American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Some new bounds analogous to generalized proportional fractional integral operator with respect to another function
  • DCDS-S Home
  • This Issue
  • Next Article
    The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data
doi: 10.3934/dcdss.2021001

Numerical solutions for a Timoshenko-type system with thermoelasticity with second sound

Makram Hamouda 1,, , Ahmed Bchatnia 2, and  Mohamed Ali Ayadi 3,

1. 

Department of Basic Sciences, Deanship of Preparatory Year and Supporting Studies, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 34212, Saudi Arabia
2. 

UR ANALYSE NON-LINÉAIRE ET GÉOMETRIE, UR13ES32 Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El-Manar 2092 El Manar II, Tunisia
3. 

UR ANALYSE NON-LINÉAIRE ET GÉOMETRIE, UR13ES32 ESPRIT School of Engineering. 1, 2 rue André Ampère 2083 - Pôle Technologique, El Ghazala

* Corresponding author: Makram Hamouda

Dedicated to the memory of Professor Ezzedine Zahrouni

Received  June 2020 Revised  November 2020 Published  January 2021

We consider in this article a nonlinear vibrating Timoshenko system with thermoelasticity with second sound. We first recall the results obtained in [2] concerning the well-posedness, the regularity of the solutions and the asymptotic behavior of the associated energy. Then, we use a fourth-order finite difference scheme to compute the numerical solutions and we prove its convergence. The energy decay in several cases, depending on the stability number $ \mu $, are numerically and theoretically studied.

Keywords: Asymptotic behavior of solutions, asymptotic stability, finite difference methods, stability and convergence of numerical methods, thermoelasticity, Timoshenko system.
Mathematics Subject Classification: Primary:65M06, 65L20, 35B35, 35B40, 93D20.
Citation: Makram Hamouda, Ahmed Bchatnia, Mohamed Ali Ayadi. Numerical solutions for a Timoshenko-type system with thermoelasticity with second sound. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021001
References:
[1]

K. Ammari, A. Bchatnia and K. El Mufti, Non-uniform decay of the energy of some dissipative evolution systems, Z. Anal. Anwend., 36 (2017), 239-251.  doi: 10.4171/ZAA/1587.  Google Scholar

[2]

M. A. Ayadi, A. Bchatnia, M. Hamouda and S. Messaoudi, General decay in a Timoshenko-type system with thermoelasticity with second sound, Adv. Nonlinear Anal., 4 (2015), 263-284.  doi: 10.1515/anona-2015-0038.  Google Scholar

[3]

A. Bchatnia, S. Chebbi, M. Hamouda and A. Soufyane, Lower bound and optimality for a nonlinearly damped Timoshenko system with thermoelasticity, Asymptot. Anal., 114 (2019), 73-91.  doi: 10.3233/ASY-191519.  Google Scholar

[4]

S. Chebbi and M. Hamouda, Discrete energy behavior of a damped Timoshenko system, Comput. Appl. Math., 39 (2020), Paper No. 4, 19 pp. doi: 10.1007/s40314-019-0982-6.  Google Scholar

[5]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194 (2009), 221-251.  doi: 10.1007/s00205-009-0220-2.  Google Scholar

[6]

Z. Gao and S. Xie, Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations, Appl. Numer. Math., 61 (2011), 593-614.  doi: 10.1016/j.apnum.2010.12.004.  Google Scholar

[7]

A. Guesmia and S. A. Messaoudi, General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping, Math. Meth. Appl. Sci., 32 (2009), 2102-2122.  doi: 10.1002/mma.1125.  Google Scholar

[8]

A. Guesmia and and S. A. Messaoudi, On the control of a viscoelastic damped Timoshenko-type system, Appl. Math. Compt., 206 (2008), 589-597.  doi: 10.1016/j.amc.2008.05.122.  Google Scholar

[9]

M. S. Ismail and F. Mosally, A fourth order finite difference method for the good Boussinesq equation, Abstr. Appl. Anal., (2014), Art. ID 323260, 10 pp. doi: 10.1155/2014/323260.  Google Scholar

[10]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429.  doi: 10.1137/0325078.  Google Scholar

[11]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[12]

S. A. Messaoudi and M. I. Mustafa, On the stabilization of the Timoshenko system by a weak nonlinear dissipation, Math. Meth. Appl. Sci., 32 (2009), 454-469.  doi: 10.1002/mma.1047.  Google Scholar

[13]

