{"title":"不适定非齐次抛物方程的正则小波解","authors":"Jinru Wang, Yuan Zhou","doi":"10.1109/IWCFTA.2012.12","DOIUrl":null,"url":null,"abstract":"We consider the nonhomogeneous problem uxx(x, t) = ut(x, t) + f(x, t), 0 ≤ x <; 1, t ≥ 0, where the Cauchy data g(t) is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g(t) can produce a big alteration on its solution (if it exists). In this paper, we shall define a Meyer wavelet solution to obtain well-posed solution in the scaling space Vj. We shall also show that under certain conditions this regularized solution is convergent to the exact solution. In the previous papers, most of the theoretical results concerning the error estimate are about the homogeneous equation, i.e., f(x, t) ≡ 0.","PeriodicalId":354870,"journal":{"name":"2012 Fifth International Workshop on Chaos-fractals Theories and Applications","volume":"25 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2012-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Regularized Wavelet Solutions for Ill-posed Nonhomogeneous Parabolic Equations\",\"authors\":\"Jinru Wang, Yuan Zhou\",\"doi\":\"10.1109/IWCFTA.2012.12\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the nonhomogeneous problem uxx(x, t) = ut(x, t) + f(x, t), 0 ≤ x <; 1, t ≥ 0, where the Cauchy data g(t) is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g(t) can produce a big alteration on its solution (if it exists). In this paper, we shall define a Meyer wavelet solution to obtain well-posed solution in the scaling space Vj. We shall also show that under certain conditions this regularized solution is convergent to the exact solution. In the previous papers, most of the theoretical results concerning the error estimate are about the homogeneous equation, i.e., f(x, t) ≡ 0.\",\"PeriodicalId\":354870,\"journal\":{\"name\":\"2012 Fifth International Workshop on Chaos-fractals Theories and Applications\",\"volume\":\"25 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2012-10-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2012 Fifth International Workshop on Chaos-fractals Theories and Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/IWCFTA.2012.12\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2012 Fifth International Workshop on Chaos-fractals Theories and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/IWCFTA.2012.12","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Regularized Wavelet Solutions for Ill-posed Nonhomogeneous Parabolic Equations
We consider the nonhomogeneous problem uxx(x, t) = ut(x, t) + f(x, t), 0 ≤ x <; 1, t ≥ 0, where the Cauchy data g(t) is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g(t) can produce a big alteration on its solution (if it exists). In this paper, we shall define a Meyer wavelet solution to obtain well-posed solution in the scaling space Vj. We shall also show that under certain conditions this regularized solution is convergent to the exact solution. In the previous papers, most of the theoretical results concerning the error estimate are about the homogeneous equation, i.e., f(x, t) ≡ 0.