Recent zbMATH articles in MSC 41https://zbmath.org/atom/cc/412021-01-08T12:24:00+00:00WerkzeugChebyshev polynomial regularization method for solving the unknown source on Poisson equation.https://zbmath.org/1449.653042021-01-08T12:24:00+00:00"Dai, Pei"https://zbmath.org/authors/?q=ai:dai.pei"Xiong, Xiangtuan"https://zbmath.org/authors/?q=ai:xiong.xiangtuanSummary: The problem of determining unknown source on Poisson equation is a severely ill-posed problem. Because the standard Tikhonov regularization method has saturation restriction, in this paper, we use Chebyshev polynomial regularization method to obtain the regularization solution for the unknown source. The error estimates under an a priori parameter choice rule and an a posteriori parameter choice rule are given, respectively.Gene level association analysis based on functional linear model.https://zbmath.org/1449.920322021-01-08T12:24:00+00:00"Guo, Hao"https://zbmath.org/authors/?q=ai:guo.hao"Liu, Xiangtao"https://zbmath.org/authors/?q=ai:liu.xiangtao"Gong, Haobo"https://zbmath.org/authors/?q=ai:gong.haobo"Huang, Jianfei"https://zbmath.org/authors/?q=ai:huang.jianfeiSummary: When the functional linear model is used to carry out the gene level association analysis, it is necessary to numerically approximate the values of genetic variants in each gene. In order to improve the accuracy of approximations and reduce the time-consuming problem by using the traditional spline function in deducing functional linear model, this paper proposes a functional linear model based on Legendre polynomials, which can enhance the accuracy and efficiency of approximations in getting functional linear model due to the orthogonality. By analyzing the simulated genetic data, it is known that the proposed method can keep the reasonable type 1 error, enhance the statistical power, and reduce the computational time. Therefore, the functional linear model with Legendre polynomials has more practical application values than the traditional functional linear model in gene level association analysis.A note on farthest point problem in Banach spaces.https://zbmath.org/1449.460192021-01-08T12:24:00+00:00"Som, Sumit"https://zbmath.org/authors/?q=ai:som.sumit"Savas, Ekrem"https://zbmath.org/authors/?q=ai:savas.ekremSummary: Farthest point problem states that ``Must every uniquely remotal set in a Banach space be singleton?'' In this paper we introduce the notion of partial ideal statistical continuity of a function which is way weaker than continuity of a function. We give an example to show that partial ideal statistical continuity is weaker than continuity. In this paper we use ideal summability to give some answers to FPP problem which improves some former results. We prove that if \(E\) is a non-empty, bounded, uniquely remotal subset in a real Banach space \(X\) such that \(E\) has a Chebyshev center \(c\) and the farthest point map \(F:X\rightarrow E\) restricted to \([c,F(c)]\) is partially ideal statistically continuous at \(c\) then \(E\) is singleton.A note on the Dagum-II sigmoid function with applications to wealth data. Other applications.https://zbmath.org/1449.410142021-01-08T12:24:00+00:00"Kyurkchiev, Nikolay"https://zbmath.org/authors/?q=ai:kyurkchiev.nikolay-v"Iliev, Anton"https://zbmath.org/authors/?q=ai:iliev.anton-iUpper and lower estimates for the Hausdorff approximation of the shifted Heaviside function are obtained. To do this the authors introduce a class of Dagum-II cumulative distribution function (DCDF). They introduce a family of four parametric sigmoidal functions based on DCDF. The results are illustrated by numerical examples. The work has applications in income theory.
Reviewer: Angela Slavova (Sofia)A kernel-based technique to solve three-dimensional linear Fredholm integral equations of the second kind over general domains.https://zbmath.org/1449.653582021-01-08T12:24:00+00:00"Esmaeili, Hamid"https://zbmath.org/authors/?q=ai:esmaeili.hamid"Moazami, Davoud"https://zbmath.org/authors/?q=ai:moazami.davoudSummary: In this article, we study a kernel-based method to solve three-dimensional linear Fredholm integral equations of the second kind over general domains. The radial kernels are utilized as a basis in the discrete collocation method to reduce the solution of linear integral equations to that of a linear system of algebraic equations. Integrals appeared in the scheme are approximately computed by the Gauss-Legendre and Monte Carlo quadrature rules. The method does not require any background mesh or cell structures, so it is mesh free and accordingly independent of the domain geometry. Thus, for the three-dimensional linear Fredholm integral equation, an irregular domain can be considered. The convergence analysis is also given for the method. Finally, numerical examples are presented to show the efficiency and accuracy of the technique.