Determination of source term for the fractional Rayleigh–Stokes equation with random data

Binh, Tran Thanh; Băleanu, Dumitru; Luc, Nguyen Hoang; Can, Nguyen-H

doi:10.1186/s13660-019-2262-9

Cited by 8 publications

(12 citation statements)

References 14 publications

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“…The formula is also contained in papers [10], equation (2.2), [14], equation (2.6) and [15], equation (8). Again, since these works are devoted to the study of other problems, the above issues were not considered.…”

Section: Forward Problemmentioning

confidence: 99%

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Backward and non-local problems for the Rayleigh-Stokes equation

Ashurov¹,

Vaisova²

2022

Preprint

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The Fourier method is used to find conditions on the right-hand side and on the initial data in the Rayleigh-Stokes problem, which ensure the existence and uniqueness of the solution. Then, in the Rayleigh-Stokes problem, instead of the initial condition, consider the non-local condition: u(x, T ) = βu(x, 0) + ϕ(x), where β is either zero or one. It is well known that if β = 0, then the corresponding problem, called the backward problem, is ill-posed in the sense of Hadamard, i.e. a small change in u(x, T ) leads to large changes in the initial data. Nevertheless, we will show that if we consider sufficiently smooth current information, then the solution exists and it is unique and stable. It will also be shown that if β = 1, then the corresponding non-local problem is well-posed and coercive type inequalities are valid.

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Section: Forward Problemmentioning

confidence: 99%

“…The inverse problems of determining the right-hand side (the heat source density) of the Rayleigh-Stokes equation have been considered by a number of authors (see, e.g., [8], [9], [10] and the bibliography therein). However, there is no general closed theory for the abstract case of the source function F (x, t).…”

mentioning

confidence: 99%

Backward and non-local problems for the Rayleigh-Stokes equation

Ashurov¹,

Vaisova²

2022

Preprint

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“…A review of some works in this direction is contained in the above-mentioned paper [1]. See also recent papers [13], [14] and references therein; 4) Many works are devoted to the study of the inverse problem of determining the right-hand side of the Rayleigh-Stokes equation (see, for example, [3], [15], [16], and the bibliography cited there). Since this inverse problem is Hadamard ill-posed, various regularization methods and numerical methods for finding the right-hand side of the equation are proposed in these works; 5) If the initial condition u(x, 0) = ϕ(x) in the problem (1.1) is replaced by u(x, T ) = ϕ(x), then the resulting problem is called backward problem.…”

Section: Introductionmentioning

confidence: 99%

Inverse problem of determining the order of the fractional derivative in the Rayleigh-Stokes equation

Ashurov¹,

Mukhiddinova²

2023

Preprint

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In recent years, much attention has been paid to the study of forward and inverse problems for the Rayleigh-Stokes equation in connection with the importance of this equation for applications. This equation plays an important role, in particular, in the study of the behavior of certain non-Newtonian fluids. The equation includes a fractional derivative of order α, which is used to describe the viscoelastic behavior of the flow. In this paper, we study the behavior of the solution of such equations depending on the parameter α. In particular, it is proved that for sufficiently large t the norm ||u(x, t)|| L 2 (Ω) of the solution decreases with respect to α. Moreover the inverse problem of determining the order of the derivative α is solved uniquely.Key words and phrases. The Rayleigh-Stokes problem, dependence of solution on the order of derivetive, inverse problem, determination of order of derivative, Fourier method.

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“…Besides that, the study of problem (1.1) with random noise data also began to receive the attention of mathematicians. In [16], using the truncation method and some new techniques, the authors showed the regularized solution, and convergence rates were established. In [17], Triet et al investigated an inverse source problem (1.1) by a general filter method for random noise, the results for the study of problem (1.1) were rare.…”

Section: Introductionmentioning

confidence: 99%

Reconstructing the right-hand side of the Rayleigh–Stokes problem with nonlocal in time condition

Duc

Binh²,

Long³

et al. 2021

Adv Differ Equ

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In this paper, the problem of finding the source function for the Rayleigh–Stokes equation is considered. According to Hadamard’s definition, the sought solution of this problem is both unstable and independent of continuous data. By using the fractional Tikhonov method, we give the regularized solutions and then deal with a priori error estimate between the exact solution and its regularized solutions. Finally, the proposed regularized methods have been verified by simple numerical experiments to check error estimate between the sought solution and the regularized solution.

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Determination of source term for the fractional Rayleigh–Stokes equation with random data

Cited by 8 publications

References 14 publications

Backward and non-local problems for the Rayleigh-Stokes equation

Backward and non-local problems for the Rayleigh-Stokes equation

Inverse problem of determining the order of the fractional derivative in the Rayleigh-Stokes equation

Reconstructing the right-hand side of the Rayleigh–Stokes problem with nonlocal in time condition

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