Partial Moduli Research Articles

Approximation properties of bivariate extension of blending type operators

The present article is in the continuation of our previous work [26], where we have improved the order of approximation of ??Bernstein P?lt?nea operators. In the given note, we study the bivariate extension of first order modification of these operators and their approximation properties such as convergence, error of approximation in terms of complete and partial modulus of continuity and their asymptotic formula. We present numerical examples to show the convergence of functions of two variables with the help of MATLsoftware. Also, we construct the GBS operators associated to the bivariate extension and present their approximation behavior.

Bridge scour monitoring using smart magnetic rock

The bridge local scour monitoring is characterized as harsh service environment, great danger and difficult in sensor arrangement, etc. So a new smart magnetic rock is raised as a dynamic scour sensor for bridge scour monitoring. The monitoring system consists of smart magnetic rock and magnetic gradient measurement. The magnetic gradient’s partial modulus is used to track the position of smart magnetic rock, by which the influence of environmental magnetic field is greatly minimized. The simplified method and corresponding formula are given to gauge smart magnetic rock’s magnetic moment. The monitoring method is verified by localization experiment and the relative error is discussed. The results show that magnetic gradient’s partial modulus is highly correlated with distance. Field tests are conducted at a bridge pier, and the results show that the scour depth measured by smart magnetic rock is basically consistent with those by sounding lead and ultrasonic depth finder.

Interrelations between Partial Moduli of Smoothness in Various Mixed Metrics

Interrelations between Partial Moduli of Smoothness in Various Mixed Metrics

Approximation of Functions by Generalized Parametric Blending-Type Bernstein Operators

In this paper, we introduce a new family of generalized blending-type bivariate Bernstein operators which depends on four parameters $$s_{1}$$ , $$s_{2}$$ , $$\alpha _{1}$$ and $$\alpha _{2}$$ . Approximation properties of these operators are studied, and we obtain the rate of convergence in terms of mixed and partial modulus of continuities. Moreover, we prove a Korovkin- and a Voronovskaja-type theorems for these operators. The last part of the paper is devoted to the associated GBS operators. In this part, we study degree of approximation of the GBS operators in terms of mixed modulus of continuity. GBS operators obtained here give better approximation than the original operators to the function f(x, y). Finally, approximation properties of the suggested operators and their associated GBS operators are discussed on graphs, for some numerical examples to show how GBS operator gives better approximation to f(x, y). Also, approximation properties of the suggested operators for different values of parameters $$s_{1}$$ , $$s_{2}$$ , $$\alpha _{1}$$ and $$\alpha _{2}$$ are illustrated on graphs. It should be mentioned that any increase in $$\alpha _{i}$$ values or any decrease in $$s_{i}$$ values gives better approximation of the suggested operators to f(x, y).

Rebar corrosion monitoring using magnetic gradient and partial modulus

Abstract The rebar corrosion monitoring using magnetic method is a hot research area and it is necessary to develop effective methods using the rebar’s own magnetic characteristics with considerations of environmental magnetic field. In this paper, the magnetic gradient and partial modulus is proposed for rebar corrosion monitoring. The effectiveness of corrosion monitoring method is discussed on the basis of experimental results from eight rebars and four reinforced concrete beams before and after corrosion. The other three galvanized round steels are used to investigate the reliability and stability of the method. The results show that the percentage loss calculated by the magnetic gradient and partial modulus is basically consistent with the actual weight loss ratio and cross-sectional area loss ratio for uniform corrosion and the effects of environmental magnetic field can be negligible by using magnetic gradient and partial modulus, which shows promise for medium- and long-term monitoring in rebar corrosion.

Estimates of Partial Moduli of Smoothness in Metrics of Lp1∞ and L∞p2 through Partial Moduli of Smoothness in Metrics of Lp1p2

Interrelation between partial moduli of smoothness of positive order considered in metrics and Lp1∞, L∞p2, and Lp1p2 is studied.

Quantitative estimates for the tensor product (p,q)-Balázs-Szabados operators and associated generalized Boolean sum operators

In this study, we give some approximation results for the tensor product of (p,q)-Bal?zs-Szabados operators associated generalized Boolean sum (GBS) operators. Firstly, we introduce tensor product (p,q)-Bal?zs-Szabados operators and give an uniform convergence theorem of these operators on compact rectangular regions with an illustrative example. Then we estimate the approximation for the tensor product (p,q)-Bal?zs-Szabados operators in terms of the complete modulus of continuity, the partial modulus of continuity, Lipschitz functions and Petree?s K-functional corresponding to the second modulus of continuity. After that, we introduce the GBS operators associated the tensor product (p,q)-Bal?zs-Szabados operators. Finally, we improve the rate of smoothness by the mixed modulus of smoothness and Lipschitz class of B?gel continuous functions for the GBS operators.

Bivariate α,q-Bernstein–Kantorovich Operators and GBS Operators of Bivariate α,q-Bernstein–Kantorovich Type

In this paper, we introduce a family of bivariate α , q -Bernstein–Kantorovich operators and a family of G B S (Generalized Boolean Sum) operators of bivariate α , q -Bernstein–Kantorovich type. For the former, we obtain the estimate of moments and central moments, investigate the degree of approximation for these bivariate operators in terms of the partial moduli of continuity and Peetre’s K-functional. For the latter, we estimate the rate of convergence of these G B S operators for B-continuous and B-differentiable functions by using the mixed modulus of smoothness.