\(C^1\)-conforming quadrilateral spectral element method for fourth-order equations.https://zbmath.org/1449.653342021-01-08T12:24:00+00:00"Li, Huiyuan"https://zbmath.org/authors/?q=ai:li.huiyuan"Shan, Weikun"https://zbmath.org/authors/?q=ai:shan.weikun"Zhang, Zhimin"https://zbmath.org/authors/?q=ai:zhang.zhiminSummary: This paper is devoted to Professor Benyu Guo's open question on the \(C^1\)-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the \(C^1\)-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories -- interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global \(C^1\)-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the \(C^1\)-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a \(C^1\)-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.Using Bernstein multi-scaling polynomials to obtain numerical solution of Volterra integral equations system.https://zbmath.org/1449.653712021-01-08T12:24:00+00:00"Yaghoobnia, A. R."https://zbmath.org/authors/?q=ai:yaghoobnia.a-r"Ezzati, R."https://zbmath.org/authors/?q=ai:ezzati.rezaSummary: In this paper, first, Bernstein multi-scaling polynomials (BMSPs), which are generalization of Bernstein polynomials (BPs), are introduced and some of their properties are explained. Then, a new method based on BMSPs to achieved numerical solution for system of nonlinear integral equations is proposed. The proposed method converted the system of integral equations to a nonlinear system. To evaluate the efficiency of the proposed method, some systems of nonlinear integral equations are solved, and their numerical solutions are compared with other similar methods.Conformable fractional approximation by max-product operators.https://zbmath.org/1449.410202021-01-08T12:24:00+00:00"Anastassiou, George A."https://zbmath.org/authors/?q=ai:anastassiou.george-aSummary: Here we study the approximation of functions by a big variety of Max-product operators under conformable fractional differentiability. These are positive sublinear operators. Our study is based on our general results about positive sublinear operators. We produce Jackson type inequalities under conformable fractional initial conditions. So our approach is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of a high order conformable fractional derivative of the function under approximation.Numerical approach for solving the Riccati and logistic equations via QLM-rational Legendre collocation method.https://zbmath.org/1449.651652021-01-08T12:24:00+00:00"Khader, Mohamed M."https://zbmath.org/authors/?q=ai:khader.mohamed-m"Adel, M."https://zbmath.org/authors/?q=ai:adel.mohamed-hSummary: In the presented study, we are presenting the approximate solutions of two important equations, the Riccati and Logistic equations; the presented technique is based on the rational Legendre function. Since all the studied models are nonlinear, we convert these nonlinear equations to a sequence of linear ordinary differential equations (ODEs), then by applying the quasi-linearization method (QLM) on these resulting ODEs at each iteration, these ODEs will be converted to a simple linear system of algebraic equations which can be solved. Two numerical examples are presented and we compared between the approximate and the exact solutions.Rate of convergence of Bézier Durrmeyer type \(\lambda \)-Bernstein operators.https://zbmath.org/1449.410122021-01-08T12:24:00+00:00"Cai, Qingbo"https://zbmath.org/authors/?q=ai:cai.qingbo"Chen, Shuni"https://zbmath.org/authors/?q=ai:chen.shuniSummary: In this paper, we introduce Bézier Durrmeyer type \(\lambda \)-Bernstein operators \(D_{n, \lambda}^{ (\alpha)} (f; x)\) with a parameter \(\lambda \in [-1, 1]\). We establish a global approximation theorem in terms of the second-order modulus of continuity and a direct approximation theorem by means of the Ditzian-Totik modulus of smoothness. We also combine the Bojanic-Cheng's decomposition method jointly with some analysis techniques to derive an asymptotically estimate on the rate of convergence for some absolutely continuous functions of \(D_{n, \lambda}^{ (\alpha)} (f; x)\). Finally, we present an example to show the convergence of \(D_{n, \lambda}^{ (\alpha)} (f; x)\) to \(f (x)\) with some given \(f\).On a new class of Bernstein type operators based on Beta function.https://zbmath.org/1449.470332021-01-08T12:24:00+00:00"Bhatt, Dhawal J."https://zbmath.org/authors/?q=ai:bhatt.dhawal-j"Mishra, Vishnu Narayan"https://zbmath.org/authors/?q=ai:mishra.vishnu-narayan"Jana, Ranjan Kumar"https://zbmath.org/authors/?q=ai:jana.ranjan-kumarSummary: We develop Bernstein type operators using the Beta function and study their approximation properties. By using Korovkin's theorem, we achieve the uniform convergence of sequences of these operators. We obtain the rate of convergence in terms of modulus of continuity and establish the Voronovskaja type asymptotic result for these operators. At last, the graphical comparison of these newly defined operators with few of the fundamental but significant operators is discussed.Weighted approximation of Hermite interpolation operators in Orlicz spaces.https://zbmath.org/1449.410022021-01-08T12:24:00+00:00"Wang, Yaru"https://zbmath.org/authors/?q=ai:wang.yaru"Wu, Garidi"https://zbmath.org/authors/?q=ai:wu.garidiSummary: In this paper, we mainly consider the approximation problem of Hermite interpolation operators which is based on the zeros of Chebyshev polynomials of the second kinds in Orlicz spaces. The degree of approximation of the interpolation operators is given by using Hölder inequality, Hardy-Littlewoods maximal functions mode and convexity of \(N\)-function in Orlicz spaces.A bounds Tauberian theorem.https://zbmath.org/1449.400102021-01-08T12:24:00+00:00"Stenger, Allen"https://zbmath.org/authors/?q=ai:stenger.allenSummary: We weaken the hypothesis and the conclusion of a Hardy-Littlewood Tauberian theorem, and apply the new theorem to deduce asymptotic behavior of the coefficients of an exponentiated lacunary series.On integral inequalities of Hermite-Hadamard type for coordinated \(r\)-mean convex functions.https://zbmath.org/1449.260292021-01-08T12:24:00+00:00"Gao, Dan-Dan"https://zbmath.org/authors/?q=ai:gao.dan-dan"Xi, Bo-Yan"https://zbmath.org/authors/?q=ai:xi.boyan"Wu, Ying"https://zbmath.org/authors/?q=ai:wu.ying"Guo, Bai-Ni"https://zbmath.org/authors/?q=ai:guo.bai-niSummary: