace operator if the jump length distribution of the particles is power law instead of gaussian the space derivative should be replaced with fractional laplacian this paper focuses on solving the two state fokker planck system with fractional laplacian we first provide a priori estimate for this system under different regularity assumptions on the initial data then we use l 1 scheme to discretize the time fractional derivatives and finite element method to approximate the fractional laplacian operators furthermore we give the error estimates for the space semidiscrete and fully discrete schemes without any assumption on regularity of solutions finally the effectiveness of the designed scheme is verified by one and two dimensional numerical experiments

generalized bosbach and riecan states based on relative negations in residuated lattices bosbach and riecan states on residuated lattices both are generalizations of probability measures on boolean algebras recently two types of generalized bosbach states on residuated lattices were introduced by georgescu and muresan through replacing the standard mv algebra in the original definition with arbitrary residuated lattices as codomains however several interesting problems there remain still open the first part of the present paper gives positive answers to these open problems it is proved that every generalized bosbach state of type ii is of type i and the similarity cauchy completion of a residuated lattice endowed with an order preserving generalized bosbach state of type i is unique up to homomorphisms preserving similarities where the codomain of the type i state is assumed to be cauchy complete consequently many existing results about generalized bosbach states can be further strengthened the second part of the paper introduces the notion of relative negation with respect to a given element called relative element in residuated lattices and then many issues with the canonical negation such as glivenko property semi divisibility generalized riecan state of residuated lattices can be extended to the context of such relative negations in particular several necessary and sufficient conditions for the set of all relatively regular elements of a residuated lattice to be special residuated lattices are given of which one extends the well known glivenko theorem and it is also proved that relatively generalized riecan states vanishing at the relative element are uniquely determined by their restrictions on the mv algebra consisting of all relatively regular elements when the domain of the states is relatively semi divisible and the codomain is involutive

error analysis of a finite element method for the space fractional parabolic equation we consider an initial boundary value problem for a one dimensional fractional order parabolic equation with a space fractional derivative of riemann liouville type and order alpha in 1 2 we study a spatial semidiscrete scheme using the standard galerkin finite element method with piecewise linear finite elements as well as fully discrete schemes based on the backward euler method and the crank nicolson method error estimates in the l 2 d and h alpha 2 d norm are derived for the semidiscrete scheme and in the l 2 d norm for the fully discrete schemes these estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data extensive numerical results are presented to illustrate the theoretical results

a fourier method for the fractional diffusion equation describing sub diffusion in this paper a fractional partial differential equation fpde describing sub diffusion is considered an implicit difference approximation scheme idas for solving a fpde is presented we propose a fourier method for analyzing the stability and convergence of the idas derive the global accuracy of the idas and discuss the solvability finally numerical examples are given to compare with the exact solution for the order of convergence and simulate the fractional dynamical systems

conditional probability on σ mv algebras using the loomis sikorski representation of s mv algebras and dedekind s complete groups a conditional probability and expectation on these structures are defined as a natural generalization of the usual kolmogorovian conditioning

an optimal order estimate for eulerian lagrangian localized adjoint methods for variable coefficient advection reaction problems previously we developed eulerian lagrangian localized adjoint methods ellams to solve advection reaction problems ellams provide a systematic approach to treat boundary conditions and to conserve mass and have proven to be powerful methods for advection dominated problems in this paper the authors conduct theoretical analysis for these ellams and prove that they have an optimal order convergence rate moreover the estimates involve only the derivatives of the exact solution in space and along the characteristics in contrast many existing methods have only suboptimal order error estimates for advection reaction problems and the estimates involve the temporal derivatives of the exact solution which are usually much larger than the derivatives along the characteristics key words advection reaction equations eulerian lagrangian method convergence analysis optimal order error estimates

