Z. M. Alaofi, Talaat Sayed Ali, F. A. Alaal, S. Dragomir
{"title":"求解三阶色散偏微分方程的四次非多项式样条","authors":"Z. M. Alaofi, Talaat Sayed Ali, F. A. Alaal, S. Dragomir","doi":"10.4236/ajcm.2021.113013","DOIUrl":null,"url":null,"abstract":"In the present paper, we introduce a non-polynomial quadratic spline method for solving third-order boundary value problems. Third-order singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor theory, Newtonian fluid mechanics, optimal control, convection-diffusion proc-esses, hydrodynamics, aerodynamics, etc. These problems have various im-portant applications in fluid dynamics. The procedure involves a reduction of a third-order partial differential equation to a first-order ordinary differential equation. Truncation errors are given. The unconditional stability of the method is analysed by the Von-Neumann stability analysis. The developed method is tested with an illustrated example, and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, a graphical comparison between analytical and approximate solutions is also shown for the illustrated example.","PeriodicalId":64456,"journal":{"name":"美国计算数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Quartic Non-Polynomial Spline for Solving the Third-Order Dispersive Partial Differential Equation\",\"authors\":\"Z. M. Alaofi, Talaat Sayed Ali, F. A. Alaal, S. Dragomir\",\"doi\":\"10.4236/ajcm.2021.113013\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In the present paper, we introduce a non-polynomial quadratic spline method for solving third-order boundary value problems. Third-order singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor theory, Newtonian fluid mechanics, optimal control, convection-diffusion proc-esses, hydrodynamics, aerodynamics, etc. These problems have various im-portant applications in fluid dynamics. The procedure involves a reduction of a third-order partial differential equation to a first-order ordinary differential equation. Truncation errors are given. The unconditional stability of the method is analysed by the Von-Neumann stability analysis. The developed method is tested with an illustrated example, and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, a graphical comparison between analytical and approximate solutions is also shown for the illustrated example.\",\"PeriodicalId\":64456,\"journal\":{\"name\":\"美国计算数学期刊(英文)\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"美国计算数学期刊(英文)\",\"FirstCategoryId\":\"1089\",\"ListUrlMain\":\"https://doi.org/10.4236/ajcm.2021.113013\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"美国计算数学期刊(英文)","FirstCategoryId":"1089","ListUrlMain":"https://doi.org/10.4236/ajcm.2021.113013","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Quartic Non-Polynomial Spline for Solving the Third-Order Dispersive Partial Differential Equation
In the present paper, we introduce a non-polynomial quadratic spline method for solving third-order boundary value problems. Third-order singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor theory, Newtonian fluid mechanics, optimal control, convection-diffusion proc-esses, hydrodynamics, aerodynamics, etc. These problems have various im-portant applications in fluid dynamics. The procedure involves a reduction of a third-order partial differential equation to a first-order ordinary differential equation. Truncation errors are given. The unconditional stability of the method is analysed by the Von-Neumann stability analysis. The developed method is tested with an illustrated example, and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, a graphical comparison between analytical and approximate solutions is also shown for the illustrated example.