In the paper, the authors first introduce a concept ``\(r\)-mean convex function on coordinates'' and then establish several integral inequalities of the Hermite-Hadamard type for \(r\)-convex functions and \(r\)-mean convex functions on coordinates.Rate of convergence of Gupta-Srivastava operators based on certain parameters.https://zbmath.org/1449.410252021-01-08T12:24:00+00:00"Pratap, Ram"https://zbmath.org/authors/?q=ai:pratap.ram"Deo, Naokant"https://zbmath.org/authors/?q=ai:deo.naokantSummary: In the present paper, we consider the Bézier variant of the Gupta-Srivastava operators [\textit{R. Pratap} and \textit{N. Deo}, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 113, No. 3, 2495--2505 (2019; Zbl 1419.41002)] and discuss some direct convergence results by using of Lipschitz type spaces, Ditzian-Totik modulus of smoothness, weighted modulus of continuity and for functions whose derivatives are of bounded variation. In the end some graphical representation for comparison with other variants have been presented.Chebyshev polynomials on circular arcs.https://zbmath.org/1449.300802021-01-08T12:24:00+00:00"Schiefermayr, Klaus"https://zbmath.org/authors/?q=ai:schiefermayr.klausThe Chebyshev polynomial of degree \(N\), \(N\in\mathbb{N}\), on a compact set \(K\subset\mathbb{C}\) in the complex plane is that monic polynomial \(\hat{\mathcal{P}}_N\in\hat{\mathbb{P}}_N\) which is minimal with respect to the supremum norm on \(K\) within the set of all monic polynomials, i.e. \[\hat{\mathcal{P}}_N:=\min\{\|\hat{P}_N\|_K:\hat{P}_N\in\hat{\mathbb{P}}_N\}\,, \] where \(\|\cdot\|\) denotes the supremum norm on \(K\) and \(\hat{\mathbb{P}}_N\) denotes the set of all monic polynomials of degree \(N\). In this paper, an explicit parametric representation of the complex Chebyshev polynomials \(\hat{P}_N(z)\) on a given circular arc \(A_\alpha\), defined by \[ A_\alpha:=\{z\in\mathbb{C}: |z|=1,-\alpha\leq\arg(z)\leq\alpha\},\quad 0<\alpha\leq\pi\,,\] of the unit circle (in the complex plane) in terms of real Chebyshev polynomials \(\hat{\mathcal{T}}_{N'}(x)\) on two symmetric intervals \([-1,-a]\cup[a,1]\) (on the real line) is given. For example, let \(0<\alpha<\frac{2n\pi}{2n+1}\), \(0<c<\frac{n\pi}{2n+1}\) be fixed and \(a:=\cos(\alpha/2)\). Let \(\mathcal{T}_{2n+1}\in\mathbb{P}_{2n+1}\) and \(\mathcal{U}_{2n-2}\in\mathbb{P}_{2n-2}\) be uniquely determined by \[ \mathcal{T}_{2n+1}^2(x)+(1-x^2)(x^2-a^2)(x^2-c^2)\mathcal{U}_{2n-2}^2(x)=1\,. \] Then \[\hat{P}_{2n}(z)=L_{2n}z^{n-1/2}\left(\mathcal{T}_{2n+1}(x)+i\sqrt{1-x^2}(x^2-a^2)\mathcal{U}_{2n-2}(x)\right)\,,\] is a monic polynomial of degree \(2n\) in \(z\) with real coefficients, where \(x\) and \(z\) are connected by \[ z\mapsto\frac12\left(\sqrt{z}+\frac1{\sqrt{z}}\right)=:x\,. \] Moreover, \(\hat{P}_{2n}(z)\) is the Chebyshev polynomial of degree \(2n\) on \(A_\alpha\) with minimum deviation. The case \(N=2n-1\) is also considered. It is also considered representation of Chebyshev polynomials on \([-1,-a]\cup[a,1]\) with the help of Jacobian elliptic and theta functions, which goes back to the work of Akhiezer in the 1930's.
Reviewer: Konstantin Malyutin (Kursk)A numerical study of Newton interpolation with extremely high degrees.https://zbmath.org/1449.650102021-01-08T12:24:00+00:00"Breuss, Michael"https://zbmath.org/authors/?q=ai:breuss.michael"Kemm, Friedemann"https://zbmath.org/authors/?q=ai:kemm.friedemann"Vogel, Oliver"https://zbmath.org/authors/?q=ai:vogel.oliverSummary: In current textbooks the use of Chebyshev nodes with Newton interpolation is advocated as the most efficient numerical interpolation method in terms of approximation accuracy and computational effort. However, we show numerically that the approximation quality obtained by Newton interpolation with Fast Leja (FL) points is competitive to the use of Chebyshev nodes, even for extremely high degree interpolation. This is an experimental account of the analytic result that the limit distribution of FL points and Chebyshev nodes is the same when letting the number of points go to infinity. Since the FL construction is easy to perform and allows to add interpolation nodes on the fly in contrast to the use of Chebyshev nodes, our study suggests that Newton interpolation with FL points is currently the most efficient numerical technique for polynomial interpolation. Moreover, we give numerical evidence that any reasonable function can be approximated up to machine accuracy by Newton interpolation with FL points if desired, which shows the potential of this method.On Hermite interpolation and divided differences.https://zbmath.org/1449.410012021-01-08T12:24:00+00:00"Dubeau, François"https://zbmath.org/authors/?q=ai:dubeau.francoisSummary: This paper is a survey of topics related to Hermite interpolation. In the first part we present the standard analysis of the Hermite interpolation problem. Existence, uniqueness and error formula are included. Then some computational aspects are studied including Leibnitz' formula and devided differences for monomials. Moreover continuity and differentiation properties of divided differences are analyzed. Finally we represent Hermite polynomial with respect to different basis and give links between them.A continued fractional recurrence algorithm for generalized inverse tensor Padé approximation.https://zbmath.org/1449.410112021-01-08T12:24:00+00:00"Gu, Chuanqing"https://zbmath.org/authors/?q=ai:gu.chuanqing"Huang, Yizheng"https://zbmath.org/authors/?q=ai:huang.yizheng"Chen, Zhibing"https://zbmath.org/authors/?q=ai:chen.zhibingSummary: