Boundary value problems for a class of stochastic nonlinear fractional order differential equations

M. Omaba, L. Omenyi
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引用次数: 0

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.
一类随机非线性分数阶微分方程的边值问题
考虑一类随机非线性分数阶微分方程的两点边值问题 \(D^\alpha u(t)=\lambda\sqrt{I^\beta[\sigma^2(t,u(t))]}\dot{w}(t)\  ,0< t< 1\) 有边界条件 \(u(0)=0,\,\,u'(0)=u'(1)=0,\) 在哪里 \(\lambda>0\) 是噪声项的一个级别, \(\sigma:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) 是连续的, \(\dot{w}(t)\) 是Wiener过程(高斯白噪声)的广义导数, \(D^\alpha\) 黎曼-刘维尔分数阶微分算子是有序的吗 \(\alpha\in (3,4)\) 和 \(I^\beta,\,\,\beta>0\) 是一个分数积分算子。我们利用随机volterra型方程给出了方程的解,并利用收缩不动点定理研究了该方程在一些精确线性条件下的存在唯一性。的随机非线性二阶微分方程的上述BVP的一个例子 \(\alpha=2\) 和 \(\beta=0\) 有 \(u(0)=u(1)=0\) 也被研究过。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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自引率
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发文量
10
审稿时长
8 weeks
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