In this article, we discuss one of the subsections of finite element method (FEM), classified as the Composite Finite Element Method, abbreviated as CFE. Dimensionality reduction is the primary benefit of the CFE method as it helps to reduce the complexity for the domain space. The degrees of freedom is more in FEM, while compared to the CFE method. Here, the semilinear parabolic problem in a 2D convex polygonal domain is considered. The analysis of the semidiscrete method for the problem is carried out initially in the CFE framework. Here, the discretization would be carried out for the space co-ordinate. Then, fully discrete problem is taken into account, where both the spatial and time components get discretized. In the fully discrete case, the backward Euler method and the Crank-Nicolson method in the CFE framework is adapted for the semilinear problem. The properties of convergence are derived and the error estimates are examined. It is verified that the order of convergence is preserved. The results obtained from the numerical computations are also provided.
{"title":"Adaptation of the composite finite element framework for semilinear parabolic problems","authors":"Anjaly Anand, T. Pramanick","doi":"10.33993/jnaat531-1392","DOIUrl":"https://doi.org/10.33993/jnaat531-1392","url":null,"abstract":"In this article, we discuss one of the subsections of finite element method (FEM), classified as the Composite Finite Element Method, abbreviated as CFE. Dimensionality reduction is the primary benefit of the CFE method as it helps to reduce the complexity for the domain space. The degrees of freedom is more in FEM, while compared to the CFE method. Here, the semilinear parabolic problem in a 2D convex polygonal domain is considered. The analysis of the semidiscrete method for the problem is carried out initially in the CFE framework. Here, the discretization would be carried out for the space co-ordinate. Then, fully discrete problem is taken into account, where both the spatial and time components get discretized. In the fully discrete case, the backward Euler method and the Crank-Nicolson method in the CFE framework is adapted for the semilinear problem. The properties of convergence are derived and the error estimates are examined. It is verified that the order of convergence is preserved. The results obtained from the numerical computations are also provided.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"26 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140696577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Our aim is to answer the following question: "Among the Filon-type methods for computing oscillatory integrals, which one is the most efficient in practice?". We first discuss why we should seek the answer among the family of Filon-Clenshaw-Curtis rules. A theoretical analysis accompanied by a set of numerical experiments reveals that the plain Filon-Clenshaw-Curtis rules reach a given accuracy faster than the (adaptive) extended Filon-Clenshaw-Curtis rules. The comparison is based on the CPU run-time for certain wave numbers (medium and large).
我们的目的是回答以下问题："在计算振荡积分的菲隆类方法中，哪种方法在实践中最有效？我们首先讨论为什么要在菲隆-克伦肖-柯蒂斯规则家族中寻找答案。理论分析和一组数值实验表明，普通菲隆-克伦肖-柯蒂斯规则比（自适应）扩展菲隆-克伦肖-柯蒂斯规则更快达到给定精度。比较基于某些波数（中、大）的 CPU 运行时间。
{"title":"A comparative study of Filon-type rules for oscillatory integrals","authors":"H. Majidian","doi":"10.33993/jnaat531-1380","DOIUrl":"https://doi.org/10.33993/jnaat531-1380","url":null,"abstract":"Our aim is to answer the following question: \"Among the Filon-type methods for computing oscillatory integrals, which one is the most efficient in practice?\". We first discuss why we should seek the answer among the family of Filon-Clenshaw-Curtis rules. A theoretical analysis accompanied by a set of numerical experiments reveals that the plain Filon-Clenshaw-Curtis rules reach a given accuracy faster than the (adaptive) extended Filon-Clenshaw-Curtis rules. The comparison is based on the CPU run-time for certain wave numbers (medium and large).","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"72 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140261268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The goal in this study is to present a unified local convergence analysis of frozen Steffensen-type methods under generalized Lipschitz-type conditions for Banach space valued operators. We also use our new idea of restricted convergence domains, where we find a more precise location, where the iterates lie leading to at least as tight majorizing functions. Consequently, the new convergence criteria are weaker than in earlier works resulting to the expansion of the applicability of these methods. The conditions do not necessarily imply the differentiability of the operator involved. This way our method is suitable for solving equations and systems of equations.
