Wondwosen Gebeyaw Melesse, A. Tiruneh, G. A. Derese
{"title":"用初值法求解线性二阶奇摄动微分差分方程","authors":"Wondwosen Gebeyaw Melesse, A. Tiruneh, G. A. Derese","doi":"10.1155/2019/5259130","DOIUrl":null,"url":null,"abstract":"In this paper, an initial value method for solving a class of linear second-order singularly perturbed differential difference equation containing mixed shifts is proposed. In doing so, first, the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay and advance parameters using Taylor series expansion. From the modified problem, two explicit initial value problems which are independent of the perturbation parameter are produced; namely, the reduced problem and the boundary layer correction problem. These problems are then solved analytically and/or numerically, and those solutions are combined to give an approximate solution to the original problem. An error estimate for this method is derived using maximum norm. Several test problems are considered to illustrate the theoretical results. It is observed that the present method approximates the exact solution very well.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":"2019 1","pages":"1-10"},"PeriodicalIF":1.4000,"publicationDate":"2019-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2019/5259130","citationCount":"10","resultStr":"{\"title\":\"Solving Linear Second-Order Singularly Perturbed Differential Difference Equations via Initial Value Method\",\"authors\":\"Wondwosen Gebeyaw Melesse, A. Tiruneh, G. A. Derese\",\"doi\":\"10.1155/2019/5259130\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, an initial value method for solving a class of linear second-order singularly perturbed differential difference equation containing mixed shifts is proposed. In doing so, first, the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay and advance parameters using Taylor series expansion. From the modified problem, two explicit initial value problems which are independent of the perturbation parameter are produced; namely, the reduced problem and the boundary layer correction problem. These problems are then solved analytically and/or numerically, and those solutions are combined to give an approximate solution to the original problem. An error estimate for this method is derived using maximum norm. Several test problems are considered to illustrate the theoretical results. It is observed that the present method approximates the exact solution very well.\",\"PeriodicalId\":55967,\"journal\":{\"name\":\"International Journal of Differential Equations\",\"volume\":\"2019 1\",\"pages\":\"1-10\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2019-11-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1155/2019/5259130\",\"citationCount\":\"10\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Journal of Differential Equations\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1155/2019/5259130\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Differential Equations","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2019/5259130","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Solving Linear Second-Order Singularly Perturbed Differential Difference Equations via Initial Value Method
In this paper, an initial value method for solving a class of linear second-order singularly perturbed differential difference equation containing mixed shifts is proposed. In doing so, first, the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay and advance parameters using Taylor series expansion. From the modified problem, two explicit initial value problems which are independent of the perturbation parameter are produced; namely, the reduced problem and the boundary layer correction problem. These problems are then solved analytically and/or numerically, and those solutions are combined to give an approximate solution to the original problem. An error estimate for this method is derived using maximum norm. Several test problems are considered to illustrate the theoretical results. It is observed that the present method approximates the exact solution very well.