Regularization of the fractional Rayleigh–Stokes equation using a fractional Landweber method

Luc, Nguyen Hoang; Huynh, Lam K.; O’Regan, Donal; Can, Nguyen Huu

doi:10.1186/s13662-020-02922-4

Cited by 6 publications

(15 citation statements)

References 24 publications

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“…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. The backward problem for the Rayleigh-Stokes equation is of great importance and is aimed at determining the previous state of the physical field (for for example, at t = 0) based on his current information (see, for example, [4], [17] for the case N ≤ 3 and [18] for an arbitrary N ). However, this problem (as well as the inverse problem of finding the right-hand side of the equation) is ill-posed according to Hadamard.…”

Section: Introductionmentioning

confidence: 99%

“…However, this problem (as well as the inverse problem of finding the right-hand side of the equation) is ill-posed according to Hadamard. Therefore, the authors of [4], [17] proposed various regularization methods and tested these methods using numerical experiments. In the paper [18], along with other questions, problem (1.1) is investigated by taking the non-local condition u(x, T ) = u(x, 0)+ ϕ(x) instead of the initial condition.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

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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“…a small change in u(x, T ) leads to a large change of the solution. In papers [22], [23] (see also references therein) various regularization methods are proposed, accompanied by verification of these methods using numerical experiments. We emphasize that in these papers N < 4, and this is connected with the method used there.…”

Section: Introductionmentioning

confidence: 99%

A non-local problem for the fractional order Rayleigh-Stokes equation

Ashurov¹,

Mukhiddinova²,

Umarov³

2023

Preprint

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A nonlocal boundary value problem for the fractional version of the well known in fluid dynamics Rayleigh-Stokes equation is studied. Namely, the condition u(x, T ) = βu(x, 0) + ϕ(x), where β is an arbitrary real number, is proposed instead of the initial condition. If β = 0, then we get the inverse problem in time, called the backward problem. It is well known that the backward problem is ill-posed in the sense of Hadamard. If β = 1, then the corresponding non-local problem becomes well-posed in the sense of Hadamard, and moreover, in this case a coercive estimate for the solution can be established. The aim of this work is to find values of the parameter β, which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if β ≥ 1, or β < 0, then the problem is wellposed; if β ∈ (0, 1), then depending on the eigenvalues of the elliptic part of the equation, for the existence of a solution an additional condition on orthogonality of the right-hand side of the equation and the boundary function to some eigenfunctions of the corresponding elliptic operator may emerge.

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“…In the case when the initial condition u(x, 0) = ϕ(x) in problem ( 1) is replaced by u(x, T) = ϕ(x), then the resulting problem is called the backward problem. The backward problem for the Rayleigh-Stokes equation is of great importance and it consists in determining the previous state of the physical field (for example, at t = 0) based on its current information (see, e.g., [19,20] and the bibliography therein). However, this problem (as well as the inverse problem of finding the right-hand side of the equation) is not stable.…”

Section: Introductionmentioning

confidence: 99%

Backward and Non-Local Problems for the Rayleigh-Stokes Equation

Ashurov

Vaisova

2022

Fractal Fract

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This paper presents the method of separation of variables 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. Further, in the Rayleigh-Stokes problem, instead of the initial condition, the non-local condition is considered: u(x,T)=βu(x,0)+φ(x), where β is equal to 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, it is unique and stable. It will also be shown that if β=1, then the corresponding non-local problem is well-posed and inequalities of coercive type are satisfied.

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Regularization of the fractional Rayleigh–Stokes equation using a fractional Landweber method

Cited by 6 publications

References 24 publications

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

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

A non-local problem for the fractional order Rayleigh-Stokes equation

Backward and Non-Local Problems for the Rayleigh-Stokes Equation

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