An improved non-local boundary value problem method for a cauchy problem of the Laplace equation

Abstract:In this paper, we are concerned with the problem of approximating a solution of an ill-posed biparabolic problem in the abstract setting. In order to overcome the instability of the original problem, we propose a modi ed quasi-boundary value method to construct approximate stable solutions for the original ill-posed boundary value problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution. Moreover, numerical tests are presented to illustrate the accuracy and e ciency of this method. Keywords:Ill-posed problems, Biparabolic problem, Regularization MSC: 47A52, 65J22Formulation of the problem Throughout this paper H denotes a complex separable Hilbert space endowed with the inner product ⟨., .⟩ and the norm . , L(H) stands for the Banach algebra of bounded linear operators on H.Let A ∶ D(A) ⊂ H → H be a positive, self-adjoint operator with compact resolvent, so that A has an orthonormal basis of eigenvectors (φ n ) ⊂ H with real eigenvalues (λ n ) ⊂ R + , i.e.,In this paper, we consider the following inverse source problem of determining the unknown source term u( ) = f and the temperature distribution u(t) for ≤ t < T, of the following biparabolic problemwhere < T < ∞ and g is a given H-valued function.

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“…is the exact solution of the problem (43). Consequently, the data function is g(x) = u(x, ) = π e sin(x).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…This method has been used to solve some ill-posed problems for parabolic, hyperbolic and elliptic equations; for more details, see [22,[32][33][34][35][36][37][38][39][40][41][42][43][44] and the references therein.…”

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confidence: 99%

A modified quasi-boundary value method for an abstract ill-posed biparabolic problem

2017

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“…The most popular one is Tikhonov's regularization method [6,21,22], which transforms the original ill-posed into a well posed problem by minimizing the L 2 -norm of the solution subjected to the constraint equation. Then other methods have been proposed to regularize the Cauchy problem,we can mention for instance the alternating method [2,19], the universal method [18], the quasi-reversibility method [3,9,20], the technic of fundamental solution [12] and improved nonlocal boundary value problem method [23], etc. Nevertheless, the literature devoted to the Cauchy problem for linear elliptic equations is very rich (see for example [1,4,7,11,15,16] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

On a regularization approach for solving the inverse Cauchy Stokes problem

Chakib,

Nachaoui,

Nachaoui

et al. 2021

Preprint

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In this paper, we are interested to an inverse Cauchy problem governed by Stokes equation, called the data completion problem. It consists in determining the unspecified fluid velocity, or one of its components over a part of its boundary, by introducing given measurements on its remaining part. As it's known, this problem is one of highly ill-posed problem in the Hadamard's sense [14], it is then an interesting challenge to carry out a numerical procedure for approximating their solutions, mostly in the particular case of noisy data. To solve this problem, we propose here a regularizing approach based on a coupled complex boundary method, originally proposed in [8], for solving an inverse source problem. We show the existence of the regularization optimization problem and prove the convergence of subsequence of optimal solutions of Tikhonov regularization formulations to the solution of the Cauchy problem. Then we suggest the numerical approximation of this problem using the adjoint gradient technic and the finite elements method of P 1 − bubble/P 1 type. Finally, we provide some numerical results showing the accuracy, the effectiveness and robustness of the proposed approach.

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“…Benrabah and Boussetila considered an ill‐posed problem for the biharmonic equations. Zhang and Wei considered an improved nonlocal boundary value problem method for a cauchy problem of the Laplace equation. Khelili et al considered a modified QBVM for an abstract ill‐posed biparabolic problem.…”

Section: Introductionmentioning

confidence: 99%

Modified auxiliary boundary conditions method for an ill‐posed problem for the homogeneous biharmonic equation

Benrabah

Boussetila

Rebbani³

2019

Math Methods in App Sciences

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In this paper, we are interested in the inverse problem for the biharmonic equation posed on a rectangle, which is of great importance in many areas of industry and engineering. We show that the problem under consideration is ill-posed; therefore, to solve it, we opted for a regularization method via modified auxiliary boundary conditions. The numerical implementation is based on the application of the semidiscrete finite difference method for a sequence of well-posed direct problems depending on a small parameter of regularization. Numerical results are performed for a rectangle domain showing the effectiveness of the proposed method. KEYWORDS auxiliary boundary conditions, biharmonic equation, convergence estimate, ill-posed problems, plates, regularization MSC CLASSIFICATION 35J40; 31B30; 35B45; 74K20; 47A52 Abbreviations: BHP, biharmonic problem; MABCM, modified auxiliary boundary conditions method; QBVM, quasi-boundary value method. wileyonlinelibrary.com/journal/mma

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An improved non-local boundary value problem method for a cauchy problem of the Laplace equation

Cited by 10 publications

References 22 publications

A modified quasi-boundary value method for an abstract ill-posed biparabolic problem

A modified quasi-boundary value method for an abstract ill-posed biparabolic problem

On a regularization approach for solving the inverse Cauchy Stokes problem

Modified auxiliary boundary conditions method for an ill‐posed problem for the homogeneous biharmonic equation

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