The tensor exponential function has been widely used in cybernetics, image processing and various engineering fields. Based on the generalized matrix inverse, an effective tensor generalized inverse is defined for the first time on the scalar inner product space, and a continued fractional algorithm is constructed for the tensor Padé approximation. On the other hand, we successfully use the tensor \(t\)-product to calculate the power of the tensor, and recursively give the power series expansion of the tensor exponential function. Based on the previous two work, the continuous fractional algorithm designed in this paper is used to approximate the tensor exponential function. Its characteristic is that the algorithm can be programmed to implement recursive calculations, and in the calculation process, it is not necessary to calculate the product of the tensor and to calculate the inverse of the tensor. The numerical experiments of the two tensor exponential functions given in this paper show that comparing the continuous fractional algorithm with the commonly used truncation method, the proposed algorithm is effective without reducing the approximation order. If the dimension of the tensor is relatively large, a continuous fractional algorithm based on the generalized inverse of tensors will also have certain advantages.Approximation of functions belonging to \(L[0, \infty)\) by product summability means of its Fourier-Laguerre series.https://zbmath.org/1449.420052021-01-08T12:24:00+00:00"Khatri, Kejal"https://zbmath.org/authors/?q=ai:khatri.kejal"Mishra, Vishnu Narayan"https://zbmath.org/authors/?q=ai:mishra.vishnu-narayanSummary: In this paper, we have proved the degree of approximation of functions belonging to \(L[0, \infty)\) by harmonic-Euler means of its Fourier-Laguerre series at \(x=0\). The aim of this paper is to concentrate on the approximation properties of the functions in \(L[0, \infty)\) by harmonic-Euler means of its Fourier-Laguerre series associated with the function \(f\).Product of statistical probability convergence and its applications to Korovkin-type theorem.https://zbmath.org/1449.400032021-01-08T12:24:00+00:00"Jena, Bidu Bhusan"https://zbmath.org/authors/?q=ai:jena.bidu-bhusan"Paikray, Susanta Kumar"https://zbmath.org/authors/?q=ai:paikray.susanta-kumarSummary: In the present work, we introduce and study the notion of statistical probability convergence for sequences of random variables as well as the concept of statistical convergence for sequences of real numbers, which are defined over a Banach space via product of deferred Cesàro and deferred Nörlund summability means. We first establish a theorem presenting a connection between them. Based upon our proposed method, we then prove a Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results (in classical as well as statistical versions). Finally, an illustrative example is presented here by means of the generalized Meyer-König and Zeller operators for a positive sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.Approximation of Durrmeyer type Bernstein-Stancu polynomials in movable compact disks.https://zbmath.org/1449.410062021-01-08T12:24:00+00:00"Pang, Zhaojun"https://zbmath.org/authors/?q=ai:pang.zhaojun"Yu, Dansheng"https://zbmath.org/authors/?q=ai:yu.danshengSummary: A kind of new Durrmeyer type Bernstein-Stancu polynomials in movable compact disks is introduced. The approximation rate of the Durrmeyer type Bernstein-Stancu polynomials in the movable compact disks is given.Direct estimates for Stancu variant of Lupaş-Durrmeyer operators based on Polya distribution.https://zbmath.org/1449.410222021-01-08T12:24:00+00:00"Mishra, Lakshmi Narayan"https://zbmath.org/authors/?q=ai:mishra.lakshmi-narayan"Kumar, Alok"https://zbmath.org/authors/?q=ai:kumar.alokSummary: In this paper, we study approximation properties of a family of linear positive operators and establish the Voronovskaja type asymptotic formula, local approximation and pointwise estimates using the Lipschitz type maximal function. In the last section, we consider the King type modification of these operators to obtain better estimates.The Dunkl generalization of Stancu type \(q\)-Szász-Mirakjan-Kantorovich operators and some approximation results.https://zbmath.org/1449.410232021-01-08T12:24:00+00:00"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad-ayman|mursaleen.mohammad|mursaleen.momammad"Ahasan, Mohd."https://zbmath.org/authors/?q=ai:ahasan.mohdSummary: In this paper, a Dunkl type generalization of Stancu type \(q\)-Szász-Mirakjan-Kantorovich positive linear operators of the exponential function is introduced. With the help of well-known Korovkin's theorem, some approximation properties and also the rate of convergence for these operators in terms of the classical and second-order modulus of continuity, Peetre's \(K\)-functional and Lipschitz functions are investigated.Univariate Lidstone-type multiquadric quasi-interpolants.https://zbmath.org/1449.410042021-01-08T12:24:00+00:00"Wu, Ruifeng"https://zbmath.org/authors/?q=ai:wu.ruifeng"Li, Huilai"https://zbmath.org/authors/?q=ai:li.huilai"Wu, Tieru"https://zbmath.org/authors/?q=ai:wu.tieruSummary: In this paper, a kind of univariate multiquadric quasi-interpolants with the derivatives of approximated function is proposed by combining a univariate multiquadric quasi-interpolant with Lidstone interpolation polynomials proposed in [\textit{G. J. Lidstone}, Proc. Edinb. Math. Soc., II. Ser. 2, 16--19 (1930; JFM 56.1053.03); \textit{F. A. Costabile} and \textit{F. Dell'Accio}, Appl. Numer. Math. 52, No. 4, 