Zineb Laouar, Nouria Arar, Abdellatif Ben Makhlouf
{"title":"基于第三类Chebyshev多项式的Volterra-Fredholm分数阶积分微分方程的理论与数值研究","authors":"Zineb Laouar, Nouria Arar, Abdellatif Ben Makhlouf","doi":"10.1155/2023/6401067","DOIUrl":null,"url":null,"abstract":"<div>\n <p>In this paper, we develop an efficient numerical method to approximate the solution of fractional integro-differential equations (FI-DEs) of mixed Volterra−Fredholm type using spectral collocation method with shifted Chebyshev polynomials of the third kind (S-Cheb-3). The fractional derivative is described in the Caputo sense. A Chebyshev−Gauss quadrature is involved to evaluate integrals for more precision. Two types of equations are studied to obtain algebraic systems solvable using the Gauss elimination method for linear equations and the Newton algorithm for nonlinear ones. In addition, an error analysis is carried out. Six numerical examples are evaluated using different error values (maximum absolute error, root mean square error, and relative error) to compare the approximate and the exact solutions of each example. The experimental rate of convergence is calculated as well. The results validate the numerical approach’s efficiency, applicability, and performance.</p>\n </div>","PeriodicalId":50653,"journal":{"name":"Complexity","volume":"2023 1","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1155/2023/6401067","citationCount":"0","resultStr":"{\"title\":\"Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind\",\"authors\":\"Zineb Laouar, Nouria Arar, Abdellatif Ben Makhlouf\",\"doi\":\"10.1155/2023/6401067\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>\\n <p>In this paper, we develop an efficient numerical method to approximate the solution of fractional integro-differential equations (FI-DEs) of mixed Volterra−Fredholm type using spectral collocation method with shifted Chebyshev polynomials of the third kind (S-Cheb-3). The fractional derivative is described in the Caputo sense. A Chebyshev−Gauss quadrature is involved to evaluate integrals for more precision. Two types of equations are studied to obtain algebraic systems solvable using the Gauss elimination method for linear equations and the Newton algorithm for nonlinear ones. In addition, an error analysis is carried out. Six numerical examples are evaluated using different error values (maximum absolute error, root mean square error, and relative error) to compare the approximate and the exact solutions of each example. The experimental rate of convergence is calculated as well. The results validate the numerical approach’s efficiency, applicability, and performance.</p>\\n </div>\",\"PeriodicalId\":50653,\"journal\":{\"name\":\"Complexity\",\"volume\":\"2023 1\",\"pages\":\"\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2023-04-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1155/2023/6401067\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Complexity\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1155/2023/6401067\",\"RegionNum\":4,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Complexity","FirstCategoryId":"5","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1155/2023/6401067","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind
In this paper, we develop an efficient numerical method to approximate the solution of fractional integro-differential equations (FI-DEs) of mixed Volterra−Fredholm type using spectral collocation method with shifted Chebyshev polynomials of the third kind (S-Cheb-3). The fractional derivative is described in the Caputo sense. A Chebyshev−Gauss quadrature is involved to evaluate integrals for more precision. Two types of equations are studied to obtain algebraic systems solvable using the Gauss elimination method for linear equations and the Newton algorithm for nonlinear ones. In addition, an error analysis is carried out. Six numerical examples are evaluated using different error values (maximum absolute error, root mean square error, and relative error) to compare the approximate and the exact solutions of each example. The experimental rate of convergence is calculated as well. The results validate the numerical approach’s efficiency, applicability, and performance.
期刊介绍:
Complexity is a cross-disciplinary journal focusing on the rapidly expanding science of complex adaptive systems. The purpose of the journal is to advance the science of complexity. Articles may deal with such methodological themes as chaos, genetic algorithms, cellular automata, neural networks, and evolutionary game theory. Papers treating applications in any area of natural science or human endeavor are welcome, and especially encouraged are papers integrating conceptual themes and applications that cross traditional disciplinary boundaries. Complexity is not meant to serve as a forum for speculation and vague analogies between words like “chaos,” “self-organization,” and “emergence” that are often used in completely different ways in science and in daily life.