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

Abstract

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

Conditional stability estimate for an ill-posed elliptic equation by using nonlocal conditions

“…To the best of our knowledge, the first applications of this idea to the topic of ill-posed biharmonic equations were done in paper of Benrabah and Boussetila [16]. It should be noticed that the problem (P) with f (x) = h(x) = 0, has been studied in [17]. For readers interested in the ill-posedness of elliptical PDEs in parameter estimation, i.e.…”

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