scholarly journals Iterative Method to Solve a Data Completion Problem for Biharmonic Equation for Rectangular Domain

2017 ◽  
Vol 55 (1) ◽  
pp. 129-147
Author(s):  
Chakir Tajani ◽  
Houda Kajtih ◽  
Ali Daanoun
Keyword(s):  
Iterative Method ◽  
Biharmonic Equation ◽  
Rectangular Domain ◽  
Numerical Results ◽  
Cauchy Problems ◽  
Completion Problem ◽  
Data Completion ◽  
Ill Posed ◽  
Convergent Method ◽  
Direct Problems

AbstractIn this work, we are interested in a class of problems of great importance in many areas of industry and engineering. It is the invese problem for the biharmonic equation. It consists to complete the missing data on the inaccessible part from the measured data on the accessible part of the boundary. To solve this ill-posed problem, we opted for the alternative iterative method developed by Kozlov, Mazya and Fomin which is a convergent method for the elliptical Cauchy problems in general. The numerical implementation of the iterative algorithm is based on the application of the boundary element method (BEM) for a sequence of mixed well-posed direct problems. Numerical results are performed for a square domain showing the effectiveness of the algorithm by BEM to produce accurate and stable numerical results.

scholarly journals A Preconditioned Richardson Regularization for the Data Completion Problem and the Kozlov-Maz’ya-Fomin Method

2010 ◽  
Vol Volume 13 - 2010 - Special... ◽  
Author(s):  
Duc Thang Du ◽  
Faten Jelassi
Keyword(s):  
Inverse Problems ◽  
Stopping Rules ◽  
Completion Problem ◽  
Variational Framework ◽  
Data Completion ◽  
International Audience ◽  
Ill Posed ◽  
General Source ◽  
Richardson Iterative Method ◽  
Math Phys

International audience Using a preconditioned Richardson iterative method as a regularization to the data completion problem is the aim of the contribution. The problem is known to be exponentially ill posed that makes its numerical treatment a hard task. The approach we present relies on the Steklov-Poincaré variational framework introduced in [Inverse Problems, vol. 21, 2005]. The resulting algorithm turns out to be equivalent to the Kozlov-Maz’ya-Fomin method in [Comp. Math. Phys., vol. 31, 1991]. We conduct a comprehensive analysis on the suitable stopping rules that provides some optimal estimates under the General Source Condition on the exact solution. Some numerical examples are finally discussed to highlight the performances of the method. L’objectif est d’utiliser une méthode itérative de Richardson préconditionnée comme une technique de régularisation pour le problème de complétion de données. Le problème est connu pour être sévèrement mal posé qui rend son traitement numérique ardu. L’approche adoptée est basée sur le cadre variationnel de Steklov-Poincaré introduit dans [Inverse Problems, vol. 21, 2005].L’algorithme obtenu s’avère être équivalent à celui de Kozlov-Maz’ya-Fomin parû dans [Comp. Math. Phys., vol. 31, 1991]. Nous menons une analyse complète pour le choix du critère d’arrêt, et établissons des estimations optimales sous les Conditions Générale de Source sur la solution exacte. Nous discutons, enfin, quelques exemples numériques qui confortent les pertinence de la méthode.

scholarly journals Reconstruction of missing boundary conditions from partially overspecified data : the Stokes system

2016 ◽  
Vol Volume 23 - 2016 - Special... ◽  
Author(s):  
Amel Ben Abda ◽  
Faten Khayat
Keyword(s):  
Inverse Problem ◽  
Stokes System ◽  
Stokes Problem ◽  
Domain Boundary ◽  
The Other ◽  
First Order ◽  
Completion Problem ◽  
Data Completion ◽  
Ill Posed ◽  
Problème Inverse

We are interested in this paper with the ill-posed Cauchy-Stokes problem. We consider a data completion problem in which we aim recovering lacking data on some part of a domain boundary , from the knowledge of partially overspecified data on the other part. The inverse problem is formulated as an optimization one using an energy-like misfit functional. We give the first order opti-mality condition in terms of an interfacial operator. Displayed numerical results highlight its accuracy. Nous nous intéressons à un problème de Cauchy mal posé, celui de la complétion de données frontières pour les équations de Stokes. Nous voulons reconstituer les données manquantes sur une partie non accessible de la frontière du domaine à partir de données peu surdéterminées sur la partie accessible. Nous formulons ce problème inverse sous forme de minimisation d'une fonctionnelle de type énergie. Les conditions d'optimalité du premier ordre sont écrites en termes d'équation d'interface utilisant les opérateurs de Stecklov-Poincaré. Nous donnons des résultats numériques attestant l'efficacité de la méthode.

scholarly journals On an ill-posed problem for a biharmonic equation

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1051-1056 ◽  
Author(s):  
Tynysbek Kal’menov ◽  
Ulzada Iskakova
Keyword(s):  
Boundary Value Problem ◽  
Biharmonic Equation ◽  
Sufficient Conditions ◽  
Rectangular Domain ◽  
Problem Solution ◽  
Local Boundary ◽  
Stability Of Solution ◽  
Ill Posed ◽  
The Stability ◽  
Necessary And Sufficient

A local boundary value problem for the biharmonic equation in a rectangular domain is considered. Boundary conditions are given on all boundary of the domain. We show that the considered problem is self-adjoint. Herewith the problem is ill-posed. We show that the stability of solution to the problem is disturbed. Necessary and sufficient conditions of existence of the problem solution are found.