339--361 (2005; Zbl 1064.65008); \textit{T. Cătinaş}, J. Appl. Funct. Anal. 1, No. 4, 425--439 (2006; Zbl 1100.41001)]. For practical purposes, another kind of approximation operators without any derivative of the approximated function is given using divided differences to approximate the derivatives. Some error bounds and the convergence rates of new operators are derived, which demonstrates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter \(c\) and a non-negative integer \(n\). Finally, we make extensive comparison with the other existing methods and give some numerical examples. Moreover, the associated algorithm is easily implemented.Approximation of weighted Müntz rational in Orlicz space.https://zbmath.org/1449.410092021-01-08T12:24:00+00:00"Gao, Ya"https://zbmath.org/authors/?q=ai:gao.ya"Wu, Garidi"https://zbmath.org/authors/?q=ai:wu.garidiSummary: In this paper, the Hardy-Littlewood maximal function, the weighted modulus of continuity, the convexity of \(N\) function, and the skills of inequality were applied to study the approximation rate of weighted Müntz rational approximation of smooth function in weighted Orlicz space. Furthermore, the approximation of the rational function in the changed system was considered, in which the approximation speed was better than the approximation of normal Müntz rational function.On orthogonal second order finite functions associated with triangular meshes and their application in mathematical modeling.https://zbmath.org/1449.420702021-01-08T12:24:00+00:00"Leont'ev, Viktor Leonidovich"https://zbmath.org/authors/?q=ai:leontev.v-l"Kochulimov, Aleksandr Valer'evich"https://zbmath.org/authors/?q=ai:kochulimov.aleksandr-valerevichSummary: Second order orthogonal finite functions are studied, their compact supports are local sets of grid triangles. The theorem on approximating the properties of sequences of sets of such functions is formulated. The applications in algorithms of mixed numerical methods of boundary value problems solution, in algorithms of linear approximation of surfaces are considered.On approximation by some Bernstein-Kantorovich exponential-type polynomials.https://zbmath.org/1449.410052021-01-08T12:24:00+00:00"Aral, Ali"https://zbmath.org/authors/?q=ai:aral.ali"Otrocol, Diana"https://zbmath.org/authors/?q=ai:otrocol.diana"Raşa, Ioan"https://zbmath.org/authors/?q=ai:rasa.ioanThe authors introduce the integral extension in Kantorovich sense of a special case of the modified Bernstein operators defined in [\textit{S. Morigi} and \textit{M. Neamtu}, Adv. Comput. Math. 12, No. 2--3, 133--149 (2000; Zbl 1044.42500)]. It is proved that these operators form an approximation process in \(C[0,1],\) and in the exponentially weighted space \(L_{p,\mu}[0,1],\) where \(1 \leq p < \infty\) and \(\mu > 0.\) Quantitative results in terms of appropriate moduli of smoothness and \(K\)-functionals, and a quantitative Voronovskaja type result are obtained.
Reviewer: Zoltán Finta (Cluj-Napoca)Approximation by generalized Stancu type integral operators involving Sheffer polynomials.https://zbmath.org/1449.410242021-01-08T12:24:00+00:00"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad"Rahman, Shagufta"https://zbmath.org/authors/?q=ai:rahman.shagufta"Ansari, Khursheed J."https://zbmath.org/authors/?q=ai:ansari.khursheed-jamalSummary: In this article, we give a generalization of integral operators which involves Sheffer polynomials introduced by \textit{S. Sucu} and \textit{İ. Büyükyazici} [Bull. Math. Anal. Appl. 4, No. 4, 56--66 (2012; Zbl 1314.41014)]. We obtain approximation properties of our operators with the help of the universal Korovkin's theorem and study convergence properties by using modulus of continuity, the second order modulus of smoothness and Peetre's \(K\)-functional. We have also established Voronovskaja type asymptotic formula. Furthermore, we study the convergence of these operators in weighted spaces of functions on the positive semi-axis and estimate the approximation by using weighted modulus of continuity.Sequential approximation of functions in Sobolev spaces using random samples.https://zbmath.org/1449.410262021-01-08T12:24:00+00:00"Wu, Kailiang"https://zbmath.org/authors/?q=ai:wu.kailiang"Xiu, Dongbin"https://zbmath.org/authors/?q=ai:xiu.dongbinSummary: We present an iterative algorithm for approximating an unknown function sequentially using random samples of the function values and gradients. This is an extension of the recently developed sequential approximation (SA) method, which approximates a target function using samples of function values only. The current paper extends the development of the SA methods to the Sobolev space and allows the use of gradient information naturally. The algorithm is easy to implement, as it requires only vector operations and does not involve any matrices. We present tight error bound of the algorithm, and derive an optimal sampling probability measure that results in fastest error convergence. Numerical examples are provided to verify the theoretical error analysis and the effectiveness of the proposed SA algorithm.Reverse Markov inequality on the unit interval for polynomials whose zeros lie in the upper unit half-disk.https://zbmath.org/1449.410072021-01-08T12:24:00+00:00"Komarov, M. A."https://zbmath.org/authors/?q=ai:komarov.mikhail-aSummary: We prove that there is an absolute constant \(A>0\) such that \[\max_{-1\leq x\leq 1}\vert P'(x)\vert \geq A\sqrt{n}\cdot \max_{-1\leq x\leq 1}\vert P'(x)\vert \] for an arbitraray algebraic polynomial \(P\) of degree \(n\) whose zeros lie in the half-disk \(\{z:\vert