{"title":"Local convergence analysis of frozen Steffensen-type methods under generalized conditions","authors":"I. Argyros, S. George","doi":"10.33993/jnaat522-1160","DOIUrl":"https://doi.org/10.33993/jnaat522-1160","url":null,"abstract":"The goal in this study is to present a unified local convergence analysis of frozen Steffensen-type methods under generalized Lipschitz-type conditions for Banach space valued operators. We also use our new idea of restricted convergence domains, where we find a more precise location, where the iterates lie leading to at least as tight majorizing functions. Consequently, the new convergence criteria are weaker than in earlier works resulting to the expansion of the applicability of these methods. The conditions do not necessarily imply the differentiability of the operator involved. This way our method is suitable for solving equations and systems of equations.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"30 52","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139148215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend the Cheney-Sharma operators of the first kind using Stancu type technique and we study some approximation properties of the new operator. We calculate the moments, we study local approximation with respect to a K-functional and the preservation of the Lipschitz constant and order.
我们利用斯坦库型技术扩展了切尼-夏尔马第一类算子，并研究了新算子的一些近似性质。我们计算矩，研究 K 函数的局部逼近以及 Lipschitz 常量和阶的保留。
{"title":"An extension of the Cheney-Sharma operator of the first kind","authors":"Teodora Cătinaş, Iulia Buda","doi":"10.33993/jnaat522-1373","DOIUrl":"https://doi.org/10.33993/jnaat522-1373","url":null,"abstract":"We extend the Cheney-Sharma operators of the first kind using Stancu type technique and we study some approximation properties of the new operator. We calculate the moments, we study local approximation with respect to a K-functional and the preservation of the Lipschitz constant and order.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"2 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139151669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Our aim in this work is to extend the primal-dual interior point method based on a kernel function for linear fractional problem. We apply the techniques of kernel function-based interior point methods to solve a standard linear fractional program. By relying on the method of Charnes and Cooper [3], we transform the standard linear fractional problem into a linear program. This transformation will allow us to define the associated linear program and solve it efficiently using an appropriate kernel function. To show the efficiency of our approach, we apply our algorithm on the standard linear fractional programming found in numerical tests in the paper of A. Bennani et al. [4], we introduce the linear programming associated with this problem. We give three interior point conditions on this example, which depend on the dimension of the problem. We give the optimal solution for each linear program and each linear fractional program. We also obtain, using the new algorithm, the optimal solutions for the previous two problems. Moreover, some numerical results are illustrated to show the effectiveness of the method.
我们在这项工作中的目标是扩展基于核函数的初等二元内点法，以解决线性分数问题。我们将基于核函数的内点法技术应用于求解标准线性分数程序。依靠 Charnes 和 Cooper [3] 的方法，我们将标准线性分数问题转化为线性程序。通过这种转换，我们可以定义相关的线性程序，并使用适当的核函数高效地求解。为了展示我们方法的效率，我们将算法应用于 A. Bennani 等人的论文[4]中数值测试发现的标准线性分数程序，并介绍了与该问题相关的线性程序。我们给出了与问题维度相关的三个内点条件。我们给出了每个线性规划和每个线性分数规划的最优解。我们还利用新算法得到了前两个问题的最优解。此外，我们还通过一些数值结果来说明该方法的有效性。
{"title":"Extension of primal-dual interior point method based on a kernel function for linear fractional problem","authors":"Mousaab Bouafia, Adnan Yassine","doi":"10.33993/jnaat522-1349","DOIUrl":"https://doi.org/10.33993/jnaat522-1349","url":null,"abstract":"Our aim in this work is to extend the primal-dual interior point method based on a kernel function for linear fractional problem. We apply the techniques of kernel function-based interior point methods to solve a standard linear fractional program. By relying on the method of Charnes and Cooper [3], we transform the standard linear fractional problem into a linear program. This transformation will allow us to define the associated linear program and solve it efficiently using an appropriate kernel function. To show the efficiency of our approach, we apply our algorithm on the standard linear fractional programming found in numerical tests in the paper of A. Bennani et al. [4], we introduce the linear programming associated with this problem. We give three interior point conditions on this example, which depend on the dimension of the problem. We give the optimal solution for each linear program and each linear fractional program. We also obtain, using the new algorithm, the optimal solutions for the previous two problems. Moreover, some numerical results are illustrated to show the effectiveness of the method.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"6 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139148678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish a fuzzy Korovkin type approximation theorem by using (eq-stat^{D}_{CE})(deferred Ces'{a}ro and deferred Euler equi-statistical) convergence proposed by Saini et al. for continuous functions over ([a,b]). Further, we determine the rate of convergence via fuzzy modulus of continuity.
