On a terminal value problem for stochastic space‐time fractional wave equations. (22nd July 2022)

• Record Type: Journal Article
• Authors: Tran Bao, Ngoc; Nane, Erkan; Huy Tuan, Nguyen
• Is Part Of: Mathematical methods in the applied sciences. Volume 46:Number 1(2023)
• Page Start: 1206
• Page End: 1226
• Publication Date: 2022-07-22
• Subjects: existence and uniqueness -- fractional stochastic wave equation -- regularization method -- terminal value problem; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8573
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : This work is to investigate terminal value problem for a stochastic time fractional wave equation, driven by a cylindrical Wiener process on a Hilbert space. A representation of the solution is obtained by basing on the terminal value data u ( T, x ) = φ ( x ) $$ u\left(T, x\right)&amp;amp;amp;#x0003D;\varphi (x) $$ and the spectrum of the fractional Laplacian operator ( − Δ ) s / 2 $$ {\left(-\Delta \right)}&amp;amp;amp;#x0005E;{s/2} $$ (in a bounded domain 𝕏 ⊂ ℝ d, 0 < s < 2 $$ 0&amp;amp;lt;s&amp;amp;lt;2 $$ ). First, we show the existence and uniqueness of a mild solution in L p ( 0, T ; L 2 ( Ω, V ) ) ∩ C ( ( 0, T ] ; L 2 ( Ω, L 2 ( 𝕏 ) ) ), for a suitable sub‐space V $$ V $$ of L 2 ( 𝕏 ) . A limitation of this result is the lack of time continuity at t = 0 $$ t&amp;amp;amp;#x0003D;0 $$ . Second, we study the inverse problem (IP) of recovering u ( 0, x ) $$ u\left(0, x\right) $$ when the terminal value data φ $$ \varphi $$ and the source f $$ f $$ are given. We give an explanation why the time continuity of the solution at t = 0 $$ t&amp;amp;amp;#x0003D;0 $$ could not derived. The main reason comes from unboundedness of a solution operator, so the problem (IP) is then ill‐posed, that is, recovery u ( 0, x ) $$ u\left(0, x\right) $$ cannot be obtained in general. Hence, we propose a truncation regularization method with a suitable choice of the regularization parameter. Finally, we present a numerical example to demonstrate our proposed method.