z\vert \leq 1,\)\; Im\(z\geq 0\}\).An \(hp\)-version Chebyshev spectral collocation method for nonlinear Volterra integro-differential equations with weakly singular kernels.https://zbmath.org/1449.652742021-01-08T12:24:00+00:00"Jia, Hongli"https://zbmath.org/authors/?q=ai:jia.hongli"Yang, Yang"https://zbmath.org/authors/?q=ai:yang.yang.5|yang.yang.2|yang.yang.1|yang.yang.3|yang.yang.4"Wang, Zhongqing"https://zbmath.org/authors/?q=ai:wang.zhongqingSummary: This paper presents an \(hp\)-version Chebyshev spectral collocation method for nonlinear Volterra integro-differential equations with weakly singular kernels. The \(hp\)-version error bound of the collocation method under the \({H^1}\)-norm is established on an arbitrary mesh. Numerical experiments demonstrate the effectiveness of the proposed method.Approximation of the modified binary Gauss-Weierstrass operator in \({L_p} (\mathbb R_+^2)\).https://zbmath.org/1449.410172021-01-08T12:24:00+00:00"Guan, Xinguo"https://zbmath.org/authors/?q=ai:guan.xinguo"He, Cuiling"https://zbmath.org/authors/?q=ai:he.cuiling"Zhong, Yu"https://zbmath.org/authors/?q=ai:zhong.yu"Wang, Chunju"https://zbmath.org/authors/?q=ai:wang.chunjuSummary: Using \(K\)-functional, the approximation theorem of the modified binary Gauss-Weierstrass operator in \({L_p} (\mathbb R_+^2)\) space is obtained.Using differentiation matrices for pseudospectral method solve Duffing oscillator.https://zbmath.org/1449.651682021-01-08T12:24:00+00:00"Nhat, L. A."https://zbmath.org/authors/?q=ai:nhat.le-anhSummary: This article presents an approximate numerical solution for nonlinear Duffing Oscillators by pseudospectral (PS) method to compare boundary conditions on the interval \([-1, 1]\). In the PS method, we have been used differentiation matrix for Chebyshev points to calculate numerical results for nonlinear Duffing Oscillators. The results of the comparison show that this solution had the high degree of accuracy and very small errors. The software used for the calculations in this study was Mathematica V.10.4.\(\alpha\)-Bernstein-Kantorovich operators.https://zbmath.org/1449.410132021-01-08T12:24:00+00:00"Deo, Naokant"https://zbmath.org/authors/?q=ai:deo.naokant"Pratap, Ram"https://zbmath.org/authors/?q=ai:pratap.ramSummary: In this paper we proposed the Kantorovich form of \(\alpha\)-Bernstein operators introduced by \textit{X. Chen} et al. [J. Math. Anal. Appl. 450, No. 1, 244--261 (2017; Zbl 1357.41015)]. We give some auxiliary properties and study the direct local approximation theorem, Voronovskaya type asymptotic and function of bounded variation for \(\alpha \)-Bernstein-Kantorovich operators.Multi-domain decomposition pseudospectral method for nonlinear Fokker-Planck equations.https://zbmath.org/1449.652792021-01-08T12:24:00+00:00"Sun, Tao"https://zbmath.org/authors/?q=ai:sun.tao"Wang, Tian-Jun"https://zbmath.org/authors/?q=ai:wang.tianjunSummary: Results on the composite generalized Laguerre-Legendre interpolation in unbounded domains are established. As an application, a composite Laguerre-Legendre pseudospectral scheme is presented for nonlinear Fokker-Planck equations on the whole line. The convergence and the stability of the proposed scheme are proved. Numerical results show the efficiency of the scheme and conform well to theoretical analysis.Exact estimate of \(n\)-widths of a convolution function class in Orlicz spaces.https://zbmath.org/1449.410162021-01-08T12:24:00+00:00"Sun, Fangmei"https://zbmath.org/authors/?q=ai:sun.fangmei"Wu, Garidi"https://zbmath.org/authors/?q=ai:wu.garidiSummary: In this paper, we study the \(n\)-widths of a \(2\pi\)-periodic convolution function class defined by linear differential operators with real coefficient in Orlicz spaces, and obtain the exact values of \(n\)-K width, \(n\)-G width, \(n\)-L width, and \(n\)-B width of this function class and its corresponding optimal subspaces.Analysis of approximation accuracy for structure functions by orthogonal functions of exponential type.https://zbmath.org/1449.410152021-01-08T12:24:00+00:00"Prokhorov, S. A."https://zbmath.org/authors/?q=ai:prokhorov.s-a"Grafkin, V. V."https://zbmath.org/authors/?q=ai:grafkin.v-vSummary: The method and the result of analysis for the approximation accuracy for structure functions by orthogonal functions of exponential type are given.A new modification of Durrmeyer type mixed hybrid operators.https://zbmath.org/1449.410212021-01-08T12:24:00+00:00"Kajla, Arun"https://zbmath.org/authors/?q=ai:kajla.arun"Acar, Tuncer"https://zbmath.org/authors/?q=ai:acar.tuncerSummary: In 2008, \textit{V. Miheşan} [Creat. Math. Inform. 17, No. 3, 466--472 (2008; Zbl 1265.41055)] constructed a general class of linear positive operators generalizing the Szász operators. In this article, a Durrmeyer variant of these operators is introduced which is a method to approximate the Lebesgue integrable functions. First, we derive some indispensable auxiliary results in the second section. We present a quantitative Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. The rate of convergence for differential functions whose derivatives are of bounded variation is also obtained.Best approximation and characterization of Hilbert spaces.https://zbmath.org/1449.410272021-01-08T12:24:00+00:00"Rajabi, Setareh"https://zbmath.org/authors/?q=ai:rajabi.setarehSummary: It is well known that for any nonempty closed convex subset \(C\) of a Hilbert space, any best approximation \(y\in C\) of the point \(x\) satisfies the inequality \(\Vert x-y\Vert^{2}+\Vert z-y\Vert^{2} \leq \Vert x-z\Vert^{2}\) for all \(z\in C\). In this paper, we