Mohd Kashif , Manpal Singh , Tanmoy Som , Eduard-Marius Craciun
{"title":"使用运算矩阵和配位法对化学过程中出现的变阶模型进行数值研究","authors":"Mohd Kashif , Manpal Singh , Tanmoy Som , Eduard-Marius Craciun","doi":"10.1016/j.jocs.2024.102339","DOIUrl":null,"url":null,"abstract":"<div><p>This article introduces the fractional variable order (VO) Gray–Scott model using the notion of VO fractional derivative in the Caputo sense. An efficient numerical method has been designed based on the Vieta–Lucas polynomial and the spectral collocation method for solving this model. The designed technique converts the concerned model into a nonlinear algebraic system of equations, which can be solved by Newton’s iterative method. In this article, we have illustrated the convergence analysis of the approximation and shown that a high order of convergence can be achieved despite a smaller number of approximations. A few numerical results are presented in order to verify the reliability and accuracy of the demonstrated scheme. The results of absolute errors for the considered Gray–Scott model with its exact solution show that the technique is very suitable for finding the solutions to the said kind of complex physical problem.</p></div>","PeriodicalId":48907,"journal":{"name":"Journal of Computational Science","volume":null,"pages":null},"PeriodicalIF":3.1000,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Numerical study of variable order model arising in chemical processes using operational matrix and collocation method\",\"authors\":\"Mohd Kashif , Manpal Singh , Tanmoy Som , Eduard-Marius Craciun\",\"doi\":\"10.1016/j.jocs.2024.102339\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This article introduces the fractional variable order (VO) Gray–Scott model using the notion of VO fractional derivative in the Caputo sense. An efficient numerical method has been designed based on the Vieta–Lucas polynomial and the spectral collocation method for solving this model. The designed technique converts the concerned model into a nonlinear algebraic system of equations, which can be solved by Newton’s iterative method. In this article, we have illustrated the convergence analysis of the approximation and shown that a high order of convergence can be achieved despite a smaller number of approximations. A few numerical results are presented in order to verify the reliability and accuracy of the demonstrated scheme. The results of absolute errors for the considered Gray–Scott model with its exact solution show that the technique is very suitable for finding the solutions to the said kind of complex physical problem.</p></div>\",\"PeriodicalId\":48907,\"journal\":{\"name\":\"Journal of Computational Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.1000,\"publicationDate\":\"2024-05-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Computational Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1877750324001327\",\"RegionNum\":3,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1877750324001327","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
摘要
本文利用 Caputo 意义上的 VO 分数导数概念,介绍了分数变阶 (VO) Gray-Scott 模型。基于 Vieta-Lucas 多项式和谱配位法,设计了一种高效的数值方法来求解该模型。所设计的技术将相关模型转换为非线性代数方程系,并可通过牛顿迭代法求解。在本文中,我们阐述了近似的收敛性分析,并表明尽管近似次数较少,但仍可实现高阶收敛。为了验证所演示方案的可靠性和准确性,我们给出了一些数值结果。对所考虑的格雷-斯科特模型及其精确解的绝对误差结果表明,该技术非常适合用于寻找上述复杂物理问题的解。
Numerical study of variable order model arising in chemical processes using operational matrix and collocation method
This article introduces the fractional variable order (VO) Gray–Scott model using the notion of VO fractional derivative in the Caputo sense. An efficient numerical method has been designed based on the Vieta–Lucas polynomial and the spectral collocation method for solving this model. The designed technique converts the concerned model into a nonlinear algebraic system of equations, which can be solved by Newton’s iterative method. In this article, we have illustrated the convergence analysis of the approximation and shown that a high order of convergence can be achieved despite a smaller number of approximations. A few numerical results are presented in order to verify the reliability and accuracy of the demonstrated scheme. The results of absolute errors for the considered Gray–Scott model with its exact solution show that the technique is very suitable for finding the solutions to the said kind of complex physical problem.
期刊介绍:
Computational Science is a rapidly growing multi- and interdisciplinary field that uses advanced computing and data analysis to understand and solve complex problems. It has reached a level of predictive capability that now firmly complements the traditional pillars of experimentation and theory.
The recent advances in experimental techniques such as detectors, on-line sensor networks and high-resolution imaging techniques, have opened up new windows into physical and biological processes at many levels of detail. The resulting data explosion allows for detailed data driven modeling and simulation.
This new discipline in science combines computational thinking, modern computational methods, devices and collateral technologies to address problems far beyond the scope of traditional numerical methods.
Computational science typically unifies three distinct elements:
• Modeling, Algorithms and Simulations (e.g. numerical and non-numerical, discrete and continuous);
• Software developed to solve science (e.g., biological, physical, and social), engineering, medicine, and humanities problems;
• Computer and information science that develops and optimizes the advanced system hardware, software, networking, and data management components (e.g. problem solving environments).