{"title":"一类随机非线性分数阶微分方程的边值问题","authors":"M. Omaba, L. Omenyi","doi":"10.30538/PSRP-OMA2020.0074","DOIUrl":null,"url":null,"abstract":"Consider a class of two-point Boundary Value Problems (BVP) for a stochastic nonlinear fractional order differential equation \\(D^\\alpha u(t)=\\lambda\\sqrt{I^\\beta[\\sigma^2(t,u(t))]}\\dot{w}(t)\\ ,0< t< 1\\) with boundary conditions \\(u(0)=0,\\,\\,u'(0)=u'(1)=0,\\) where \\(\\lambda>0\\) is a level of the noise term, \\(\\sigma:[0,1]\\times\\mathbb{R}\\rightarrow\\mathbb{R}\\) is continuous, \\(\\dot{w}(t)\\) is a generalized derivative of Wiener process (Gaussian white noise), \\(D^\\alpha\\) is the Riemann-Liouville fractional differential operator of order \\(\\alpha\\in (3,4)\\) and \\(I^\\beta,\\,\\,\\beta>0\\) is a fractional integral operator. We formulate the solution of the equation via a stochastic Volterra-type equation and investigate its existence and uniqueness under some precise linearity conditions using contraction fixed point theorem. A case of the above BVP for a stochastic nonlinear second order differential equation for \\(\\alpha=2\\) and \\(\\beta=0\\) with \\(u(0)=u(1)=0\\) is also studied.","PeriodicalId":52741,"journal":{"name":"Open Journal of Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Boundary value problems for a class of stochastic nonlinear fractional order differential equations\",\"authors\":\"M. Omaba, L. Omenyi\",\"doi\":\"10.30538/PSRP-OMA2020.0074\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Consider a class of two-point Boundary Value Problems (BVP) for a stochastic nonlinear fractional order differential equation \\\\(D^\\\\alpha u(t)=\\\\lambda\\\\sqrt{I^\\\\beta[\\\\sigma^2(t,u(t))]}\\\\dot{w}(t)\\\\ ,0< t< 1\\\\) with boundary conditions \\\\(u(0)=0,\\\\,\\\\,u'(0)=u'(1)=0,\\\\) where \\\\(\\\\lambda>0\\\\) is a level of the noise term, \\\\(\\\\sigma:[0,1]\\\\times\\\\mathbb{R}\\\\rightarrow\\\\mathbb{R}\\\\) is continuous, \\\\(\\\\dot{w}(t)\\\\) is a generalized derivative of Wiener process (Gaussian white noise), \\\\(D^\\\\alpha\\\\) is the Riemann-Liouville fractional differential operator of order \\\\(\\\\alpha\\\\in (3,4)\\\\) and \\\\(I^\\\\beta,\\\\,\\\\,\\\\beta>0\\\\) is a fractional integral operator. We formulate the solution of the equation via a stochastic Volterra-type equation and investigate its existence and uniqueness under some precise linearity conditions using contraction fixed point theorem. A case of the above BVP for a stochastic nonlinear second order differential equation for \\\\(\\\\alpha=2\\\\) and \\\\(\\\\beta=0\\\\) with \\\\(u(0)=u(1)=0\\\\) is also studied.\",\"PeriodicalId\":52741,\"journal\":{\"name\":\"Open Journal of Mathematical Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-12-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Open Journal of Mathematical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.30538/PSRP-OMA2020.0074\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Open Journal of Mathematical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30538/PSRP-OMA2020.0074","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Boundary value problems for a class of stochastic nonlinear fractional order differential equations
Consider a class of two-point Boundary Value Problems (BVP) for a stochastic nonlinear fractional order differential equation \(D^\alpha u(t)=\lambda\sqrt{I^\beta[\sigma^2(t,u(t))]}\dot{w}(t)\ ,0< t< 1\) with boundary conditions \(u(0)=0,\,\,u'(0)=u'(1)=0,\) where \(\lambda>0\) is a level of the noise term, \(\sigma:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous, \(\dot{w}(t)\) is a generalized derivative of Wiener process (Gaussian white noise), \(D^\alpha\) is the Riemann-Liouville fractional differential operator of order \(\alpha\in (3,4)\) and \(I^\beta,\,\,\beta>0\) is a fractional integral operator. We formulate the solution of the equation via a stochastic Volterra-type equation and investigate its existence and uniqueness under some precise linearity conditions using contraction fixed point theorem. A case of the above BVP for a stochastic nonlinear second order differential equation for \(\alpha=2\) and \(\beta=0\) with \(u(0)=u(1)=0\) is also studied.