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Efficient numerical approaches with accelerated graphics processing unit (GPU) computations for Poisson problems and Cahn-Hilliard equations. 利用图形处理器（GPU）加速计算泊松问题和卡恩-希利亚德方程的高效数值方法。
IF 1.8 3区 数学
AIMS Mathematics Pub Date : 2024-01-01 Epub Date: 2024-09-23 DOI: 10.3934/math.20241334
Saulo Orizaga, Maurice Fabien, Michael Millard
{"title":"Efficient numerical approaches with accelerated graphics processing unit (GPU) computations for Poisson problems and Cahn-Hilliard equations.","authors":"Saulo Orizaga, Maurice Fabien, Michael Millard","doi":"10.3934/math.20241334","DOIUrl":"10.3934/math.20241334","url":null,"abstract":"<p><p>In this computational paper, we focused on the efficient numerical implementation of semi-implicit methods for models in materials science. In particular, we were interested in a class of nonlinear higher-order parabolic partial differential equations. The Cahn-Hilliard (CH) equation was chosen as a benchmark problem for our proposed methods. We first considered the Cahn-Hilliard equation with a convexity-splitting (CS) approach coupled with a backward Euler approximation of the time derivative and tested the performance against the bi-harmonic-modified (BHM) approach in terms of accuracy, order of convergence, and computation time. Higher-order time-stepping techniques that allow for the methods to increase their accuracy and order of convergence were then introduced. The proposed schemes in this paper were found to be very efficient for 2D computations. Computed dynamics in 2D and 3D are presented to demonstrate the energy-decreasing property and overall performance of the methods for longer simulation runs with a variety of initial conditions. In addition, we also present a simple yet powerful way to accelerate the computations by using MATLAB built-in commands to perform GPU implementations of the schemes. We show that it is possible to accelerate computations for the CH equation in 3D by a factor of 80, provided the hardware is capable enough.</p>","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"9 10","pages":"27471-27496"},"PeriodicalIF":1.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11466300/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142401649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fejér type inequalities for harmonically convex functions 调和凸函数的fejsamr型不等式
IF 2.2 3区 数学
AIMS Mathematics Pub Date : 2023-08-18 DOI: 10.3934/math.2022835
Muhammad Amer Latif
{"title":"Fejér type inequalities for harmonically convex functions","authors":"Muhammad Amer Latif","doi":"10.3934/math.2022835","DOIUrl":"https://doi.org/10.3934/math.2022835","url":null,"abstract":"In this study, some mappings related to the Fejér-type inequalities for harmonically convex functions are defined over $ left[ 0, 1right] $. Some Fejér-type inequalities for harmonically convex functions are proved using these mappings. Properties of these mappings are considered and consequently, refinements are obtained of some known results.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46984565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Angle in the space of $ p $-summable sequences $ p $-可和序列空间中的角
IF 2.2 3区 数学
AIMS Mathematics Pub Date : 2023-07-16 DOI: 10.3934/math.2022155
M. Nur, M. Bahri, A. Islamiyati, Harmanus Batkunde
{"title":"Angle in the space of $ p $-summable sequences","authors":"M. Nur, M. Bahri, A. Islamiyati, Harmanus Batkunde","doi":"10.3934/math.2022155","DOIUrl":"https://doi.org/10.3934/math.2022155","url":null,"abstract":"The aim of this paper is to investigate completness of $ A $ that equipped with usual norm on $ p $-summable sequences space where $ A $ is subspace in $ p $-summable sequences space and $ 1le p < infty $. We also introduce a new inner product on $ A $ and prove completness of $ A $ using a new norm that corresponds this new inner product. Moreover, we discuss the angle between two vectors and two subspaces in $ A $. In particular, we discuss the angle between $ 1 $-dimensional subspace and $ (s-1) $-dimensional subspace where $ sge 2 $ of $ A $.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48080725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Cohomologies of modified $ lambda $-differential Lie triple systems and applications 修正$ λ $-微分李三元系统的上同调及其应用
IF 2.2 3区 数学
AIMS Mathematics Pub Date : 2023-07-12 DOI: 10.3934/math.20231280
Wen Teng, Fengshan Long, Yu Zhang
{"title":"Cohomologies of modified $ lambda $-differential Lie triple systems and applications","authors":"Wen Teng, Fengshan Long, Yu Zhang","doi":"10.3934/math.20231280","DOIUrl":"https://doi.org/10.3934/math.20231280","url":null,"abstract":"In this paper, we introduce the concept and representation of modified $ lambda $-differential Lie triple systems. Next, we define the cohomology of modified $ lambda $-differential Lie triple systems with coefficients in a suitable representation. As applications of the proposed cohomology theory, we study 1-parameter formal deformations and abelian extensions of modified $ lambda $-differential Lie triple systems.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42483532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative 具有空间导数的双线性项的一维波动方程经典解的寿命
IF 2.2 3区 数学
AIMS Mathematics Pub Date : 2023-06-12 DOI: 10.3934/math.20231300
Takiko Sasaki, Shuhei Takamatsu, H. Takamura
{"title":"The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative","authors":"Takiko Sasaki, Shuhei Takamatsu, H. Takamura","doi":"10.3934/math.20231300","DOIUrl":"https://doi.org/10.3934/math.20231300","url":null,"abstract":"This paper is devoted to the lifespan estimates of small classical solutions of the initial value problems for one dimensional wave equations with semilinear terms of the spatial derivative of the unknown function. It is natural that the result is same as the one for semilinear terms of the time-derivative. But there are so many differences among their proofs. Moreover, it is meaningful to study this problem in the sense that it may help us to investigate its blow-up boundary in the near future.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47127415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Stability of stochastic dynamic systems of a random structure with Markov switching in the presence of concentration points 具有马尔可夫切换的随机结构动态系统在集中点下的稳定性
