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The use of “ideal” optimal test functions in a Petrov–Galerkin scheme guarantees the discrete stability of the variational problem. However, in practice, the computation of the ideal optimal test functions is computationally intractable. In this paper, we study the effect of using approximate, “practical” test functions on the stability of the DPG (discontinuous Petrov–Galerkin) method and the change in stability between the “ideal” and “practical” cases is analyzed by constructing a Fortin operator. We highlight the construction of an “optimal” DPG Fortin operator for H 1 and H ( div ) spaces; the continuity constant of the Fortin operator is a measure of the loss of stability between the ideal and practical DPG methods. We take a two-pronged approach: first, we develop a numerical procedure to estimate an upper bound on the continuity constant of the Fortin operator in terms of the inf–sup constant γ h of an auxiliary problem. Second, we construct a sequence of approximate Fortin operators and exactly compute the continuity constants of the approximate operators, which provide a lower bound on the exact Fortin continuity constant. Our results shed light not only on the change in stability by using practical test functions, but also indicate how stability varies with the approximation order p and the enrichment order Δ p . The latter has important ramifications when one wishes to pursue local h p -adaptivity. [ABSTRACT FROM AUTHOR]

In this paper, efficient numerical schemes are proposed for solving the fractional water wave models that describe the propagation of surface water wave. By using the weighted and shifted Grünwald–Letnikov (WSGL) formula to approximate the nonlocal fractional operators, we design a series of second order accurate difference schemes for the considered models. The existence, stability and convergence of numerical solutions of the proposed numerical schemes are established rigorously. The analysis shows that the proposed numerical schemes are unconditionally stable with second order accuracy for both temporal and spatial discretizations. Several numerical results are provided to verify the efficiency and accuracy of our theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract: To realize the basic units of fractional order controllers, it is necessary to approximate irrational differintegrators. Many methods deal with analog approximation. But not many results exist for discrete approximation. In this case, approximations are most times poor in some frequency range, or the quality of the approximation is good in a narrow frequency interval. This paper proposes an indirect discretization method that takes advantage from the particle swarm optimization (PSO) to approximate fractional order operators. The method employs an heuristic procedure to optimize the interlacing of zero–pole pairs on the real axis. Then, a discretization rule is applied to obtain the discrete approximation. Simulation results show that the frequency response obtained by PSO improves the approximation offered by other efficient and recent indirect discretization techniques. [Copyright &y& Elsevier]

Abstract: In the present paper, introducing a King type modification of the Lupaş operators, the rates of statistical convergence of this modification are studied by means of the modulus of continuity and Lipschitz class functional. We also prove that, on some subintervals of [0,1], the error estimation of these operators is better than the classical q-Bernstein operators. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to construct a new iterative scheme by the hybrid method and prove a strong convergence theorem for the approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings which is also a common solution to infinite systems of equilibrium and variational inequality problems in a uniformly smooth and strictly convex real Banach space with Kadec–Klee property using the properties of a generalized -projection operator. Using this result, we give some applications. Our results extend many known recent results in the literature. [Copyright &y& Elsevier]

Abstract: Fractional differential equations have wide applications in science and engineering. In this paper, we consider a class of control systems governed by the semilinear fractional differential equations in Hilbert spaces. By using the semigroup theory, the fractional power theory and fixed point strategy, a new set of sufficient conditions are formulated which guarantees the approximate controllability of semilinear fractional differential systems. The results are established under the assumption that the associated linear system is approximately controllable. Further, we extend the result to study the approximate controllability of fractional systems with nonlocal conditions. An example is provided to illustrate the application of the obtained theory. [Copyright &y& Elsevier]

Abstract: The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfying many interesting properties. Many types of generalized rough set models have been proposed in the literature. This paper discusses the rough approximations of Atanassov intuitionistic fuzzy sets in crisp and fuzzy approximation spaces in which both constructive and axiomatic approaches are used. In the constructive approach, concepts of rough intuitionistic fuzzy sets and intuitionistic fuzzy rough sets are defined, properties of rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are examined. Different classes of rough intuitionistic fuzzy set algebras and intuitionistic fuzzy rough set algebras are obtained from different types of fuzzy relations. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of crisp/fuzzy relations which produce the same operators. [Copyright &y& Elsevier]

