This article is concerned with a forward problem for the following sub-diffusion equation driven by standard Brownian motion \begin{align*} \left( ^{\mathcal C} \partial^\gamma_t + A \right) u(t) = f(t) + B(t) \dot{W}(t), \quad t\in J:=(0,T), \end{align*} where $^{\mathcal C} \partial^\gamma_t$ is the conformable derivative, $\gamma \in (\frac{1}{2},1].$ Under some flexible assumptions on $f,B$ and the initial data, we investigate the existence, regularity, continuity of the solution on two spaces $L^r(J;L^2(\Omega,\dot{H}^\sigma))$ and $C^\alpha(\overline{J};L^2(\Omega,H))$ separately.
📌 3 Key Takeaways
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Objective & Hypothesis:This article is concerned with a forward problem for the following sub-diffusion equation driven by standard Brownian motion \begin{align*} \left( ^{\mathcal C} \partial^\gamma_t + A \right) u(t) = f(t) + B(t) \dot{W}(t), \quad t\in J:=(0,T), \end{align*} where $^{\mathcal C} \partial^\gamma_t$ is the conformable derivative, $\gamma \in (\frac{1}{2},1].$ Under some flexible assumptions on $f,B$ and the initial data, we investigate the existence, regularity, continuity of the solution on two spaces $L^r(J;L^2(\Omega,\dot{H}^\sigma))$ and $C^\alpha(\overline{J};L^2(\Omega,H))$ separately.
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Key Findings & Data:This article is concerned with a forward problem for the following sub-diffusion equation driven by standard Brownian motion \begin{align*} \left( ^{\mathcal C} \partial^\gamma_t + A \right) u(t) = f(t) + B(t) \dot{W}(t), \quad t\in J:=(0,T), \end{align*} where $^{\mathcal C} \partial^\gamma_t$ is the conformable derivative, $\gamma \in (\frac{1}{2},1].$ Under some flexible assumptions on $f,B$ and the initial data, we investigate the existence, regularity, continuity of the solution on two spaces $L^r(J;L^2(\Omega,\dot{H}^\sigma))$ and $C^\alpha(\overline{J};L^2(\Omega,H))$ separately.
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Conclusion & Impact:This article is concerned with a forward problem for the following sub-diffusion equation driven by standard Brownian motion \begin{align*} \left( ^{\mathcal C} \partial^\gamma_t + A \right) u(t) = f(t) + B(t) \dot{W}(t), \quad t\in J:=(0,T), \end{align*} where $^{\mathcal C} \partial^\gamma_t$ is the conformable derivative, $\gamma \in (\frac{1}{2},1].$ Under some flexible assumptions on $f,B$ and the initial data, we investigate the existence, regularity, continuity of the solution on two spaces $L^r(J;L^2(\Omega,\dot{H}^\sigma))$ and $C^\alpha(\overline{J};L^2(\Omega,H))$ separately.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Diffusion equation · Standard Brownian motion · Fractional Brownian motion
Abstract
This article is concerned with a forward problem for the following sub-diffusion equation driven by standard Brownian motion \begin{align*} \left( ^{\mathcal C} \partial^\gamma_t + A \right) u(t) = f(t) + B(t) \dot{W}(t), \quad t\in J:=(0,T), \end{align*} where $^{\mathcal C} \partial^\gamma_t$ is the conformable derivative, $\gamma \in (\frac{1}{2},1].$ Under some flexible assumptions on $f,B$ and the initial data, we investigate the existence, regularity, continuity of the solution on two spaces $L^r(J;L^2(\Omega,\dot{H}^\sigma))$ and $C^\alpha(\overline{J};L^2(\Omega,H))$ separately.
HUNG, N., BİNH, H., LUC, N., KIEU, A. N. T., LONG, L. D. (2021). Stochastic sub-diffusion equation with conformable derivative driven by standard Brownian motion. Advances in the Theory of Nonlinear Analysis and its Application, Cilt: 5 - Sayı: 3, 287–299. https://ajindex.com/articles/729062
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