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Multinode Shepard method for two-dimensional elliptic boundary problems on different shaped domains

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2024年 448卷

作者： Dell'Accio, Francesco Di Tommaso, Filomena Francomano, Elisa Univ Calabria Dept Math & Comp Sci Via P Bucci Arcavacata Di Rende CS Italy Univ Palermo Dept Engn Viale Sci Palermo Italy

In this paper, we continue the study on the application of multinode Shepard method to numerically solve elliptic Partial Differential Equations (PDEs) equipped with various conditions at the boundary of domains of different shapes. In particular, for the first time, the multinode Shepard method is proposed to solve elliptic PDEs with Dirichlet and/or Neumann boundary conditions. The method has been opportunely handled to efficiently work dealing with scattered distribution of points and, to this aim, several experiments in different 2d domains have been performed. Comparisons with the analytic solution and the results generated by the Kansa's RBF solvers have been reported referring to Halton points. The results are very promising and should be of interest for applications in the real world.

关键词： approximation by rational functions Multinode Shepard method Collocation method Elliptic PDEs

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The enriched multinode Shepard collocation method for solving elliptic problems with singularities

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APPLIED NUMERICAL MATHEMATICS 2024年 205卷 87-100页

作者： Dell'Accio, Francesco Di Tommaso, Filomena Francomano, Elisa Univ Calabria Dept Math & Comp Sci I-87036 Arcavacata Di Rende CS Italy Univ Palermo Dept Engn Palermo Italy

In this paper, the multinode Shepard method is adopted for the first time to numerically solve a differential problem with a discontinuity in the boundary. Starting from previous studies on elliptic boundary value problems, here the Shepard method is employed to catch the singularity on the boundary. Enrichments of the functional space spanned by the multinode cardinal Shepard basis functions are proposed to overcome the difficulties encountered. The Motz's problem is considered as numerical benchmark to assess the method. Numerical results are presented to show the effectiveness of the proposed approach.

关键词： approximation by rational functions Multinode Shepard method Collocation method Elliptic PDEs Motz problem

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Accurate and stable numerical method based on the Floater-Hormann interpolation for stochastic Ito-Volterra integral equations

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NUMERICAL ALGORITHMS 2023年第1期94卷 275-292页

作者： Mirzaee, Farshid Naserifar, Shiva Solhi, Erfan Malayer Univ Fac Math Sci & Stat POB 65719-95863 Malayer Iran

In various fields of science and engineering, such as financial mathematics, mathematical physics models, and radiation transfer, stochastic integral equations are important and practical tools for modeling and describing problems. Due to the existence of random factors, we face a fundamental problem in solving stochastic integral equations, and that is the lack of analytical solutions or the great complexity of these solutions. Therefore, finding an efficient numerical solution is essential. In this paper, we intend to propose and study a new method based on the Floater-Hormann interpolation and the spectral collocation method for linear and nonlinear stochastic Ito-Volterra integral equations (SVIEs). The Floater-Hormann interpolation offers an approximation regardless of the distribution of the points. Therefore, this method can be mentioned as a meshless method. The presented method reduces SVIEs under consideration into a system of algebraic equations that can be solved by the appropriate method. We presented an error bound to be sure of the convergence and reliability of the method. Finally, the efficiency and the applicability of the present scheme are investigated through some numerical experiments.

关键词： Stochastic Ito integral equations Spectral collocation method Brownian motion approximation by rational functions

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New Analytic Representations of the Hypergeometric functions p+1Fp

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CONSTRUCTIVE approximation 2022年第3期55卷 891-917页

作者： Lopez, Jose L. Pagola, Pedro J. Palacios, Pablo Univ Publ Navarra Dept Estadist Informat & Matemat Pamplona 31006 Spain Univ Publ Navarra INAMAT2 Pamplona 31006 Spain

The power series expansions of the hypergeometric functions F-p+1(p) (a, b(1),..., b(p);c(1),..., c(p);z) converge either inside the unit disk vertical bar z vertical bar 1. Norlund's expansion in powers of z/(z ... 详细信息

The power series expansions of the hypergeometric functions F-p+1(p) (a, b(1),..., b(p);c(1),..., c(p);z) converge either inside the unit disk vertical bar z vertical bar < 1 or outside this disk vertical bar z vertical bar > 1. Norlund's expansion in powers of z/(z - 1) converges in the half-plane R(z) < 1/2. For arbitrary z(0) is an element of C, Buhring's expansion in inverse powers of z - z(0) converges outside the disk vertical bar z - z(0)vertical bar = max{vertical bar z(0)vertical bar, vertical bar z(0) - 1 vertical bar}. None of them converge on the whole indented closed unit disk vertical bar z vertical bar <= 1, z not equal 1. In this paper, we derive new expansions in terms of rational functions of z that converge in different regions, bounded or unbounded, of the complex plane that contain the indented closed unit disk. We give either explicit formulas for the coefficients of the expansions or recurrence relations. The key point of the analysis is the use of multi-point Taylor expansions in appropriate integral representations of F-p+1(p)(a, b(1),..., b(p);c(1),..., c(p);z). We show the accuracy of the approximations by means of several numerical experiments.

