1 Ocak 2023 Dergi makalesi Açık Erişim

Baysal, Onur; Hasanov, Alemdar; Kumarasamy, Sakthivel

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{
  "DOI": "10.1515/jiip-2023-0021", 
  "abstract": "<p>In this paper, we present a new methodology, based on the inverse problem approach, for the determination of an unknown shear force acting on the inaccessible tip of the microcantilever, which is a key component of transverse dynamic force microscopy (TDFM). The mathematical modelling of this phenomenon leads to the inverse problem of determining the shear force g(t) acting on the inaccessible boundary x = l in a system governed by the variable coefficient Euler-Bernoulli equation</p>\n<p>r(A)(x)u(tt) + mu(x)u(t) + (r(x)u(xx) + kappa(x)u(xxt))(xx) = 0, (x, t) is an element of (0, l) x (0, T),</p>\n<p>subject to the homogeneous initial conditions and the boundary conditions</p>\n<p>u(0, t) = u(0)(t), u(x)(0, t) = 0, (u(xx)(x, t) + kappa(x)u(xxt))(x=l) = 0, (-(r(x)u(xx) + kappa(x)u(xxt))x)(x=l) = g(t),</p>\n<p>from the final time measured output (displacement) u(T) (x) := u( x, T). We introduce the input-output map (Phi g)(x) := u( x, T; g), g is an element of G, and prove that it is a compact and Lipschitz continuous linear operator. Then we introduce the Tikhonov functional</p>\n<p>J(F) = 1/2 parallel to Phi g - u(T)parallel to(2)(L2(0,l))</p>\n<p>and prove the existence of a quasi-solution of the inverse problem. We derive a gradient formula for the Frechet gradient of the Tikhonov functional through the corresponding adjoint problem solution and prove that it is a Lipschitz continuous functional. The results of the numerical experiments clearly illustrate the effectiveness and feasibility of the proposed approach.</p>", 
  "author": [
    {
      "family": "Baysal", 
      "given": " Onur"
    }, 
    {
      "family": "Hasanov", 
      "given": " Alemdar"
    }, 
    {
      "family": "Kumarasamy", 
      "given": " Sakthivel"
    }
  ], 
  "container_title": "JOURNAL OF INVERSE AND ILL-POSED PROBLEMS", 
  "id": "270120", 
  "issued": {
    "date-parts": [
      [
        2023, 
        1, 
        1
      ]
    ]
  }, 
  "page": "18", 
  "title": "Determination of unknown shear force in transverse dynamic force microscopy from measured final data", 
  "type": "article-journal"
}