Numerical reconstruction of unknown boundary data in the Cauchy problem for Laplace’s equation

Mukanova, Balgaisha

doi:10.1080/17415977.2012.744405

Cited by 10 publications

(8 citation statements)

References 12 publications

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“…As another practical case, especially in geophysical applications, for a given domain some boundary data cannot be stated explicitly. This fact may be seen in .…”

Section: Introductionmentioning

confidence: 78%

“…As another practical case, especially in geophysical applications, for a given domain some boundary data cannot be stated explicitly. This fact may be seen in . Case 2: Right‐hand side identification Consider the inverse problem in which it is required to determine the unknown right‐hand sides from observation data measured at the domain boundary . Heat source estimation , heat conduction , crack identification , electromagnetic theory , Stefan design problem , geophysical prospecting, and pollutant detection are examples of area for applications of such problems .…”

Section: Introductionmentioning

confidence: 99%

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Efficient numerical methods for boundary data and right‐hand side reconstructions in elliptic partial differential equations

Rashedi

Adibi

Dehghan

2015

Numerical Methods Partial

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In this article, we discuss the application of two important numerical methods, Ritz-Galerkin and Method of Fundamental Solutions (MFS), for solving some inverse problems, arising in the context of two-dimensional elliptic equations. The main incentive for studying the considered problems is their wide applications in engineering fields. In the previous literature, the use of these methods, particularly MFS for right hand side reconstruction has been limited, partly due to stability concerns. We demonstrate that these diculties may be surmounted if the aforementioned methods are combined with techniques such as dual reciprocity method(DRM). Moreover, we incorporate some iterative regularization techniques. This fact is especially veried by taking into account the noisy data with boundary conditions. In addition, parts of this article are dedicated to the problem of boundary data approximation and the issue of numerical stability, ending with a general discussion on the advantages and disadvantages of various methods.

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“…As another practical case, especially in geophysical applications, for a given domain some boundary data cannot be stated explicitly. This fact may be seen in .…”

Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Efficient numerical methods for boundary data and right‐hand side reconstructions in elliptic partial differential equations

Rashedi

Adibi

Dehghan

2015

Numerical Methods Partial

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“…The idea of that method is described in detail in [24], so we describe here only key points of that method. Let us introduce formally a functional (Lagrangian):…”

Section: Construction Of the Quasi-solution To The Inverse Problemmentioning

confidence: 99%

“…New method of reconstruction of unknown boundary data in the Cauchy problem for Laplace's equation is proposed by Mukanova in [24]. The problem is reduced to a system consisting of mutually dependent direct and adjoint problems.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper the continuation inverse problem for a 3D steady-state diffusion model inside a cylindrical layered medium is considered. Methods presented in [5,24] for the first time are used for the numerical solution of continuation problem for 3D elliptic equation with piecewise constant coefficient. They are compared on numerical examples in three dimensional cases.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Solution of Continuation Problem for 3D Steady-State Diffusion in Cylindrically Layered Medium

Kulbay

Maussumbekova

Mukanova

2015

Abstract and Applied Analysis

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This work is based on the application of Fourier and quasi-solution methods for solving the continuation inverse problem for 3D steady-state diffusion model inside a cylindrical layered medium. The diffusion coefficient is supposed to be a piecewise constant function, Cauchy data are given on the outer boundary of the cylinder, and we seek to recover the temperature at the inner boundary of the cylinder. Numerical experiments are investigated and show the capacity of proposed method only for smooth boundary condition. Under the suitable choice of regularization parameters we recover the distribution of temperature on the inner boundary with satisfactory quality for noisy data.

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Solving Geometric Inverse Problems with a Polynomial Based Meshless Method

Nachaoui

Aboud

2023

Springer Proceedings in Mathematics &Amp; Statistics

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Numerical reconstruction of unknown boundary data in the Cauchy problem for Laplace’s equation

Cited by 10 publications

References 12 publications

Efficient numerical methods for boundary data and right‐hand side reconstructions in elliptic partial differential equations

Efficient numerical methods for boundary data and right‐hand side reconstructions in elliptic partial differential equations

Numerical Solution of Continuation Problem for 3D Steady-State Diffusion in Cylindrically Layered Medium

Solving Geometric Inverse Problems with a Polynomial Based Meshless Method

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