1 Ocak 2023 Dergi makalesi Açık Erişim

Baysal, Onur; Hasanov, Alemdar; Kumarasamy, Sakthivel

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    <subfield code="a">&lt;p&gt;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&lt;/p&gt;
&lt;p&gt;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),&lt;/p&gt;
&lt;p&gt;subject to the homogeneous initial conditions and the boundary conditions&lt;/p&gt;
&lt;p&gt;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),&lt;/p&gt;
&lt;p&gt;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&lt;/p&gt;
&lt;p&gt;J(F) = 1/2 parallel to Phi g - u(T)parallel to(2)(L2(0,l))&lt;/p&gt;
&lt;p&gt;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.&lt;/p&gt;</subfield>
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    <subfield code="a">Baysal, Onur</subfield>
    <subfield code="u">Univ Malta, Dept Math, Msida, Malta</subfield>
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    <subfield code="a">10.1515/jiip-2023-0021</subfield>
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    <subfield code="a">Determination of unknown shear force in transverse dynamic force microscopy from measured final data</subfield>
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    <subfield code="a">Hasanov, Alemdar</subfield>
    <subfield code="u">Kocaeli Univ, Dept Math, Altunsehir Str,Ayazma Villalari,22 Bahcecik Basisk, TR-41030 Izmit, Kocaeli, Turkiye</subfield>
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    <subfield code="a">Kumarasamy, Sakthivel</subfield>
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