first introduce and study a new subset of best approximations involving this inequality in general metric spaces. Then, we provide some equivalent conditions which characterize Hilbert spaces.On some applicable approximations of Gaussian type integrals.https://zbmath.org/1449.330232021-01-08T12:24:00+00:00"Chesneau, Christophe"https://zbmath.org/authors/?q=ai:chesneau.christophe"Navarro, Fabien"https://zbmath.org/authors/?q=ai:navarro.fabienSummary: In this paper, we introduce new applicable approximations for Gaussian type integrals. A key ingredient is the approximation of the function \(e^{-x^2}\) by the sum of three simple polynomial-exponential functions. Five special Gaussian type integrals are then considered as applications. Approximation of the so-called Voigt error function is investigated.On \(G\)-Banach frames.https://zbmath.org/1449.420602021-01-08T12:24:00+00:00"Rathore, Ghanshyam Singh"https://zbmath.org/authors/?q=ai:singh.ghanshyam"Mittal, Tripti"https://zbmath.org/authors/?q=ai:mittal.triptiSummary: \textit{M. R. Abdollahpour} et al. [Methods Funct. Anal. Topol. 13, No. 3, 201--210 (2007; Zbl 1144.46010)] generalized the concepts of frames for Banach
spaces and defined \(g\)-Banach frames in Banach spaces. In the present paper, we
define various types of \(g\)-Banach frames in Banach spaces. Examples and counter
examples to distinguish various types of \(g\)-Banach frames in Banach spaces have
been given. It has been proved that if a Banach space \(\mathcal{X}\) has a Banach frame, then \(\mathcal{X}\) has a normalized tight \(g\)-Banach frame for \(\mathcal{X}\). A characterization of an exact \(g\)-Banach frame has been given. Also, we consider the finite sum of \(g\)-Banach frames and give a sufficient condition for the finite sum of \(g\)-Banach frames to be a \(g\)-Banach frame. Finally, a sufficient condition for the stability of \(g\)-Banach frames in Banach spaces which provides optimal frame bounds has been given.Study on the approximation properties of Gauss-Weierstrass operator.https://zbmath.org/1449.410182021-01-08T12:24:00+00:00"Wang, Tao"https://zbmath.org/authors/?q=ai:wang.tao|wang.tao.7|wang.tao.4|wang.tao.3|wang.tao.9|wang.tao.8|wang.tao.5|wang.tao.1|wang.tao.2|wang.tao.6Summary: We use analytic methods and function decomposition technique to study bounded functions for Gauss-Weierstrass operators in probability type operators, and obtain the pointwise asymptotic expansion and preserving properties for bounded functions.Highly accurate technique for solving distributed-order time-fractional-sub-diffusion equations of fourth order.https://zbmath.org/1449.652702021-01-08T12:24:00+00:00"Abdelkawy, M. A."https://zbmath.org/authors/?q=ai:abdelkawy.mohamed-a"Babatin, Mohammed M."https://zbmath.org/authors/?q=ai:babatin.mohammed-m"Lopes, António M."https://zbmath.org/authors/?q=ai:lopes.antonio-mSummary: This paper presents a new method for calculating the numerical solution of distributed-order time-fractional-sub-diffusion equations (DO-TFSDE) of fourth order. The method extends the shifted fractional Jacobi (SFJ) collocation scheme for discretizing both the time and space variables. The approximate solution is expressed as a finite expansion of SFJ polynomials whose derivatives are evaluated at the SFJ quadrature points. The process yields a system of algebraic equations that are solved analytically. The new method is compared with alternative numerical algorithms when solving different types of DO-TFSDE. The results show that the proposed method exhibits superior accuracy with an exponential convergence rate.An indirect finite element method for variable-coefficient space-fractional diffusion equations and its optimal-order error estimates.https://zbmath.org/1449.653312021-01-08T12:24:00+00:00"Zheng, Xiangcheng"https://zbmath.org/authors/?q=ai:zheng.xiangcheng"Ervin, V. J."https://zbmath.org/authors/?q=ai:ervin.vincent-j"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.1Summary: We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension. By the representation formula of the solutions \(u(x)\) to the proposed variable coefficient models in terms of \(v(x)\), the solutions to the constant coefficient analogues, we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations \(v_h(x)\) to \(v(x)\) and then obtain the approximations \(u_h(x)\) of \(u(x)\) by plugging \(v_h(x)\) into the representation of \(u(x)\). Optimal-order convergence estimates of \(u(x)-u_h(x)\) are proved in both \(L^2\) and \(H^{\alpha /2}\) norms. Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.Multivariate homogeneous two-point Padé approximants and continued fractions.https://zbmath.org/1449.410102021-01-08T12:24:00+00:00"Chakir, Y."https://zbmath.org/authors/?q=ai:chakir.y"Abouir, J."https://zbmath.org/authors/?q=ai:abouir.jilali"Benouahmane, B."https://zbmath.org/authors/?q=ai:benouahmane.brahimSummary: The multivariate homogeneous two-point Padé approximants have been defined and studied recently. In the current work, we consider higher-order approximants and derive error formulas of these approximants using orthogonality conditions. Diverse three-term recurrence relations satisfied by the monic orthogonal polynomials are presented. Various continued fractions provided by these relations and the quotient-difference algorithm applied to a power series (positive or negative exponents) are described in terms of their relationships with the multivariate homogeneous two-point Padé table. Numerical examples are furnished to illustrate our results.The rational approximation to \(|x|^\alpha\) (\(1 \le \alpha < 2\)) at the adjusted tangent nodes.https://zbmath.org/1449.410082021-01-08T12:24:00+00:00"Cheng, Yiyuan"https://zbmath.org/authors/?q=ai:cheng.yiyuan"Zhang, Yongquan"https://zbmath.org/authors/?q=ai:zhang.yongquan"Zha, Xingxing"https://zbmath.org/authors/?q=ai:zha.xingxingSummary: Since Newman's rational operator has a good approximation for \(|x|\), we consider the approximation of \(|x|^\alpha\) by a Newman-\(\alpha\) rational operator. In this paper, we discuss the convergence rate of the operator Newman-\(\alpha\) at the adjusted tangent nodes \(X = \{{\mathrm{tan}}^2\frac{k\pi}{4n}\}_{k = 1}^n\), and finally obtain the exact approximation order \(O (\frac{1}{n^{2\alpha}})\). The result not only contains the approximation result in the case of \(\alpha = 1\), but also is better than the conclusion when the node group is selected for the first and the second type of Chebyshev nodes, equidistant nodes etc.Exponential time differencing methods for the time-space-fractional Schrödinger equation.https://zbmath.org/1449.651892021-01-08T12:24:00+00:00"Liang, Xiao"https://zbmath.org/authors/?q=ai:liang.xiao"Bhatt, Harish"https://zbmath.org/authors/?q=ai:bhatt.harish-pSummary: In this paper, exponential time differencing schemes with Padé approximation to the Mittag-Leffler function are proposed for the time-space-fractional nonlinear Schrödinger equations. Ways of increasing the efficiency of the proposed schemes are discussed. Numerical experiments are performed on the time-space-fractional nonlinear Schrödinger equations with various parameters. The accuracy, efficiency, and reliability of the proposed method are illustrated by numerical results.On the study of oscillating viscous flows by using the Adomian-Padé approximation.https://zbmath.org/1449.760412021-01-08T12:24:00+00:00"Liu, Chi-Min"https://zbmath.org/authors/?q=ai:liu.chi-minSummary: The Adomian-Padé technique is applied to examine two oscillating viscous flows, the Stokes' second problem and the pressure-driven pulsating flow. Main purposes for studying oscillating flows are not only to verify the accuracy of the approximation solution, but also to provide a basis for analyzing more problems by the present method with the help of Fourier analysis. Results show that the Adomian-Padé approximation presents a very excellent behavior in comparison with the exact solution of Stokes' second problem. For the pulsating flow, only the Adomian decomposition method is required to perform the calculation as the fluid domain is finite where the Padé approximant may not provide a better solution. Based on present results, more problems can be mathematically solved by using the Adomian-Padé technique, the Fourier analysis, and powerful computers.On the numerical treatment and analysis of two-dimensional Fredholm integral equations using quasi-interpolant.https://zbmath.org/1449.653572021-01-08T12:24:00+00:00"Derakhshan, M."https://zbmath.org/authors/?q=ai:derakhshan.maryam"Zarebnia, M."https://zbmath.org/authors/?q=ai:zarebnia.mohammadSummary: In this paper, we study the quadratic rule for the numerical solution of linear and nonlinear two-dimensional Fredholm integral equations based on spline quasi-interpolant. Also the convergence analysis of the method is given. We show that the order of the method is \(O(h_x^{m+1}) + O(h_y^{m'+1})\). The theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical part.On Voronovskaja type estimation for a Kantorovich type Bernstein-Stancu operator.https://zbmath.org/1449.410192021-01-08T12:24:00+00:00"Xia, Rongrong"https://zbmath.org/authors/?q=ai:xia.rongrong"Yu, Dansheng"https://zbmath.org/authors/?q=ai:yu.danshengSummary: Aiming at a Kantorovich type Bernstein-Stancu operator and the direct theorems of the approximation, the article further generalizes the relevant conclusions and establishes a Voronovskaja type asymptotic estimation for its approximation.A series expansion method for solving the boundary value problem connected with the Helmholtz equation.https://zbmath.org/1449.653052021-01-08T12:24:00+00:00"Du, Xinwei"https://zbmath.org/authors/?q=ai:du.xinweiSummary: A boundary value problem connected with the Helmholtz equation is studied in a smooth bounded domain. A series expansion method is proposed for obtaining an approximate solution to the problem. Tikhonov regularization is applied to the problem with noisy data. Numerical experiments are presented to show the effectiveness of the proposed method.Multiquadric quasi-interpolation operator based on integral values of successive subintervals.https://zbmath.org/1449.410032021-01-08T12:24:00+00:00"Wu, Jinming"https://zbmath.org/authors/?q=ai:wu.jinming"Shan, Tingting"https://zbmath.org/authors/?q=ai:shan.tingting"Zhu, Chungang"https://zbmath.org/authors/?q=ai:zhu.chungangSummary: In some interpolation problems, the function values at the interpolated points are not known, whereas the integral values of some successive intervals are given. How to use the integral values of successive intervals to tackle function reconstruction is an important problem. Firstly, the function values and first-order derivatives at the knots with fourth-order approximation are derived by using the linear combination of the integral values. Secondly, a kind of multiquadric quasi-interpolation operator based on integral values of successive intervals is constructed. It is called integro multiquadric quasi-interpolation operator. Finally, its global error is given and it also possesses fourth-order approximation. Numerical experiments illustrate that the proposed method is flexible and effective.