A note on a Cauchy problem for the Laplace equation: Regularization and error estimates

Tuan, Nguyen Huy; Trong, Dang Duc; Quan, Pham Hoang

doi:10.1016/j.amc.2010.09.019

Cited by 29 publications

(28 citation statements)

References 11 publications

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“…H is a linear densely defined self-adjoint and positivedefinite operator with compact inverse on H . Following the work [5] of Hào, Duc Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 5/30/15 11:53 AM 145 and Lesnic, in [13] Tuan, Trong and Quan used a Fourier truncated method to treat the following Cauchy problem of an elliptic equation with nonhomogeneous Dirichlet and Neumann data: and where they improved the results of convergence estimates in [5]. In [12] Tuan and Trong also adopt this method to solve a semi-linear backward heat conduction problem (SBHCP).…”

Section: Introductionmentioning

confidence: 99%

A Fourier truncated regularization method for a Cauchy problem of a semi-linear elliptic equation

Zhang

Wei

2013

Journal of Inverse and Ill-Posed Problems

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In this paper, a Cauchy problem for a semi-linear elliptic equation is considered. This problem is well known to be severely ill-posed and regularization methods are required. We use a Fourier truncated regularization method to deal with it, and a convergence estimate for the regularized solution is obtained under an a-priori bound assumption for the exact solution. Some numerical results are given to show the effectiveness of truncated method.

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Section: Introductionmentioning

confidence: 99%

A Fourier truncated regularization method for a Cauchy problem of a semi-linear elliptic equation

Zhang

Wei

2013

Journal of Inverse and Ill-Posed Problems

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“…This fact will be proved by the contraction principle. We claim by mathematical induction with respect to m = 1, 2, .. 35) where…”

Section: )mentioning

confidence: 99%

“…Nevertheless, the literature devoted to the Cauchy problem for linear homogeneous elliptic equations is very rich, see e.g. [4,5,7,9,12,13,16,21,23,29,33,35] and the references therein. Recently, a linear inhomogeneous version of Helmholtz equation (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Alessandrini et al [1] provided optimal stability results under minimal assumptions, whilst Reginska and Tautenhahn [32] presented some stability estimates and a regularization method for a Cauchy problem for Helmholtz equation. Many methods have been proposed to solve the Cauchy problem for linear homogeneous elliptic equations, such as the method of successive iterations [10], the alternating method [26], the conjugate gradient method [11,24], the iterative regularization method [15], the quasi-reversibility method [23,28], the fourth-order modified method [30], the Fourier truncation regularized (or spectral regularized method) [17,35], etc. Nevertheless, the literature devoted to the Cauchy problem for linear homogeneous elliptic equations is very rich, see e.g.…”

Section: Introductionmentioning

confidence: 99%

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A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source

Tuan

Thang

Lesnic

2016

Journal of Mathematical Analysis and Applications

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Up to now, studies on the semi-linear Cauchy problem for elliptic partial differential equations needed to assume that the source term present in the governing equation is a global Lipschitz function. The current paper is the first investigation to not only the more general but also the more practical case of interest when the souce term is only a local Lipschitz function. In such a situation, the methods of solution from the previous studies with a global Lipschitz source term are not directly applicable and therefore, novel ideas and techniques need to be developed to tackle the local Lipschitz nonlinearity. This locally Lipschitz source arises in many applications of great physical interest governed by, for example, the sine-Gordon, Lane-Emden, Allen-Cahn and Liouville equations. The inverse problem is severely ill-posed in the sense of Hadamard by violating the continuous dependence upon the input Cauchy data. Therefore, in order to obtain a stable solution we consider theoretical aspects of regularization of the problem by a new generalized filter method. Under some priori assumptions on the exact solution, we prove and obtain rigorously convergence estimates.

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“…(1.1), see e.g. [3,4,5,6,7,9,10,13,14,16,17] to mention only a few. On the other hand, the Cauchy problem for nonlinear elliptic equations has been much less investigated, [11,21], and it is the purpose of this study to make advances into the semi-linear problem (1.1).…”

Section: Introductionmentioning

confidence: 99%

On the Cauchy problem for semilinear elliptic equations

Tuan

Binh

Viet

et al. 2015

Journal of Inverse and Ill-Posed Problems

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We study the Cauchy problem for non-linear (semilinear) elliptic partial differential equations in Hilbert spaces. The problem is severely ill-posed in the sense of Hadamard. Under a weak a priori assumption on the exact solution, we propose a new regularization method for stabilising the ill-posed problem. These new results extend some earlier works on Cauchy problems for nonlinear elliptic equations. Numerical results are presented and discussed.

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A note on a Cauchy problem for the Laplace equation: Regularization and error estimates

Cited by 29 publications

References 11 publications

A Fourier truncated regularization method for a Cauchy problem of a semi-linear elliptic equation

A Fourier truncated regularization method for a Cauchy problem of a semi-linear elliptic equation

A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source

On the Cauchy problem for semilinear elliptic equations

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