关键词： Generalized hypergeometric functions approximation by rational functions Two and three-point Taylor expansions

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Nested sequences of rational spaces: Bernstein approximation, dimension elevation, and Polya-type theorems on positive polynomials

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JOURNAL OF approximation THEORY 2018年 234卷 20-50页

作者： Ait-Haddou, Rachid Mazure, Marie-Laurence Render, Hermann Osaka Univ Cybermedia Ctr 6F 1-32 Machikaneyama Toyonaka Osaka 5600043 Japan Grenoble Alpes Univ Lab Jean Kuntzmann CNRS UMR 5224 F-38000 Grenoble France Univ Coll Dublin Sch Math & Stat Dublin 4 Ireland

On a given closed bounded interval, an infinite nested sequence of Extended Chebyshev spaces containing the constants automatically generates an infinite sequence of positive linear operators of Bernstein-type. Unlike the polynomial framework, this situation does not guarantee convergence of the corresponding approximation process. Obviously, convergence cannot be obtained without the density of the union of all the involved spaces in the set of continuous functions equipped with the uniform norm. The initial purpose of this work was to answer the following question: conversely, is density sufficient to guarantee convergence? Addressing this issue is all the more natural as density was indeed proved to imply convergence in the special case of nested sequences of Miintz spaces on positive intervals. In this paper we give a negative answer to the aforementioned question by considering nested sequences of rational spaces defined by infinite sequences of real poles outside the given interval. Surprisingly, in this rational context, we show that ensuring convergence is equivalent to determining all Polya positive sequences, in the sense of all infinite sequences of positive numbers which guarantee Polya-type results for the positivity of univariate polynomials on the non-negative axis. This interesting connection with Polya positive sequences enables us to produce a simple necessary and sufficient condition for the poles to ensure convergence, thanks to results by Baker and Handelman on strongly positive sequences of polynomials. (C) 2018 Elsevier Inc. All rights reserved.

关键词： approximation by rational functions Bernstein-type operators Extended Chebyshev spaces Dimension elevation Positive polynomials Polya-type theorems

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SLICE REGULAR MALMQUIST-TAKENAKA SYSTEM IN THE QUATERNIONIC HARDY SPACES

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ANALYSIS MATHEMATICA 2018年第1期44卷 99-114页

作者： Pap, M. Univ Pecs Fac Sci Ifjusag Ut 6 H-7634 Pecs Hungary

In this paper the slice regular analogue of the Malmquist-Takenaka system is introduced. It is proved that, under certain restrictions regarding to the parameters of the system, they form a complete orthonormal system in the quaternionic Hardy spaces of the unit ball. The properties of associated projection operator are also studied.

关键词： function of hypercomplex variables quaternionic Hardy space series expansion interpolation approximation by rational functions quaternionic Malmquist-Takenaka system

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Sharp Markov-type inequalities for rational functions on several intervals

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2016年第2期436卷 1017-1022页

作者： Akturk, M. A. Lukashov, A. Istanbul Univ Dept Engn Sci TR-34320 Istanbul Turkey Saratov NG Chernyshevskii State Univ Dept Mech & Math Saratov 410012 Russia

In this paper, we give sharp Rusak- and Markov-type inequalities for rational functions on several intervals when the system of intervals is a "rational function inverse image" of an interval and those funct... 详细信息

关键词： Inequalities in approximation approximation by rational functions

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approximation of Poincare's new functions by rational functions

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JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS 2013年第9期19卷 1452-1462页

作者： Nishioka, Seiji Yamagata Univ Dept Math Sci Yamagata 9908560 Japan

In this paper, we introduce a way to approximate meromorphic functions belonging to Poincare's new' class by rational functions. The class consists of meromorphic functions having a theorem de multiplication' with a suitable condition, which is a system of difference equations on the transform sending to , where . Since graphing rational functions is somewhat easy, we are able to know rough graphs of those new' functions by our result. Our method uses repeated substitutions of the difference equations.

关键词： Poincare's multiplication formula double-angle formula q-difference equation meromorphic function approximation by rational functions

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Discrete Orthogonality of the Malmquist Takenaka System of the Upper Half Plane and rational Interpolation

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2014年第1期20卷 1-16页

作者： Eisner, Timea Pap, Margit Univ Pecs H-7634 Pecs Hungary

In the representation of non periodic signals the use of special rational orthogonal systems is more efficient. One of these bases is the Malmquist-Takenaka system for the upper half plane. We will prove the discrete orthogonality of this system. Based on the discretization we introduce a new rational interpolation operator and we will study the properties of this operator. A finite sampling theorem for a special subset of non periodic analytic signals will be presented.

关键词： Hardy spaces Malmquist-Takenaka systems Discrete orthogonality approximation by rational functions rational interpolation Sampling

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New series expansions of the Gauss hypergeometric function

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2013年第2期39卷 349-365页

作者： Lopez, Jose L. Temme, Nico M. Univ Publ Navarra Dept Ingn Matemat & Informat Pamplona 31006 Spain CWI NL-1098 XG Amsterdam Netherlands

The Gauss hypergeometric function (2) F (1)(a,b,c;z) can be computed by using the power series in powers of . With these expansions, (2) F (1)(a,b,c;z) is not completely computable for all complex values of z. As pointed out in Gil et al. (2007, A 2.3), the points z = e (+/- i pi/3) are always excluded from the domains of convergence of these expansions. Buhring (SIAM J Math Anal 18:884-889, 1987) has given a power series expansion that allows computation at and near these points. But, when b -aEuro parts per thousand a is an integer, the coefficients of that expansion become indeterminate and its computation requires a nontrivial limiting process. Moreover, the convergence becomes slower and slower in that case. In this paper, we obtain new expansions of the Gauss hypergeometric function in terms of rational functions of z for which the points z = e (+/- i pi/3) are well inside their domains of convergence. In addition, these expansions are well defined when b -aEuro parts per thousand a is an integer and no limits are needed in that case. Numerical computations show that these expansions converge faster than Buhring's expansion for z in the neighborhood of the points e (+/- i pi/3), especially when b -aEuro parts per thousand a is close to an integer number.

关键词： Gauss hypergeometric function approximation by rational functions Two- and three-point Taylor expansions

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