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

representation theorem for probabilities on ifs events riecan gave in riecan 2004 9 an axiomatic characterization of a probability on ifs events and proved in riecan 2006 10 a representation theorem for it in rencova 9 rencova introduced the notions of additive f probability and strongly additive f probability and gave a representation theorem for strongly additive f probabilities in this paper we give a generalization of the axiomatic characterization given by riecan in riecan 2006 10 from this the representation theorem for strongly additive f probabilities will follow

states on quantum and algebraic structures and their integral representation we review the notion of a state as a finitely additive functional on algebraic and quantum structures such as mv algebras effect algebras and bl algebras and their non commutative generalizations we show how a state can be represented as a standard integral through a additive borel probability measure

states on commutative basic algebras the paper deals with states on commutative basic algebras that are a non associative generalization of mv algebras or in other words the algebraic semantics for a fuzzy logic which generalizes the lukasiewicz logic in that the conjunction is not associative states are defined in the same way as mundici s states on mv algebras as normalized finitely additive 0 1 valued functions and some results analogous to the results that are known from mv algebras are proved

a characteristics mixed finite element method for advection dominated transport problems we define a new finite element method called the characteristics mixed method for approximating the solution to an advection dominated transport problem the method is based on a space time variational form of the advection diffusion equation our test functions are piecewise constant in space and in time they approximately follow the characteristics of the advective i e hyperbolic part of the equation thus the scheme uses a characteristic approximation to handle advection in time this is combined with a low order mixed finite element spatial approximation of the equation boundary conditions are incorporated in a natural and mass conservative fashion the scheme is completely locally conservative in fact on the discrete level fluid is transported along the approximate characteristics a postprocessing step is included in the scheme in which the approximation to the scalar unknown is improved by utilizing the approximate vector flux this has the effect of improving the rate of convergence of

variational formulation of time fractional parabolic equations abstract we consider initial boundary value problems for parabolic pde t u u f with fractional caputo derivative t of order 1 2 1 as time derivative and the usual laplacian as space derivative also called fractional diffusion equations in the literature we prove well posedness of corresponding variational formulations based entirely on fractional sobolev bochner spaces and extend existing results for possible choices of the initial value for u at t 0

uncertainty measures problems concerning additivity additivity of an uncertainty measure on an mv algebra has a clear meaning if the divisibility is dropped we come up to a so called girard algebra there we discuss strong resp weak additivity based on so called divisible disjoint unions resp on additivity for all sub mv algebras we obtain a description of those extensions from additive measures on an mv algebra to the canonical girard algebra extension of pairs which are strongly additive and valuation measures finally we prove the non existence of strongly additive measure extensions if the underlying mv algebra is a finite chain with more than two non trivial elements

time discretization of fractional subdiffusion equations via fractional resolvent operators abstract in this work we study time discretization of subdiffusion equations that is fractional differential equations of order 0 1 assuming that a is the generator of a fractional resolvent family s t t 0 which allows to write the solution to the subdiffusion equation t u t a u t f t as a variation of constants formula we find an interesting connection between s t t 0 and a discrete resolvent family s n n n and then by using the properties of s t t 0 we study the existence of solutions to the discrete subdiffusion equation c u n a u n f n n n where based on the backward euler method for a 0 given c u n is an approximation of t u t at time t n n we study simultaneously the fractional derivative in the caputo and riemann liouville sense we also provide error estimates and some experiments to illustrate the results