{"title":"Fuzzy Korovkin type Theorems via deferred Cesaro and deferred Euler equi-statistical convergence","authors":"Purshottam Agrawal, Behar Baxhaku","doi":"10.33993/jnaat522-1350","DOIUrl":"https://doi.org/10.33993/jnaat522-1350","url":null,"abstract":"We establish a fuzzy Korovkin type approximation theorem by using (eq-stat^{D}_{CE})(deferred Ces'{a}ro and deferred Euler equi-statistical) convergence proposed by Saini et al. for continuous functions over ([a,b]). Further, we determine the rate of convergence via fuzzy modulus of continuity.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"305 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139152357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Quantitative Korovkin-type theorems for approximation by bounded linear operators defined on (C(X,d)) are given, where ((X,d)) is a compact metric space. Special emphasis is on positive linear operators.As is known from previous work of Newman and Shapiro, Jimenez Pozo, Nishishiraho and the author, among others, there are two possible ways to obtain error estimates for bounded linear operator approximation: the so-called direct approach, and the smoothing technique.We give various generalizations and refinements of earlier results which were obtained by using both techniques. Furthermore, it will be shown that, in a certain sense, none of the two methods is superior to the other one.
{"title":"The rate of convergence of bounded linear processes on spaces of continuous functions","authors":"H. Gonska","doi":"10.33993/jnaat522-1326","DOIUrl":"https://doi.org/10.33993/jnaat522-1326","url":null,"abstract":"Quantitative Korovkin-type theorems for approximation by bounded linear operators defined on (C(X,d)) are given, where ((X,d)) is a compact metric space. Special emphasis is on positive linear operators.As is known from previous work of Newman and Shapiro, Jimenez Pozo, Nishishiraho and the author, among others, there are two possible ways to obtain error estimates for bounded linear operator approximation: the so-called direct approach, and the smoothing technique.We give various generalizations and refinements of earlier results which were obtained by using both techniques. Furthermore, it will be shown that, in a certain sense, none of the two methods is superior to the other one.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"241 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139152771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we aim to generalize an existing result by obtaining localized solutions within bounded convex sets, while also relaxing specific initial assumptions. To achieve this, we employ an iterative scheme that combines a fixed-point argument based on the Minty-Browder Theorem with a modified version of the Ekeland variational principle for bounded sets. An application to a system of second-order differential equations with Dirichlet boundary conditions is presented.
{"title":"Localization of Nash-type equilibria for systems with partial variational structure","authors":"Andrei Stan","doi":"10.33993/jnaat522-1356","DOIUrl":"https://doi.org/10.33993/jnaat522-1356","url":null,"abstract":"In this paper, we aim to generalize an existing result by obtaining localized solutions within bounded convex sets, while also relaxing specific initial assumptions. To achieve this, we employ an iterative scheme that combines a fixed-point argument based on the Minty-Browder Theorem with a modified version of the Ekeland variational principle for bounded sets. An application to a system of second-order differential equations with Dirichlet boundary conditions is presented.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"64 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139151859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct optimal nonlinear extrapolation estimates of (pi) based on random cyclic polygons generated from symmetric Dirichlet distributions. While the semiperimeter ( S_n ) and the area ( A_n ) of such random inscribed polygons and the semiperimeter (and area) ( S_n' ) of the corresponding random circumscribing polygons are known to converge to ( pi ) w.p.(1) and their distributions are also asymptotically normal as ( n to infty ), we study in this paper nonlinear extrapolations of the forms ( mathcal{W}_n = S_n^{alpha} A_n^{beta} S_n'^{, gamma} ) and ( mathcal{W}_n (p) = ( alpha S_n^p + beta A_n^p + gamma S_n'^{, p} )^{1/p} ) where ( alpha + beta + gamma = 1 ) and ( p neq 0 ). By deriving probabilistic asymptotic expansions with carefully controlled error estimates, we show that ( mathcal{W}_n ) and ( mathcal{W}_n (p) ) also converge to ( pi ) w.p.(1) and are asymptotically normal. Furthermore, to minimize the approximation error associated with ( mathcal{W}_n ) and ( mathcal{W}_n (p) ), the parameters must satisfy the optimality condition ( alpha + 4 beta - 2 gamma = 0 ). Our results generalize previous work on nonlinear extrapolations of ( pi ) which employ inscribed polygons only and the vertices are also assumed to be independently and uniformly distributed on the unit circle.
