Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian

Koba, Hajime; Matsuoka, Hiroatsu

doi:10.1515/anly-2014-1262

Cited by 8 publications

(2 citation statements)

References 18 publications

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“…For the inverse problems for space-fractional diffusion equations, we can refer to [17], in which the authors gave the inverse time-dependent source from one-point measurement data by using the method of fundamental solutions. Besides, one can refer to two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem by Zheng and Wei [18], and generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian [19].…”

Section: Introductionmentioning

confidence: 99%

Simultaneous determination of source term and the initial value in the space-fractional diffusion problem by a novel modified quasi-reversibility regularization method

Wen¹,

Ren²,

Wang³

2023

Phys. Scr.

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This paper deals with an inverse problem of determining source term and initial data simultaneously for a space-fractional diffusion equation in a strip domain, with the aid of extra measurement data at a fixed time. The uniqueness results are obtained by a simple trick based on the linear property of the proposed equation. Since this problem is ill-posed, a modified quasi-reversibility method is obtained by employing the Fourier transform. Error estimates for source term and initial value are obtained from a suitable parameter choice rule. Finally, several numerical examples show that the proposed regularization method is effective and stable.

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Section: Introductionmentioning

confidence: 99%

Simultaneous determination of source term and the initial value in the space-fractional diffusion problem by a novel modified quasi-reversibility regularization method

Wen¹,

Ren²,

Wang³

2023

Phys. Scr.

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“…If the noise is considered as a deterministic quantity, it is natural to study the worst-case error. In the literature a number of efficient methods for the solution of (1) have been developed: see, for example, [3,8], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Approximate Solutions of Inverse Problems for Nonlinear Space Fractional Diffusion Equations with Randomly Perturbed Data

Nane¹,

Tuan²

2018

SIAM/ASA J. Uncertainty Quantification

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This paper is concerned with backward problem for nonlinear space fractional diffusion with additive noise on the right-hand side and the final value. To regularize the instable solution, we develop some new regularized method for solving the problem. In the case of constant coefficients, we use the truncation methods. In the case of perturbed time dependent coefficients, we apply a new quasi-reversibility method. We also show the convergence rate between the regularized solution and the sought solution under some a priori assumption on the sought solution.Keywords: Inverse problem for fractional heat equation, truncation method, approximate solu-2 recently considered by Tuan and Nane [19].Next we give some details about our methods for the following two cases The first case: a(t) = 1. First, we transform the problem (1) into a nonlinear integral equation, then we apply the Fourier truncation method(using the eigenfunctions cos(px), p = 0, 1, 2 · · · of the Laplacian in the interval (0, π) with Neumann boundary conditions) associated with some techniques in nonparametric regression to establish a first regularized solution U Mn,n (x, t) which satisfies (33). To obtain the estimate between U Mn,n and u, we need some stronger assumptions on u, such as (27) and (29). The main result is Theorem 2.6. However, as pointed out in Remark 2.5, the assumptions (27) and (29) are difficult to come up in practice. Motivated by this, when g = 0 we develop a second regularized solution U Mn,n defined by (69) to obtain the estimate for u ∈ C([0, T ]; H γ (Ω)) (see Remark 2.5 for more details). The main result for the second type of regularization is given in Theorem 2.8. It is important to realize that the second regularized solution is a modification of the first regularized solution. Our methods in this paper can be applied to solve many ill-posed problems of nonlinear PDEs such as Cauchy problem for nonlinear elliptic, nonlinear ultraparabolic, nonlinear strongly damped wave equations, and many others. The second case: a(t) depend on t and is perturbed. Note that if the coefficient a in the main equation of (1) is not noisy then the Fourier truncation method in [21] can be applied to Problem (1). However, the difficulty occurs for (86) when the time dependent coefficient a(t) is noisy. Indeed, we assume that a(t) is noisy by observed random data a(t) which satisfy thatwhere ξ(t) is Brownian motion. If we have used a Fourier truncation solution for (1), then the regularized solution would contain some terms such as exp p 2β T t s t a(τ )dτ ds , which would lead to some complex computations. Hence, we don't follow the truncation method as in [21], instead we develop a new method to find a regularized solution. We will apply a new quasireversibility method for solving the problem. Further details of this method can be found in Tuan [21]. In this case our main results are Theorems 3.2 and 3.7.In this paper, we only study the upper bound of the convergence rate. In a future work, we will study the minimax rate of convergence for fi...

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The Regularized Solution Approximation of Forward/Backward Problems for a Fractional Pseudo-Parabolic Equation with Random Noise

Rong

2022

Acta Math Sci

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Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian

Cited by 8 publications

References 18 publications

Simultaneous determination of source term and the initial value in the space-fractional diffusion problem by a novel modified quasi-reversibility regularization method

Simultaneous determination of source term and the initial value in the space-fractional diffusion problem by a novel modified quasi-reversibility regularization method

Approximate Solutions of Inverse Problems for Nonlinear Space Fractional Diffusion Equations with Randomly Perturbed Data

The Regularized Solution Approximation of Forward/Backward Problems for a Fractional Pseudo-Parabolic Equation with Random Noise

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