a new analysis for the convergence of the gradient discretization method for multidimensional time fractional diffusion and diffusion wave equations abstract we consider as discretization in space the gdm gradient discretization method developed recently in droniou et al 2018 to approximate multidimensional time fractional diffusion and diffusion wave equations where the fractional order of the time derivative is respectively satisfying 0 1 and 1 2 the time fractional derivative is given in the caputo sense the time discretization is performed using a uniform mesh for the time fractional diffusion equation we derive an implicit scheme in addition to an l l 2 a priori estimate which can be derived from a reasoning in bradji 2018 we present and prove a new l 2 h 0 1 a priori estimate for the discrete problem under the assumption that the exact solution is sufficiently smooth these a priori estimates allow to prove error estimates in discrete norms of l 2 h 0 1 and l l 2 the convergence order in these estimates is optimal in the sense of two points of view the first one is that the order in space is the same one proved in droniou et al 2018 for elliptic equation whereas the second point of view is that the particular case of this order when 1 is the same one obtained in droniou et al 2018 for the standard heat equation for the time fractional diffusion wave equation we derive two implicit schemes a full convergence analysis is carried out for both schemes in particular we develop new a priori estimates which yield error estimates in several discrete norms for each scheme the convergence is proved to be unconditional and optimal the convergence results are obtained thanks to a comparison with appropriately chosen auxiliary gradient schemes and to the stated a priori estimates these results improve the ones of bradji 2017 in which a conditional convergence is proved for a sushi scheme using stabilization and hybrid interfaces approximating a time fractional diffusion wave equation for both cases of time fractional diffusion and diffusion wave equations we show the well posedness in the sense that the discrete solutions depend continuously on the data of the considered problems the stated results can be extended to multi term time fractional diffusion and diffusion wave equations one of the main features when using gdm is that their results hold for all the schemes within the framework of gdm conforming and nonconforming finite element mixed finite element hybrid mixed mimetic family some multi point flux approximation finite volume schemes and some discontinuous galerkin schemes some examples of schemes recovered by the gss gradient schemes presented in this work are sketched among these examples we quote the one presented in jin et al 2015 to approximate time fractional diffusion equations this work extends and improves some results presented in brief or stated without proof in the notes bradji 2018 we present some numerical tests using sushi introduced in eymard et al 2010 to support the theoretical results

discrete maximal regularity of time stepping schemes for fractional evolution equations in this work we establish the maximal ell p regularity for several time stepping schemes for a fractional evolution model which involves a fractional derivative of order alpha in 0 2 alpha ne 1 in time these schemes include convolution quadratures generated by backward euler method and second order backward difference formula the l1 scheme explicit euler method and a fractional variant of the crank nicolson method the main tools for the analysis include operator valued fourier multiplier theorem due to weis math ann 319 735 758 2001 doi 10 1007 pl00004457 and its discrete analogue due to blunck stud math 146 157 176 2001 doi 10 4064 sm146 2 3 these results generalize the corresponding results for parabolic problems

conditioning on mv algebras and additive measures i abstract we present a lattice ordered semigroup approach for the foundation of conditional events which covers the special situations where the underlying unconditional events are boolean or fuzzy respectively our proposal is quite different from other ring theoretical approaches the problem of extending additivity of uncertainty measures from unconditional to conditional events will be discussed

conditional probability on mv algebras an appropriate definition of a conditional probability on an mv algebra was an open problem mentioned in riecan and mundici handbook of measure theory north holland amsterdam 2002 we propose some concept of conditional probability state on a complete mv algebra with product its basic properties will be proven and it will also be demonstrated that the conditional probability in classical probability theory is a special case of this definition moreover the paper contains also a discussion of the interpretation of fuzzy sets and conditioning in fuzzy probability theory

an analysis of the modified l1 scheme for time fractional partial differential equations with nonsmooth data we introduce a modified l1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous c

finite difference spectral approximations for the distributed order time fractional reaction diffusion equation on an unbounded domain the numerical approximation of the distributed order time fractional reaction diffusion equation on a semi infinite spatial domain is discussed in this paper a fully discrete scheme based on finite difference method in time and spectral approximation using laguerre functions in space is proposed the scheme is unconditionally stable and convergent with order o 2 2 n 1 m 2 where n and m are the time step size step size in distributed order variable polynomial degree and regularity in the space variable of the exact solution respectively a pseudospectral scheme is also proposed and analyzed some numerical examples are presented to demonstrate the efficiency of the proposed scheme

numerical algorithm for calculating the generalized mittag leffler function a numerical algorithm for calculating the generalized mittag leffler mathrm e alpha beta z function for arbitrary complex argument z and real parameters alpha 0 and beta in mathbb r is presented the algorithm uses the taylor series the exponentially improved asymptotic series and integral representations to obtain optimal stability and accuracy of the algorithm special care is applied to the limits of validity of the different schemes to avoid instabilities in the algorithm