Abstract: In this paper we present the sequence of linear Bernstein-type operators defined for by , being the classical Bernstein operators and being any function that is continuously differentiable times on , such that , and for . We investigate its shape preserving and convergence properties, as well as its asymptotic behavior and saturation. Moreover, these operators and others of King type are compared with each other and with . We present as an interesting byproduct sequences of positive linear operators of polynomial type with nice geometric shape preserving properties, which converge to the identity, which in a certain sense improve in approximating a number of increasing functions, and which, apart from the constant functions, fix suitable polynomials of a prescribed degree. The notion of convexity with respect to plays an important role. [Copyright &y& Elsevier]

Abstract: The first part of this paper represents a continuation of the study of approximations in a hypergroup. We analyse the lower and upper approximations of a subset in a hypergroup, with respect to some operators, particularly the completeness operator. In the last part, we introduce and analyse the lower and upper approximation of a subset in a fuzzy hypergroup. [Copyright &y& Elsevier]

Abstract: In this paper, we present an new approach to construct the so-called shape preserving interpolation curves. The basic idea is first to approximate the set of interpolated points with a class of MQ quasi-interpolation operators and then pass through the set with the use of multivariate interpolation by using compactly supported radial basis functions. This approach possesses the advantages of certain shape preserving and good approximation behaviors. The proposed algorithm is easy to implement. [Copyright &y& Elsevier]

Abstract: In this paper, using the concept of statistical -convergence which is stronger than statistical convergence, we obtain a statistical approximation theorem for a general sequence of max-product operators, including Shepard type operators, although its classical limit fails. We also compute the corresponding statistical rates of the approximation. [Copyright &y& Elsevier]

Abstract: In this paper we introduce the notion of equi-statistical -convergence which is stronger than the uniform convergence (in the ordinary sense), statistical uniform convergence and statistical uniform -convergence. Then, we also give its use in the Korovkin-type approximation theory. We also compute the rate of equi-statistical -convergence of a sequence of positive linear operators. [Copyright &y& Elsevier]

Abstract: This paper deals with approximating properties of the newly defined -generalization of the Szász–Mirakjan operators in the case . Quantitative estimates of the convergence, the Voronovskaja’s theorem and saturation of convergence for complex -Szász–Mirakjan operators attached to analytic functions in compact disks are given. In particular, it is proved that for functions analytic in , , the rate of approximation by the -Szász–Mirakjan operators () is of order versus for the classical Szász–Mirakjan operators. [ABSTRACT FROM AUTHOR]

Abstract: It is a useful method in research of group theory to construct a new group by using known groups. Lower and upper approximation operators of rough sets are applied into group theory and so the notion of a rough group has been introduced. In this paper, we first point out that there are still some incomplete propositions in [N. Kuroki, P.P. Wang, The lower and upper approximations in a fuzzy group, Inform. Sci. 90 (1996) 203–220] although some authors have showed several incorrect statements in the literature. We then present improved versions of the incomplete propositions and continue to study the image and inverse image of rough approximations of a subgroup with respect to a homomorphism between two groups. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate convergence and approximation properties of a Chlodowsky type generalization of Stancu polynomials (we called Stancu–Chlodowsky polynomials).The rates of convergence of this generalization are obtained by means of modulus of continuity and by using the -functional of Peetre. We also present and prove theorems on weighted approximation and the order of approximation of continuous functions by these operators on all positive semi-axis. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to investigate the error which results from the method of approximation operators with logarithmic sigmoidal function. By means of the method of extending functions, a class of feed-forward neural network operators is introduced. Using these operators as approximation tools, the upper bounds of errors, in uniform norm, approximating continuous functions, are estimated. Also, a class of quasi-interpolation operators with logarithmic sigmoidal function is constructed for approximating continuous functions defined on the total real axis. [Copyright &y& Elsevier]

Abstract: Recently, the quasi-interpolants of some classical operators were introduced. Mache and Müller gave Baskakov quasi-interpolants and obtained approximation equivalence theorem with In this paper, we extend the above result with modulus (0 ≤ λ ≤ 1) which unified classical modulus and Ditzian-Totik modulus. [Copyright &y& Elsevier]