On a criterion for the solvability of one ill-posed problem for the biharmonic equation

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Tynysbek S. Kal’menov ◽  
Makhmud A. Sadybekov ◽  
Ulzada A. Iskakova
Keyword(s):  
Biharmonic Equation ◽  
Sufficient Conditions ◽  
Adjoint Problem ◽  
Rectangular Domain ◽  
Solution Stability ◽  
Well Posedness ◽  
Local Boundary ◽  
Ill Posed ◽  
Necessary And Sufficient ◽  
Isolated Point

AbstractA local boundary value problem for an inhomogeneous biharmonic equation in a rectangular domain is considered. Boundary conditions are given on the whole boundary of the domain. It is shown that the problem turns out to be self-adjoint. And herewith the problem is ill-posed. An example is constructed demonstrating that the solution stability to the problem is violated. Necessary and sufficient conditions of the existence of a strong solution to the investigated problem are found. The idea of the method is that the solution to the problem is constructed in the form of an expansion on eigenfunctions of this self-adjoint problem. This problem has an isolated point of continuous spectrum in zero. It is shown that there exists a series (a sequence) of eigenvalues converging to zero. Asymptotics of these eigenvalues is found. Namely this asymptotics defines a reason for the ill-posedness of the investigated problem. A space of well-posedness for the investigated problem is constructed.

Global stability result for parabolic Cauchy problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mourad Choulli ◽  
Masahiro Yamamoto
Keyword(s):  
New York ◽  
Partial Differential Equations ◽  
Global Stability ◽  
Classical Problem ◽  
Stability Result ◽  
Cauchy Problems ◽  
Smooth Solutions ◽  
Linear Partial Differential Equations ◽  
Ill Posed ◽  
The Stability

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

BENDING OF ORTHOTROPIC RECTANGULAR CANTILEVER PLATE

2021 ◽  
pp. 2-9
Author(s):  
M.V. Sukhoterin ◽  
◽  
A.M. Maslennikov ◽  
T.P. Knysh ◽  
I.V. Voytko ◽  
...  
Keyword(s):  
Iterative Method ◽  
Boundary Conditions ◽  
Trigonometric Series ◽  
Bending Moment ◽  
Partial Solution ◽  
Numerical Results ◽  
Cantilever Plate ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Beam Function

Abstract. An iterative method of superposition of correcting functions is proposed. The partial solution of the main differential bending equation is represented by a fourth-degree polynomial (the beam function), which gives a residual only with respect to the bending moment on parallel free faces. This discrepancy and the subsequent ones are mutually compensated by two types of correcting functions-hyperbolic-trigonometric series with indeterminate coefficients. Each function satisfies only a part of the boundary conditions. The solution of the problem is achieved by an infinite superposition of correcting functions. For the process to converge, all residuals must tend to zero. When the specified accuracy is reached, the process stops. Numerical results of the calculation of a square ribbed plate are presented.

scholarly journals A modified truncation singular value decomposition method for solving ill-posed problems

2018 ◽  
Vol 13 ◽  
pp. 174830181881360 ◽  
Author(s):  
Zhenyu Zhao ◽  
Riguang Lin ◽  
Zehong Meng ◽  
Guoqiang He ◽  
Lei You ◽  
...  
Keyword(s):  
Singular Value Decomposition ◽  
Decomposition Method ◽  
Singular Value ◽  
Numerical Results ◽  
New Method ◽  
Truncated Singular Value Decomposition ◽  
Singular Value Decomposition Method ◽  
Ill Posed ◽  
Value Decomposition

A modified truncated singular value decomposition method for solving ill-posed problems is presented in this paper, in which the solution has a slightly different form. Both theoretical and numerical results show that the limitations of the classical TSVD method have been overcome by the new method and very few additive computations are needed.

scholarly journals Numerical Simulation of Cubic-Quartic Optical Solitons with Perturbed Fokas–Lenells Equation Using Improved Adomian Decomposition Algorithm

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 138
Author(s):  
Alyaa A. Al-Qarni ◽  
Huda O. Bakodah ◽  
Aisha A. Alshaery ◽  
Anjan Biswas ◽  
Yakup Yıldırım ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Iterative Method ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Optical Solitons ◽  
Numerical Results ◽  
Decomposition Algorithm ◽  
Adomian Decomposition ◽  
High Level ◽  
Do So

The current manuscript displays elegant numerical results for cubic-quartic optical solitons associated with the perturbed Fokas–Lenells equations. To do so, we devise a generalized iterative method for the model using the improved Adomian decomposition method (ADM) and further seek validation from certain well-known results in the literature. As proven, the proposed scheme is efficient and possess a high level of accuracy.

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