Affiliation:
1. Department of Mathematics , University of Malta , Msida , Malta
2. Department of Mathematics , Kocaeli University , Izmit - Kocaeli ; and Şehit Ekrem Dsitrict, Altunşehir Str., Ayazma Villalari, No: 22. Bahčecik - Başiskele, Kocaeli, 41030 Türkiye
3. Department of Mathematics , Indian Institute of Space Science and Technology (IIST) , Trivandrum , India
Abstract
Abstract
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
)
{g(t)}
acting on the inaccessible boundary
x
=
ℓ
{x=\ell}
in a system governed by the variable coefficient Euler–Bernoulli equation
ρ
A
(
x
)
u
t
t
+
μ
(
x
)
u
t
+
(
r
(
x
)
u
x
x
+
κ
(
x
)
u
x
x
t
)
x
x
=
0
,
(
x
,
t
)
∈
(
0
,
ℓ
)
×
(
0
,
T
)
,
\rho_{A}(x)u_{tt}+\mu(x)u_{t}+(r(x)u_{xx}+\kappa(x)u_{xxt})_{xx}=0,\quad(x,t)%
\in(0,\ell)\times(0,T),
subject to the homogeneous initial conditions and the boundary conditions
u
(
0
,
t
)
=
u
0
(
t
)
,
u
x
(
0
,
t
)
=
0
,
(
u
x
x
(
x
,
t
)
+
κ
(
x
)
u
x
x
t
)
x
=
ℓ
=
0
,
(
-
(
r
(
x
)
u
x
x
+
κ
(
x
)
u
x
x
t
)
x
)
x
=
ℓ
=
g
(
t
)
,
u(0,t)=u_{0}(t),\quad u_{x}(0,t)=0,\quad(u_{xx}(x,t)+\kappa(x)u_{xxt})_{x=\ell%
}=0,\quad\bigl{(}-(r(x)u_{xx}+\kappa(x)u_{xxt})_{x}\bigr{)}_{x=\ell}=g(t),
from the final time measured output (displacement)
u
T
(
x
)
:=
u
(
x
,
T
)
{u_{T}(x):=u(x,T)}
. We introduce the input-output map
(
Φ
g
)
(
x
)
:=
u
(
x
,
T
;
g
)
{(\Phi g)(x):=u(x,T;g)}
,
g
∈
𝒢
{g\in\mathcal{G}}
, and prove that it is a compact and Lipschitz continuous linear operator. Then we introduce the Tikhonov functional
J
(
F
)
=
1
2
∥
Φ
g
-
u
T
∥
L
2
(
0
,
ℓ
)
2
J(F)=\frac{1}{2}\lVert\Phi g-u_{T}\rVert_{L^{2}(0,\ell)}^{2}
and prove the existence of a quasi-solution of the inverse problem. We derive a gradient formula for the Fréchet 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.
Funder
National Board for Higher Mathematics
Cited by
1 articles.
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