This paper presents a numerical method for Rayleigh beams with variable coefficients and indefinite damping under a force control in moment and velocity. The system is discretized using the Galerkin method with cubic Hermite basis functions. We are interested in the effect of the damper on the stability of the beam and also focused on the numerical study of the system’s spectrum. The convergence rate in the norm is of order two for both space and time. Numerical computations are carried out to support the theoretical findings.

Rayleigh beam, variable coefficients, Galerkin approximation, finite elements, a priori estimates, viscous damping

Received: January 4, 2025; Accepted: February 11, 2025; Published: April 14, 2025

How to cite this article: Yapi Serge Alain Joresse, Kouassi Ayo Ayébié Hermith, Diop Fatou N. and Touré Kidjégbo Augustin, Numerical method for variable coefficients Rayleigh beams with indefinite damping under a force control in moment and velocity, International Journal of Numerical Methods and Applications 25(2) (2025), 251-279. https://doi.org/10.17654/0975045225011

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References:
[1] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Springer, New York, 2008.
[2] M. Bzeih, T. El Arwadi and M. Hindi, Numerical analysis and simulation for Rayleigh beam equation with dynamical boundary controls, Arab. J. Math. 10 (2021), 331-349. https://doi.org/10.1007/s40065-021-00310-8.
[3] S. M. Choo, S. K. Chung and R. Kannan, Finite element Galerkin solutions for the strongly damped extensible beam equations, Korean Journal of Computational and Applied Mathematics 9(1) (2002), 27-43.
[4] B. Z. Guo, Basis property of a Rayleigh beam with boundary stabilization, J. Optim. Theory Appl. 112(3) (2002), 529-547.
[5] B. G. Jean-Marc, Y. S. A. Joresse and T. Augustin, Numerical approximation of the dissipativity of energy and spectrum for a damped Euler-Bernoulli beam with variable coefficients, J. Nonlinear Sci. Appl. 16 (2023), 123-144.
[6] Y. S. A. Joresse, B. G. Jean-Marc, Y. Gozo and T. K. Augustin, A numerical study of a homogeneous beam with a tip mass, Advances in Mathematics: Scientific Journal 10(6) (2021), 2731-2753.
[7] B. P. Rao, Optimal energy decay rate in a damped Rayleigh beam, Discrete Contin. Dyn. Syst. 4 (1998), 721-734.
[8] D. L. Russell, Mathematical models for the elastic beam and their control theoretic implications, Semigroups, Theory and Applications, H. Brezis, M. G. Crandell and F. Kapell, eds., Vol. II, Longman Scientific and Technical, Harlow, 1986, pp. 177-216.
[9] G. Strang and G. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, N.J., 1973.
[10] Touré Kidjégbo, Adama Cherif, Hermith Kouassi and J. Adou, Exponential stability of variable coefficients Rayleigh beams with indefinite under a force control in moment and velocity, Far East J. Math. Sci. (FJMS) 99(7) (2016), 993-1020. 10.17654/MS099070993.
[11] J. M. Wang, G. Q. Xu and S. P. Yung, Exponential stability of variable coefficients Rayleigh beams under boundary feedback control: a Riesz basis approach, Systems Control Lett. 51 (2004), 33-50.
[12] J. M. Wang and S. P. Yung, Stability of a nonuniform Rayleigh beam with indefinite damping, Systems Control Lett. 55 (2006), 863-870.