Abstract
In this paper, the classical Euler-Bernoulli beam equation is considered by utilizing fractional calculus. Such an equation is called the time-fractional EulerBernoulli beam equation. The problem of determining the time-dependent coefficient for the fractional Euler-Bernoulli beam equation with homogeneous boundary conditions and an additional measurement is considered, and the existence and uniqueness theorem of the solution is proved by means of the contraction principle on a sufficiently small time interval. Numerical experiments are also provided to verify the theoretical findings.
Highlights
Fractional derivatives and integrals are very important in many areas of mathematics, physics and nano-engineering [11,13,19], and are widely used to capture natural and physical phenomena which cannot be predicted by classical integral and differential models.Consider a partial differential equation (PDE) with a fractional derivative in time t∂tαw(x, t) + wxxxx(x, t) = F (x, t; a, w), (x, t) ∈ DT, (1.1)Copyright c 2021 The Author(s)
We focus on the timefractional Euler-Bernoulli beam equation which is a more general model than the classical Euler-Bernoulli beam equation in the aforementioned work
The paper considers the problem of determining the time-dependent coefficient for the fractional Euler-Bernoulli equation with homogeneous boundary conditions and an additional measurement
Summary
Fractional derivatives and integrals are very important in many areas of mathematics, physics and nano-engineering [11,13,19], and are widely used to capture natural and physical phenomena which cannot be predicted by classical integral and differential models. The following papers are some important studies of the inverse problems for the classical Euler-Bernoulli equation. The inverse problem for the determination of the time-dependent force function in the Euler-Bernoulli beam equation, with the periodic boundary condition and an additional integral condition, was investigated in [9]. The inverse problem for finding the time-dependent potential for the Euler-Bernoulli beam equation was studied in [24]. The authors in [9] proposed a direct numerical method based on the finite difference method, to solve an inverse problem for the Euler-Bernoulli equation with a periodic boundary condition. The objective of this paper is to determine the time-dependent coefficient in the fractional Euler-Bernoulli equation using the additional information in (1.4), and prove the existence and uniqueness theorem for small T by means of the contraction principle.
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