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Dunkl-Weinstein Multiplier Operators and Applications to reproducing kernel theory

MEDITERRANEAN JOURNAL OF MATHEMATICS 2024年第3期21卷 80-80页

作者： Soltani, Fethi Maktouf, Ibrahim Univ Tunis El Manar Fac Sci Tunis Lab Anal Math & Applicat LR11ES11 Tunis 2092 Tunisia Univ Carthage Ecole Natl Ingenieurs Carthage Tunis 2035 Tunisia

In this paper, we have interested on the Dunkl-Weinstein multiplier operators Mk,beta,a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {M}_{k,\beta ,a}$$\end{document} on the Dunkl-Weinstein-type Paley-Wiener space Hs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {H}_s$$\end{document}. We give for these operators an application to the reproducing kernel theory;and we deduce for them best approximation on the Paley-Wiener space Hs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {H}_s$$\end{document}. Finally, we give two applications in two particular cases, the first is a Weinstein case (k=0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k=0)$$\end{document}, and the second is the Dunkl case when G=Z2d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=\mathbb {Z}_2<^>d$$\end{document}. More precisely, we write the solution of the Tikhonov regularization problem in these cases as a convolution product of two functions.

关键词： Dunkl-Weinstein multiplier operators Dunkl-Weinstein-type Paley-Wiener space reproducing kernel theory

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NUMERICAL SOLUTIONS OF RIESZ FRACTIONAL DIFFUSION AND ADVECTION-DISPERSION EQUATIONS IN POROUS MEDIA USING ITERATIVE reproducing kernel ALGORITHM

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JOURNAL OF POROUS MEDIA 2020年第8期23卷 783-804页

作者： Abu Arqub, Omar Al-Smadi, Mohammed Al Balqa Appl Univ Dept Math Fac Sci Salt 19117 Jordan Al Balqa Appl Univ Ajloun Coll Dept Appl Sci Ajloun 26816 Jordan

This paper presents an iterative reproducing kernel algorithm for obtaining the numerical solutions of Riesz fractional diffusion and advection-dispersion equations in porous media on a finite domain. The representation of the exact and the numerical solutions is given in the W (Omega) and H (Omega) inner product spaces. The computation of the required grid points relies on the R-(y,R-s) (x, t) and r((y,s)) (x,t) reproducing kernel functions. An efficient construction is given to obtain the numerical solution together with an existence proof of the exact solution based upon the reproducing kernel theory. Numerical solution of such Riesz fractional equations is acquired by interrupting the n-term of the exact solution. In this approach, numerical examples were analyzed to illustrate the design procedure and confirm the performance of the proposed algorithm in the form of tabulated data, numerical comparisons, and graphical results. Finally, the utilized results show the significant improvement of the algorithm while saving the convergence accuracy and time.

关键词： reproducing kernel theory Riesz fractional derivative diffusion equation advection-dispersion equation porous media

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Simplified iterative reproducing kernel method for handling time-fractional BVPs with error estimation

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AIN SHAMS ENGINEERING JOURNAL 2018年第4期9卷 2517-2525页

作者： Al-Smadi, Mohammed Al Balqa Appl Univ Ajloun Coll Dept Appl Sci Ajloun 26816 Jordan

In this article, we introduce a novel numerical scheme, the iterative reproducing kernel method (IRKM), for providing numerical approximate solutions of a certain class of time-fractional boundary value problem within favorable aspects of the reproducing kernel Hilbert space in Caputo sense. The algorithm methodology is based on generating an orthonormal basis from the reproducing kernel function to formulate the solution in form of uniformly convergent series, accordance with the constraint conditions in the space omega(5)[0,1]. Error estimates in the omega(5)-norm are obtained as well as numerical experiments are described to represent the hypothesis, and to confirm the design procedure of the proposed algorithm. The numerical results indicate that the IRKA is a significant development tool for handling such issues arising in computer, physics and engineering fields. (C) 2017 Ain Shams University.

关键词： Fractional differential equations reproducing kernel theory Inner product spaces Error estimation and error bound Numerical approximation

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Numerical solution of nonlinear Volterra integro-differential equations of fractional order by the reproducing kernel method

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APPLIED MATHEMATICAL MODELLING 2015年第16期39卷 4871-4876页

作者： Jiang, Wei Tian, Tian Harbin Inst Technol Dept Math Weihai 264209 Shandong Peoples R China

Fractional calculus is a extension of derivatives and integrals to non-integer orders. It has been used widely to model scientific and engineering problems. In this article, the reproducing kernel theory is applied to solve a kind of nonlinear fractional order Volterra integro-differential equation. The fraction derivatives are described in Caputo sense. In order to solve this kind of equation, we discuss and derive the approximate solution in the form of series with easily computable terms in the reproducing kernel space, by introducing a simple algorithm to implement this process. Some numerical examples are given to demonstrate the validity and applicability of the technique. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Fractional order Volterra integro-differential equation reproducing kernel theory

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A NEW NUMERICAL METHOD FOR SOLVING TWO-DIMENSIONAL VARIABLE-ORDER ANOMALOUS SUB-DIFFUSION EQUATION

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THERMAL SCIENCE 2016年第Sup3期20卷 S701-S710页

作者： Jiang, Wei Guo, Beibei Harbin Inst Technol Weihai Dept Math Weihai Shandong Peoples R China

The novelty and innovativeness of this paper are the combination of reproducing kernel theory and spline, this leads to a new simple but effective numerical method for solving variable-order anomalous sub-diffusion equation successfully. This combination overcomes the weaknesses of piecewise polynomials that can not be used to solve differential equations directly because of lack of the smoothness. Moreover, new bases of reproducing kernel spaces are constructed. On the other hand, the existence of any epsilon-approximate solution is proved and an effective method for obtaining the epsilon-approximate solution is established. A numerical example is given to show the accuracy and effectiveness of theoretical results.