IF 2.2 3区 数学
AIMS Mathematics Pub Date : 2023-05-19 DOI: 10.3934/math.20231245
T. Lukashiv, I. Malyk, Maryna K. Chepeleva, P. Nazarov
{"title":"Stability of stochastic dynamic systems of a random structure with Markov switching in the presence of concentration points","authors":"T. Lukashiv, I. Malyk, Maryna K. Chepeleva, P. Nazarov","doi":"10.3934/math.20231245","DOIUrl":"https://doi.org/10.3934/math.20231245","url":null,"abstract":"This article aims to investigate sufficient conditions for the stability of the trivial solution of stochastic differential equations with a random structure, particularly in contexts involving the presence of concentration points. The proof of asymptotic stability leverages the use of Lyapunov functions, supplemented by additional constraints on the magnitudes of jumps and jump times, as well as the Markov property of the system solutions. The findings are elucidated with an example, demonstrating both stable and unstable conditions of the system. The novelty of this work is in the consideration of jump concentration points, which are not considered in classical works. The assumption of the existence of concentration points leads to additional constraints on jumps, jump times and relations between them.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45589800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Mixed radial-angular bounds for Hardy-type operators on Heisenberg group Heisenberg群上hardy型算子的混合径向-角界
IF 2.2 3区 数学
AIMS Mathematics Pub Date : 2023-04-24 DOI: 10.3934/math.20231070
Zhongci Hang, Xiang Li, D. Yan
{"title":"Mixed radial-angular bounds for Hardy-type operators on Heisenberg group","authors":"Zhongci Hang, Xiang Li, D. Yan","doi":"10.3934/math.20231070","DOIUrl":"https://doi.org/10.3934/math.20231070","url":null,"abstract":"In this paper, we study $ n $-dimensional Hardy operator and its dual in mixed radial-angular spaces on Heisenberg group and obtain their sharp bounds by using the rotation method. Furthermore, the sharp bounds of $ n $-dimensional weighted Hardy operator and weighted Cesàro operator are also obtained.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41319140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations 具有随机波动的时变SIS流行病模型的hamilton系统的精确解和叠加规则
IF 2.2 3区 数学
AIMS Mathematics Pub Date : 2023-04-18 DOI: 10.3934/math.20231225
R. Campoamor-Stursberg, Eduardo Fernández-Saiz, F. J. Herranz
{"title":"Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations","authors":"R. Campoamor-Stursberg, Eduardo Fernández-Saiz, F. J. Herranz","doi":"10.3934/math.20231225","DOIUrl":"https://doi.org/10.3934/math.20231225","url":null,"abstract":"Using the theory of Lie-Hamilton systems, formal generalized time-dependent Hamiltonian systems that extend a recently proposed SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above-mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamiltonian models based on the book and oscillator algebras, denoted by $ mathfrak{b}_2 $ and $ mathfrak{h}_4 $, respectively. The last generalization corresponds to an SIS system possessing the so-called two-photon algebra symmetry $ mathfrak{h}_6 $, according to the embedding chain $ mathfrak{b}_2subset mathfrak{h}_4subset mathfrak{h}_6 $, for which an exact solution cannot generally be found but a nonlinear superposition rule is explicitly given.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43783486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Further characterizations and representations of the Minkowski inverse in Minkowski space 闵可夫斯基逆在闵可夫斯基空间中的进一步表征与表示
IF 2.2 3区 数学
AIMS Mathematics Pub Date : 2023-03-24 DOI: 10.3934/math.20231189
Jiale Gao, Qingwen Wang, Kezheng Zuo, Jiabao Wu
{"title":"Further characterizations and representations of the Minkowski inverse in Minkowski space","authors":"Jiale Gao, Qingwen Wang, Kezheng Zuo, Jiabao Wu","doi":"10.3934/math.20231189","DOIUrl":"https://doi.org/10.3934/math.20231189","url":null,"abstract":"This paper serves to identify some new characterizations and representations of the Minkowski inverse in Minkowski space. First of all, a few representations of $ {1, 3^{mathfrak{m}}} $-, $ {1, 2, 3^{mathfrak{m}}} $-, $ {1, 4^{mathfrak{m}}} $- and $ {1, 2, 4^{mathfrak{m}}} $-inverses are given in order to represent the Minkowski inverse. Second, some famous characterizations of the Moore-Penrose inverse are extended to that of the Minkowski inverse. Third, using the Hartwig-Spindelböck decomposition, we present a representation of the Minkowski inverse. And, based on this result, an interesting characterization of the Minkowski inverse is showed by a rank equation. Finally, we obtain several new representations of the Minkowski inverse in a more general form, by which the Minkowski inverse of a class of block matrices is given.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43360831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some identities on degenerate hyperbolic functions arising from $ p $-adic integrals on $ mathbb{Z}_p $ $mathbb上$p$-dic积分引起的退化双曲函数的一些恒等式{Z}_p$
IF 2.2 3区 数学
AIMS Mathematics Pub Date : 2023-03-04 DOI: 10.3934/math.20231298
Taekyun Kim, Dae San Kim, H. Kim
{"title":"Some identities on degenerate hyperbolic functions arising from $ p $-adic integrals on $ mathbb{Z}_p $","authors":"Taekyun Kim, Dae San Kim, H. Kim","doi":"10.3934/math.20231298","DOIUrl":"https://doi.org/10.3934/math.20231298","url":null,"abstract":"The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic $ p $-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45478811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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