{"title":"求解Helmholtz方程的两种方法:广义Darboux变换和/spl部分/~-问题方法","authors":"E. Gutshabash","doi":"10.1109/DD.2003.238133","DOIUrl":null,"url":null,"abstract":"Two approaches to solution of the two-dimensional Helmholtz equation with a \"wave number\" are proposed. The results can be applied both in numerical areas of physics and in the theory of nonlinear equations. The first approach is based on the requirement of the covariance of equation under the generalized Darboux transformation (Moutard transformation). It allows to construct a new solution of equation, using a given initial solution of the equation. Simultaneously we obtain the \"dressing\" relation for the \"wave number\". The simplest examples of the approach are considered in detail. In the second approach the Green-Oauchy formula (the /spl part/~ -method) is applied to reduce the solution of the equation to the solution of a system of singular integral equations.","PeriodicalId":332604,"journal":{"name":"International Seminar Day on Diffraction, 2003. Proceedings.","volume":"15 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2003-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Two approaches for Helmholtz equation: generalized Darboux transformation and the method of /spl part/~-problem\",\"authors\":\"E. Gutshabash\",\"doi\":\"10.1109/DD.2003.238133\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Two approaches to solution of the two-dimensional Helmholtz equation with a \\\"wave number\\\" are proposed. The results can be applied both in numerical areas of physics and in the theory of nonlinear equations. The first approach is based on the requirement of the covariance of equation under the generalized Darboux transformation (Moutard transformation). It allows to construct a new solution of equation, using a given initial solution of the equation. Simultaneously we obtain the \\\"dressing\\\" relation for the \\\"wave number\\\". The simplest examples of the approach are considered in detail. In the second approach the Green-Oauchy formula (the /spl part/~ -method) is applied to reduce the solution of the equation to the solution of a system of singular integral equations.\",\"PeriodicalId\":332604,\"journal\":{\"name\":\"International Seminar Day on Diffraction, 2003. Proceedings.\",\"volume\":\"15 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2003-06-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Seminar Day on Diffraction, 2003. Proceedings.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/DD.2003.238133\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Seminar Day on Diffraction, 2003. Proceedings.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/DD.2003.238133","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Two approaches for Helmholtz equation: generalized Darboux transformation and the method of /spl part/~-problem
Two approaches to solution of the two-dimensional Helmholtz equation with a "wave number" are proposed. The results can be applied both in numerical areas of physics and in the theory of nonlinear equations. The first approach is based on the requirement of the covariance of equation under the generalized Darboux transformation (Moutard transformation). It allows to construct a new solution of equation, using a given initial solution of the equation. Simultaneously we obtain the "dressing" relation for the "wave number". The simplest examples of the approach are considered in detail. In the second approach the Green-Oauchy formula (the /spl part/~ -method) is applied to reduce the solution of the equation to the solution of a system of singular integral equations.
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