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

numerical analysis of nonlinear subdiffusion equations we present a general framework for the rigorous numerical analysis of time fractional nonlinear parabolic partial differential equations with a fractional derivative of order alpha in 0 1 in time it relies on three technical tools a fractional version of the discrete gronwall type inequality discrete maximal regularity and regularity theory of nonlinear equations we establish a general criterion for showing the fractional discrete gronwall inequality and verify it for the l1 scheme and convolution quadrature generated by backward difference formulas further we provide a complete solution theory e g existence uniqueness and regularity for a time fractional diffusion equation with a lipschitz nonlinear source term together with the known results of discrete maximal regularity we derive pointwise l 2 omega norm error estimates for semidiscrete galerkin finite element solutions and fully discrete solutions which are of order o h 2 up to a logarithmic factor and o tau alpha

a complete characterization of all weakly additive measures and of all valuations on the canonical extension of any finite mv chain we consider extensions of the unique additive measure on a finite mv chain to uncertainty measures on its canonical girard algebra extension if the underlying mv chain has more than two non trivial elements in a previous paper we have proved the non existence of strongly additive measure extensions where strong additivity is defined as additivity not for all disjoint unions but only restricted to the so called divisible disjoint unions this negative result motivates to look for weakly additive measure extensions which are defined to be additive only on all mv subalgebras of the canonical girard algebra extension we obtain a characterization of all such mv subalgebras which are in fact mv chains and using this result we come up to a characterization of all weakly additive measure extensions moreover we characterize also all the valuation extensions and we prove the incompatibility of these two types of measure extensions except for underlying mv chains with one or two or four non trivial elements

a second order characteristic finite element scheme for convection diffusion problems a new characteristic finite element scheme is presented for it is of second order accuracy in time increment symmetric and unconditionally stable optimal error estimates are proved in the framework of l 2 theory numerical results are presented for two examples which show the advantage of the scheme

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

analysis of a temporal discretization for a semilinear fractional diffusion equation abstract this paper considers a temporal discretization of a semilinear fractional diffusion equation with nonsmooth initial data with appropriately graded temporal grids first order temporal accuracy in the norm of l 2 0 t l 2 is derived by a new discrete gronwall s inequality and then first order temporal accuracy in the norm of l 0 t l 2 is established for 1 2 1 finally numerical results are provided to verify the theoretical results

convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics abstract a nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium conservation of mass for the mixture leads to an elliptic equation for pressure and conservation for the displacing fluid yields a convectiondominated parabolic equation for the concentration of that fluid a sequential implicit time stepping procedure is defined in which the pressure and darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the concentration is approximated by a combination of a galerkin finite element method and the method of characteristics optimal order convergence in l2 is proved time truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the concentration equation temporal and spatial errors are lessened by direct computation of the velocity in the mixed method as opposed to differentiation of the pressure several extensions of these results are outlined

a logic for reasoning about the probability of fuzzy events in this paper we present the logic fp l n l which allows to reason about the probability of fuzzy events formalized by means of the notion of state in a mv algebra this logic is defined starting from a basic idea exposed by hajek metamathematics of fuzzy logic kluwer dordrecht 1998 two kinds of semantics have been introduced namely the class of weak and strong probabilistic models the main result of this paper is a completeness theorem for the logic fp l n l w r t both weak and strong models we also present two extensions of fp l n l the first one is the logic fp l n rpl obtained by expanding the fp l n l language with truth constants for the rationals in 0 1 while the second extension is the logic fcp l n email protected allowing to reason about conditional states

every state on interval effect algebra is integral we show that every state on an interval effect algebra is an integral through some regular borel probability measure defined on the borel algebra of a compact hausdorff simplex this is true for every effect algebra satisfying riesz decomposition property or for every many valued mv algebra in addition we show that each state on an effect subalgebra of an interval effect algebra e can be extended to a state on e our method represents also every state on the set of effect operators of a hilbert space as an integral