Fuzzy bivariate triangular functions with application to nonlinear fuzzy Fredholm–Volterra integral equations in two dimensions

In this paper, an iterative numerical method based on two-dimensional triangular basis functions has been presented to obtain the numerical solution of fuzzy Fredholm–Volterra integral equations in two dimensions. Error estimation of the proposed method has been gained in terms of uniform and partial modulus of continuity. Also, numerical stability with respect to choice of the starting point of iteration has been also proved under the any theorem. Finally, some numerical examples have been included to demonstrate the convergence and accuracy of the proposed method.

Bivariate q-Bernstein–Chlodowsky–Durrmeyer type operators and the associated GBS operators

This paper deals with the approximation properties of the [Formula: see text]-bivariate Bernstein–Chlodowsky operators of Durrmeyer type. We investigate the approximation degree of the [Formula: see text]-bivariate operators for continuous functions in Lipschitz space and also with the help of partial modulus of continuity. Further, the Generalized Boolean Sum (GBS) operator of these bivariate [Formula: see text]–Bernstein–Chlodowsky–Durrmeyer operators is introduced and the rate of convergence in the Bögel space of continuous functions by means of the Lipschitz class and the mixed modulus of smoothness is examined. Furthermore, the convergence and its comparisons are shown by illustrative graphics for the [Formula: see text]-bivariate operators and the associated GBS operators to certain functions using Maple algorithms.

New approach to solve two-dimensional Fredholm integral equations

In this paper, we present an efficient iterative method based on quadrature formula to solve two-dimensional nonlinear Fredholm integral equations. We investigate the convergence analysis of the proposed method. The error estimation is given in terms of uniform and partial modulus of continuity. In addition, we show the numerical stability analysis of the method with respect to the choice of the first iteration. Finally, some numerical experiments confirm the theoretical results and illustrate the accuracy of the method.

The Approximation of Bivariate Blending Variant Szász Operators Based Brenke Type Polynomials

We have constructed a new sequence of positive linear operators with two variables by using Szasz-Kantorovich-Chlodowsky operators and Brenke polynomials. We give some inequalities for the operators by means of partial and full modulus of continuity and obtain a Lipschitz type theorem. Furthermore, we study the convergence of Szasz-Kantorovich-Chlodowsky-Brenke operators in weighted space of function with two variables and estimate the rate of approximation in terms of the weighted modulus of continuity.

Towards the accuracy of iterative numerical methods for fuzzy Hammerstein–Fredholm integral equations

An iterative numerical method based on the three-point quadrature formula is applied to solve nonlinear fuzzy Hammerstein–Fredholm integral equations of the second kind and provides more accurate results than the trapezoidal quadrature iterative method. Also, we obtain the error estimation of the iterative method in terms of the uniform and partial modulus of continuity. Moreover, we investigate the concept of numerical stability of the iterative method with respect to the choice of the initial approximation. Illustrative examples are included to demonstrate the accuracy and efficiency of the iterative method.

Conjugate Functions and the Modulus of Smoothness of Fractional Order

In the present paper, estimates of the partial moduli of smoothness of fractional order of the conjugate functions of several variables are obtained in the space C(Tn). The accuracy of the obtained estimates is established by appropriate examples.

Kantorovich variant of a new kind of q‐Bernstein–Schurer operators

Ren and Zeng (2013) introduced a new kind of q‐Bernstein–Schurer operators and studied some approximation properties. Acu et al. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q‐Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K‐functional. Next, we introduce the bivariate case of q‐Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K‐functional. Finally, we define the generalized Boolean sum operators of the q‐Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd.

Degree of Approximation for Bivariate Chlodowsky–Szasz–Charlier Type Operators

We consider a combination of Chlodowsky polynomials with generalized Szasz operators involving Charlier polynomials. We give the degree of approximation for these bivariate operators by means of the complete and partial modulus of continuity, and also by using weighted modulus of continuity. Furthermore, we construct a GBS (Generalized Boolean Sum) operator of bivariate Chlodowsky–Szasz–Charlier type and estimate the order of approximation in terms of mixed modulus of continuity.

New error estimate in the iterative numerical method for nonlinear fuzzy Hammerstein–Fredholm integral equations

In this paper, we obtain the error estimation of the iterative method based on quadrature formula to solve nonlinear fuzzy Fredholm integral equations of the second kind given in Fuzzy Sets & Syst. 245 (2014) 1–17, in terms of uniform and partial modulus of continuity. Moreover, we extend in the context of using the modulus of continuity, the notion of numerical stability of the solution with respect to the first iteration.

On certain family of mixed summation integral type two-dimensional [formula omitted]-Lupaş–Phillips–Bernstein operators

In the present paper, we introduce a two dimensional mixed summation–integral type q-Lupaş–Phillips–Bernstein operators on a rectangular domain □=[0,1]×[0,1] and investigate their Korovkin type approximation properties. We compute the rate of convergence of these new operators by means of the full and partial modulus of continuity. We also establish the order of approximation for the operators by using the Peetre K-functional. In last section, we get some numerical examples for operator.

Iterative method for numerical solution of two-dimensional nonlinear fuzzy integral equations

In this paper, we prove the convergence of the method of successive approximations used to approximate the solution of two-dimensional nonlinear fuzzy Fredholm integral equations of the second kind. Also, we present an iterative procedure based on quadrature formula to solve such equations. The error estimation of the proposed method is given in terms of uniform and partial modulus of continuity. Finally, some numerical experiments confirm the theoretical results and illustrate the accuracy of the method.

Numerical solution of nonlinear fuzzy Fredholm integral equations using iterative method

In this paper, we propose an iterative procedure based on quadrature formula to solve nonlinear fuzzy Fredholm integral equations of the second kind. Moreover, the error estimation of the proposed method in terms of uniform and partial modulus of continuity is given. Finally, illustrative examples are included in order to demonstrate the accuracy and the convergence of the proposed method.