Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data

Tuan, Nguyen Huy; Zhou, Yong; Thach, Tran Ngoc; Can, Nguyen Huu

doi:10.1016/j.cnsns.2019.104873

Cited by 40 publications

(19 citation statements)

References 7 publications

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“…The Riemann approach to integration

⟨ ρ, ϕ_{i} ⟩ = \int_{0}^{π} ρ (x) ϕ_{i} (x) d x \approx \frac{π}{n} \sum_{k = 1}^{n} ρ (x_{k}) ϕ_{i} (x_{k}) = : ρ_{n; i} .

Moreover, it is possible to find an explicit formula for the relationship between ⟨ ρ , ϕ i ⟩ and ρ n ; i , as we will show next. For the proof of this result we refer to other studies 12,20–22 …”

Section: Preliminariesmentioning

confidence: 87%

“…For the proof of this result we refer to other studies. 12,[20][21][22] Lemma 2.4 (Approximation of Fourier coefficients). For i = 1, … , n − 1, the residual is defined as the difference between the ith Fourier coefficient of ∈ C 1 () and n, i can be represented as…”

Section: Approximation Of Fourier Coefficientsmentioning

confidence: 99%

“…According to some literature, 11–13 for random noise factors, the authors usually assume that they are drawn from the original normal distribution

scriptN false(0, σ^{2} false)

. This leads to being stuck in studying the existence and uniqueness of our regularized solution.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

Triet

Binh²,

Phương

et al. 2020

Math Methods in App Sciences

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In this work, we study the final value problem for a system of parabolic diffusion equations. In which, the final value functions are derived from a random model. This problem is severely ill‐posed in the sense of Hadamard. By nonparametric estimation and truncation methods, we offer a new regularized solution. We also investigate an estimate of the error and a convergence rate between a mild solution and its regularized solutions. Finally, some numerical experiments are constructed to confirm the efficiency of the proposed method.

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“…The Riemann approach to integration

⟨ ρ, ϕ_{i} ⟩ = \int_{0}^{π} ρ (x) ϕ_{i} (x) d x \approx \frac{π}{n} \sum_{k = 1}^{n} ρ (x_{k}) ϕ_{i} (x_{k}) = : ρ_{n; i} .

Section: Preliminariesmentioning

confidence: 87%

Section: Approximation Of Fourier Coefficientsmentioning

confidence: 99%

See 1 more Smart Citation

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

Triet

Binh²,

Phương

et al. 2020

Math Methods in App Sciences

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show abstract

“…In contract to the initial value problem, the study of the terminal value problem is still limited. We can list here some papers concerned with the deterministic model of this kind of problem [19,20,28].…”

mentioning

confidence: 99%

On initial value and terminal value problems for subdiffusive stochastic Rayleigh-Stokes equation

Caraballo¹,

Ngoc²,

Thach³

et al. 2021

DCDS-B

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In this paper, we study two stochastic problems for time-fractional Rayleigh-Stokes equation including the initial value problem and the terminal value problem. Here, two problems are perturbed by Wiener process, the fractional derivative are taken in the sense of Riemann-Liouville, the source function and the time-spatial noise are nonlinear and satisfy the globally Lipschitz conditions. We attempt to give some existence results and regularity properties for the mild solution of each problem.

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“…Bazhlekove et al 16 obtained the solutions of fractional Rayleigh-Stokes problem for a generalized second grade fluid by using eigenfunction expansion and Laplace transform. The existence and regularity for fractional Rayleigh-Stokes equation are derived by Nguyen et al 17,18 For more results, we can refer to other papers. [19][20][21] In this paper, we consider fractional Rayleigh-Stokes problem ∂ t u − ð1 þ γ∂ α t ÞΔu ¼ f ðt; uÞ; x ∈ Ω; t ∈ ð0; bÞ; uðx; tÞ ¼ 0;…”

mentioning

confidence: 99%

The nonlinear Rayleigh‐Stokes problem with Riemann‐Liouville fractional derivative

Zhou

Wang

2019

Math Methods in App Sciences

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The Rayleigh‐Stokes problem has gained much attention with the further study of non‐Newtonain fluids. In this paper, we are interested in discussing the existence of solutions for nonlinear Rayleigh‐Stokes problem for a generalized second grade fluid with Riemann‐Liouville fractional derivative. We firstly show that the solution operator of the problem is compact and continuous in the uniform operator topology. Furtherly, we give an existence result of mild solutions for the nonlinear problem.

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Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data

Cited by 40 publications

References 7 publications

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

On initial value and terminal value problems for subdiffusive stochastic Rayleigh-Stokes equation

The nonlinear Rayleigh‐Stokes problem with Riemann‐Liouville fractional derivative

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