关键词： variable-order anomalous sub-dffusion equation spline reproducing kernel theory approximate solution

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A splitting iterative method for solving second kind integral equations in reproducing kernel spaces

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017年 326卷 204-216页

作者： Babolian, Esmail Hamedzadeh, Danial Islamic Azad Univ Dept Math Sci & Res Branch Tehran Iran

In the present paper, we propose a new iterative method to solve integral equations of the second kind in reproducing kernel Hilbert spaces (RKHS). At first, we make appropriate splitting in second kind integral equations and according to this splitting the iterative method will be constructed;then, bases of RKHS and reproducing kernel spaces properties are used to convert this problem to linear system of equations. We move between reproducing kernel spaces by changing bases in order to achieve more accurate approximate solutions. Classically, in iterative RKHS method, the number of iterations should be the same as the number of points;here, we present a type of iterative RKHS method without this limitation. Convergence of the proposed method is investigated, and the efficiency of the method is demonstrated through various examples. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Iterative methods Integral equation of second kind reproducing kernel theory reproducing kernel space

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An approximate solution for a neutral functional-differential equation with proportional delays

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APPLIED MATHEMATICS AND COMPUTATION 2015年 260卷 27-34页

作者： Cheng, Xue Chen, Zhong Zhang, Qingpu Harbin Inst Technol Harbin 150001 Peoples R China Harbin Inst Technol Weihai Weihai 264209 Shandong Peoples R China

In this paper, a novel algorithm based on reproducing kernel theory for neutral functional-differential equation with proportional delays is proposed. The advantages of the presented method are the establishment of complete e-approximate solution theory and high precision of absolute error. Some examples are given to demonstrate the validity and applicability of the new method and some comparisons are made with the existing results. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Neutral functional-differential equation Proportional delay epsilon-approximate solution reproducing kernel theory

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Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems

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SOFT COMPUTING 2017年第23期21卷 7191-7206页

作者： Abu Arqub, Omar Al-Smadi, Mohammed Momani, Shaher Hayat, Tasawar Al Balqa Appl Univ Fac Sci Dept Math Salt 19117 Jordan Al Balqa Appl Univ Ajloun Coll Dept Appl Sci Ajloun 26816 Jordan Univ Jordan Fac Sci Dept Math Amman 11942 Jordan Quaid I Azam Univ Fac Sci Dept Math Islamabad 45320 Pakistan

In this paper, we investigate the analytic and approximate solutions of second-order, two-point fuzzy boundary value problems based on the reproducing kernel theory under the assumption of strongly generalized differentiability. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their r-cut representation in the space . An efficient computational algorithm is provided to guarantee the procedure and to confirm the performance of the proposed method. Results of numerical experiments are provided to illustrate the theoretical statements in order to show potentiality, generality, and superiority of our algorithm for solving such fuzzy equations. Graphical results, tabulated data, and numerical comparisons are presented and discussed quantitatively to illustrate the possible fuzzy solutions.

关键词： Fuzzy boundary value problems reproducing kernel theory Strongly generalized differentiability Gram-Schmidt process

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The RKHS method for numerical treatment for integrodifferential algebraic systems of temporal two-point BVPs

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NEURAL COMPUTING & APPLICATIONS 2018年第8期30卷 2595-2606页

作者： Abu Arqub, Omar Rashaideh, Hasan Al Balqa Appl Univ Dept Math Fac Sci Salt 19117 Jordan Al Balqa Appl Univ PABG Fac Informat Technol Dept Comp Sci Salt 19117 Jordan

Many problems arising in different fields of sciences and engineering can be reduced, by applying some appropriate discretization, either to a system of integrodifferential algebraic equations or to a sequence of such systems. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of integrodifferential algebraic systems of temporal two-point boundary value problems. Two extended inner product spaces W[0,1] and H[0,1] are constructed in which the boundary conditions of the systems are satisfied, while two smooth kernel functions R-t(s) and r(t)(s) are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed.

关键词： Integrodifferential algebraic systems Temporal boundary value problems reproducing kernel theory Volterra operator

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Solving a system of linear Volterra integral equations using the new reproducing kernel method

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APPLIED MATHEMATICS AND COMPUTATION 2013年第20期219卷 10225-10230页

作者： Jiang, Wei Chen, Zhong Harbin Inst Technol Weihai Dept Math Weihai 264209 Shandong Peoples R China

In this paper, we present a new numerical technique to obtain the approximation solution for linear Volterra integral equations of the second kind based on reproducing kernel theory. The approximation solution is expressed by n-term summation of reproducing kernel functions. The merit of the new method includes (1) it is easy to implement this method;(2) high accuracy. The numerical examples compared with other methods show that the new method is more efficient. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Volterra integral equations of the second kind reproducing kernel theory reproducing kernel function Approximate solutions

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