Integral equations for biharmonic data completion

Abstract

A boundary integral based method for the stable reconstruction of missing boundary data is presented for the biharmonic equation. The solution (displacement) is known throughout the boundary of an annular domain whilst the normal derivative and bending moment are specified only on the outer boundary curve. A recent iterative method is applied for the data completion solving mixed problems throughout the iterations. The solution to each mixed problem is represented as a biharmonic single-layer potential. Matching against the given boundary data, a system of boundary integrals is obtained to be solved for densities over the boundary. This system is discretised using the Nyström method. A direct approach is also given representing the solution of the ill-posed problem as a biharmonic single-layer potential and applying the similar techniques as for the mixed problems. Tikhonov regularization is employed for the solution of the corresponding discretised system. Numerical results are presented for several annular domains showing the efficiency of both data completion approaches.

Shape reconstruction for an inverse biharmonic problem from partial Cauchy data

“…Many works are devoted to these problems and modified versions to the determination of sub-boundary for the Laplace case using the direct and indirect boundary integral equation method, detection of corrosion [11,12,19], in Wentzell-type (GIBC) boundary condition [20]. Various methods exist for solving the problems in applied science; to complete the missing Cauchy data, there are several methods, such as the iterative method [6], Tikhonov method [4], and the method of fundamental solution (MFS) in combination with the Tikhonov method [7]. For the reconstruction of the boundary, see [11,19].…”

“…The bi‐Laplace operator $$$ {\Delta}^2 $$$ is the prototype of fourth‐order elliptic operator that appears in many practical areas of science and engineering [1]. Several scientific studies have been devoted to the application of biharmonic problems in science and engineering, such as the deformation of thin plates, the motion of fluids, determining an unknown boundary, detecting the corrosion, and the problems related to blending surface [2–7].…”