{"title":"用准线性化和 Legendre-collocation 方法数值求解非线性 cordial Volterra 积分方程的广义形式","authors":"","doi":"10.1016/j.apnum.2024.09.013","DOIUrl":null,"url":null,"abstract":"<div><p>In this article, we propose a numerical method for a general form of nonlinear cordial Volterra integral equations. We discuss conditions that under them the problem has solutions. Since the existence of solutions for the problem depends on the solvability of a scalar equation and also a linear form of the problem, then we employ quasilinearization technique in which solving a nonlinear problem is reduced to solve a sequence of linear equations. The existence of solutions of linear equations and their quadratically convergence to the solutions of the nonlinear problem is considered. For the numerical solution of the produced linear equations we apply Legendre-collocation method along with a regularization technique for the quadrature formulas. We discuss the error analysis of the collocation method considering that the cordial Volterra integral operators are noncompact. To test the efficiency and accuracy of the proposed method, the solution of different cases of numerical examples are reported.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Numerical solution for a generalized form of nonlinear cordial Volterra integral equations using quasilinearization and Legendre-collocation methods\",\"authors\":\"\",\"doi\":\"10.1016/j.apnum.2024.09.013\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this article, we propose a numerical method for a general form of nonlinear cordial Volterra integral equations. We discuss conditions that under them the problem has solutions. Since the existence of solutions for the problem depends on the solvability of a scalar equation and also a linear form of the problem, then we employ quasilinearization technique in which solving a nonlinear problem is reduced to solve a sequence of linear equations. The existence of solutions of linear equations and their quadratically convergence to the solutions of the nonlinear problem is considered. For the numerical solution of the produced linear equations we apply Legendre-collocation method along with a regularization technique for the quadrature formulas. We discuss the error analysis of the collocation method considering that the cordial Volterra integral operators are noncompact. To test the efficiency and accuracy of the proposed method, the solution of different cases of numerical examples are reported.</p></div>\",\"PeriodicalId\":8199,\"journal\":{\"name\":\"Applied Numerical Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Numerical Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0168927424002496\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Numerical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168927424002496","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Numerical solution for a generalized form of nonlinear cordial Volterra integral equations using quasilinearization and Legendre-collocation methods
In this article, we propose a numerical method for a general form of nonlinear cordial Volterra integral equations. We discuss conditions that under them the problem has solutions. Since the existence of solutions for the problem depends on the solvability of a scalar equation and also a linear form of the problem, then we employ quasilinearization technique in which solving a nonlinear problem is reduced to solve a sequence of linear equations. The existence of solutions of linear equations and their quadratically convergence to the solutions of the nonlinear problem is considered. For the numerical solution of the produced linear equations we apply Legendre-collocation method along with a regularization technique for the quadrature formulas. We discuss the error analysis of the collocation method considering that the cordial Volterra integral operators are noncompact. To test the efficiency and accuracy of the proposed method, the solution of different cases of numerical examples are reported.
期刊介绍:
The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are:
(i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments.
(ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers.
(iii) Short notes, which present specific new results and techniques in a brief communication.