{"title":"Nonlinear random extrapolation estimates of (pi) under Dirichlet distributions","authors":"Shasha Wang, Zecheng Li, Wen-Qing Xu","doi":"10.33993/jnaat522-1360","DOIUrl":"https://doi.org/10.33993/jnaat522-1360","url":null,"abstract":"We construct optimal nonlinear extrapolation estimates of (pi) based on random cyclic polygons generated from symmetric Dirichlet distributions. While the semiperimeter ( S_n ) and the area ( A_n ) of such random inscribed polygons and the semiperimeter (and area) ( S_n' ) of the corresponding random circumscribing polygons are known to converge to ( pi ) w.p.(1) and their distributions are also asymptotically normal as ( n to infty ), we study in this paper nonlinear extrapolations of the forms ( mathcal{W}_n = S_n^{alpha} A_n^{beta} S_n'^{, gamma} ) and ( mathcal{W}_n (p) = ( alpha S_n^p + beta A_n^p + gamma S_n'^{, p} )^{1/p} ) where ( alpha + beta + gamma = 1 ) and ( p neq 0 ). By deriving probabilistic asymptotic expansions with carefully controlled error estimates, we show that ( mathcal{W}_n ) and ( mathcal{W}_n (p) ) also converge to ( pi ) w.p.(1) and are asymptotically normal. Furthermore, to minimize the approximation error associated with ( mathcal{W}_n ) and ( mathcal{W}_n (p) ), the parameters must satisfy the optimality condition ( alpha + 4 beta - 2 gamma = 0 ). Our results generalize previous work on nonlinear extrapolations of ( pi ) which employ inscribed polygons only and the vertices are also assumed to be independently and uniformly distributed on the unit circle.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139148826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, some new sufficient conditions for the unique solvability of a new class of Sylvester-like absolute value matrix equation (AXB - vert CXD vert =F) are given. This work is distinct from the published work by Li [Journal of Optimization Theory and Application, 195(2), 2022]. Some new conditions were also obtained, which were not covered by Li. We also provided an example in support of our result.
本文给出了一类新的西尔维斯特类绝对值矩阵方程 (AXB -vert CXD vert =F/)的唯一可解性的一些新的充分条件。这项工作有别于 Li [Journal of Optimization Theory and Application, 195(2), 2022] 已发表的工作。我们还得到了一些新的条件，这些条件是 Li 没有涉及到的。我们还提供了一个例子来支持我们的结果。
{"title":"New sufficient conditions for the solvability of a new class of Sylvester-like absolute value matrix equation","authors":"Shubham Kumar, Deepmala, Roshan Lal Keshtwal","doi":"10.33993/jnaat522-1321","DOIUrl":"https://doi.org/10.33993/jnaat522-1321","url":null,"abstract":"In this article, some new sufficient conditions for the unique solvability of a new class of Sylvester-like absolute value matrix equation (AXB - vert CXD vert =F) are given. This work is distinct from the published work by Li [Journal of Optimization Theory and Application, 195(2), 2022]. Some new conditions were also obtained, which were not covered by Li. We also provided an example in support of our result.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"11 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139152031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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