Inverse problem for the time-fractional Euler-Bernoulli beam equation
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.
Keyword : Euler-Bernoulli beam, inverse coefficient problem, existence and uniqueness
How to Cite
Tekin, I., & Yang, H. (2021). Inverse problem for the time-fractional Euler-Bernoulli beam equation. Mathematical Modelling and Analysis, 26(3), 503-518. https://doi.org/10.3846/mma.2021.13289
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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I. Tekin. Reconstruction of a time-dependent potential in a pseudo-hyperbolic equation. UPB Scientific Bulletin-Series A-Applied Mathematics and Physics, 81:115–124, 2019.
H. Yang. An inverse problem for the sixth-order linear Boussinesq-type equation. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 82(2):27– 36, 2020.
R. Ansari, M. Oskouie, M. Faraji and R. Gholami. Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory. Physica E: Low-Dimensional Systems and Nanostructures, 75:266–271, 2016. https://doi.org/10.1016/j.physe.2015.09.022
K. Cao, D. Lesnic and M.I. Ismailov. Determination of the time-dependent thermal grooving coefficient. Journal of Applied Mathematics and Computing, pp. 1–23, 2020. https://doi.org/10.1007/s12190-020-01388-7
M. Caputo and M. Fabrizio. Damage and fatigue described by a fractional derivative model. Journal of Computational Physics, 293:400–408, 2015. https://doi.org/10.1016/j.jcp.2014.11.012
G.M.L. Gladwell. The inverse problem for the Euler-Bernoulli beam. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 407(1832):199–218, 1986. https://doi.org/10.1098/rspa.1986.0093
M. Abu Hamed and A.A. Nepomnyashchy. Groove growth by surface subdiffusion. Physica D: Nonlinear Phenomena, 298:42–47, 2015. https://doi.org/10.1016/j.physd.2015.02.001
X.Q. He, S. Kitipornchai and K.M. Liew. Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. Journal of the Mechanics and Physics of Solids, 53(2):303–326, 2005. https://doi.org/10.1016/j.jmps.2004.08.003
J. Hristov. Fourth-order fractional diffusion model of thermal grooving: integral approach to approximate closed form solution of the Mullins model. Mathematical Modelling of Natural Phenomena, 13(6):1–14, 2018. https://doi.org/10.1051/mmnp/2017080
F. Kanca and I. Baglan. Inverse problem for Euler-Bernoulli equation with periodic boundary condition. Filomat, 32(16):5691–5705, 2018. https://doi.org/10.2298/FIL1816691K
A.A. Kilbas, M.H. Srivastava and J.J. Trujillo. Theory and applications of fractional differential equations, volume 204. Elsevier, 2006.
M. Klimek. Fractional sequential mechanicsmodels with symmetric fractional derivative. Czechoslovak Journal of Physics, 51(12):1348–1354, 2001. https://doi.org/10.1023/A:1013378221617
D. Lesnic, L. Elliott and D.B. Ingham. Analysis of coefficient identification problems associated to the inverse Euler-Bernoulli beam theory. IMA journal of applied mathematics, 62(2):101–116, 1999. https://doi.org/10.1093/imamat/62.2.101
B.B. Mandelbrot. Geometry of turbulence: Intermittency. The fractal geometry of nature, 173:97–105, 1983.
T.T. Marinov and A.S. Vatsala. Inverse problem for coefficient identification in the Euler–Bernoulli equation. Computers & Mathematics with Applications, 56(2):400–410, 2008. https://doi.org/10.1016/j.camwa.2007.11.048
W.W. Mullins. Theory of thermal grooving. Journal of Applied Physics, 28(3):333–339, 1957. https://doi.org/10.1063/1.1722742
T. Natsuki, Q.Q. Ni and M. Endo. Wave propagation in single-and double-walled carbon nanotubes filled with fluids. Journal of applied physics, 101(3):034319, 2007. https://doi.org/10.1063/1.2432025
A. Pirrotta, S. Cutrona and S. Di Lorenzo. Fractional visco-elastic timoshenko beam from elastic Euler–Bernoulli beam. Acta Mechanica, 226(1):179–189, 2015. https://doi.org/10.1007/s00707-014-1144-y
I. Podlubny. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, 1998.
F. Riewe. Mechanics with fractional derivatives. Physical Review E, 55(3):3581, 1997. https://doi.org/10.1103/PhysRevE.55.3581
E. Sousa. How to approximate the fractional derivative of order 1 < α ≤ 2. International Journal of Bifurcation and Chaos, 22(04):1250075, 2012. https://doi.org/10.1142/S0218127412500757
W. Sumelka, T. Blaszczyk and C. Liebold. Fractional Euler– Bernoulli beams: Theory, numerical study and experimental validation. European Journal of Mechanics-A/Solids, 54:243–251, 2015. https://doi.org/10.1016/j.euromechsol.2015.07.002
W. Sumelka and G.Z. Voyiadjis. A hyperelastic fractional damage material model with memory. International Journal of Solids and Structures, 124:151–160, 2017. https://doi.org/10.1016/j.ijsolstr.2017.06.024
W. Sumelka, R. Zaera and J. Fernández-Sáez. A theoretical analysis of the free axial vibration of non-local rods with fractional continuum mechanics. Meccanica, 50(9):2309–2323, 2015. https://doi.org/10.1007/s11012-015-0157-5
I. Tekin. Reconstruction of a time-dependent potential in a pseudo-hyperbolic equation. UPB Scientific Bulletin-Series A-Applied Mathematics and Physics, 81:115–124, 2019.
H. Yang. An inverse problem for the sixth-order linear Boussinesq-type equation. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 82(2):27– 36, 2020.