S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear damped Timoshenko systems with second sound–global existence and exponential stability, Math. Meth. Appl. Sci., 32 (2009), 505-534.  doi: 10.1002/mma.1049.  Google Scholar

[14]

S. A. Messaoudi and A. Soufyane, Boundary stabilization of solutions of a nonlinear system of Timoshenko type, Nonlinear Anal., 67 (2007), 2107-2121.  doi: 10.1016/j.na.2006.08.039.  Google Scholar

[15]

J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278.  doi: 10.1016/S0022-247X(02)00436-5.  Google Scholar

[16]

B. V. Numerov, A method of extrapolation of perturbations, Monthly Notices of the Royal Astronomical Society., 84 (1924), 592-601.  doi: 10.1093/mnras/84.8.592.  Google Scholar

[17]

B. V. Numerov, Note on the numerical integration of $d^2x/dt^2 = f(x,t)$, Astronomische Nachrichten, 230 (1927), 359-364.  doi: 10.1002/asna.19272301903.  Google Scholar

[18]

C. A. Raposo, J. A. D. Chuquipoma, J. A. J. Avila and M. L. Santos, Exponential decay and numerical solution for a Timoshenko system with delay term in the internal feedback, International Journal of Analysis and Applications., 3 (2013), 1-13.   Google Scholar

[19]

D. M. Serre, Theory and applications, Translated from the 2001 French original. Graduate Texts in Mathematics., 216. Springer-Verlag, New York, 2002.  Google Scholar

[20]

A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differential Equations., (2003), No. 29, 14 pp.  Google Scholar

[21]

B. Wang, T. Sun and D. Liang, The conservative and fourth-order compact finite difference schemes for regularized long wave equation, J. Comput. Appl. Math., 356 (2019), 98-117.  doi: 10.1016/j.cam.2019.01.036.  Google Scholar

[22]

E. Zauderer, Partial Differential Equations of Applied Mathematics, Pure and Applied Mathematics (New York), Wiley-Interscience, John Wiley & Sons, Hoboken, NJ., 2006. doi: 10.1002/9781118033302.  Google Scholar

show all references

References:
[1]

K. Ammari, A. Bchatnia and K. El Mufti, Non-uniform decay of the energy of some dissipative evolution systems, Z. Anal. Anwend., 36 (2017), 239-251.  doi: 10.4171/ZAA/1587.  Google Scholar

[2]

M. A. Ayadi, A. Bchatnia, M. Hamouda and S. Messaoudi, General decay in a Timoshenko-type system with thermoelasticity with second sound, Adv. Nonlinear Anal., 4 (2015), 263-284.  doi: 10.1515/anona-2015-0038.  Google Scholar

[3]

A. Bchatnia, S. Chebbi, M. Hamouda and A. Soufyane, Lower bound and optimality for a nonlinearly damped Timoshenko system with thermoelasticity, Asymptot. Anal., 114 (2019), 73-91.  doi: 10.3233/ASY-191519.  Google Scholar

[4]

S. Chebbi and M. Hamouda, Discrete energy behavior of a damped Timoshenko system, Comput. Appl. Math., 39 (2020), Paper No. 4, 19 pp. doi: 10.1007/s40314-019-0982-6.  Google Scholar

[5]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194 (2009), 221-251.  doi: 10.1007/s00205-009-0220-2.  Google Scholar

[6]

Z. Gao and S. Xie, Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations, Appl. Numer. Math., 61 (2011), 593-614.  doi: 10.1016/j.apnum.2010.12.004.  Google Scholar

[7]

A. Guesmia and S. A. Messaoudi, General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping, Math. Meth. Appl. Sci., 32 (2009), 2102-2122.  doi: 10.1002/mma.1125.  Google Scholar

[8]

A. Guesmia and and S. A. Messaoudi, On the control of a viscoelastic damped Timoshenko-type system, Appl. Math. Compt., 206 (2008), 589-597.  doi: 10.1016/j.amc.2008.05.122.  Google Scholar

[9]

M. S. Ismail and F. Mosally, A fourth order finite difference method for the good Boussinesq equation, Abstr. Appl. Anal., (2014), Art. ID 323260, 10 pp. doi: 10.1155/2014/323260.  Google Scholar

[10]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429.  doi: 10.1137/0325078.  Google Scholar

[11]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[12]

S. A. Messaoudi and M. I. Mustafa, On the stabilization of the Timoshenko system by a weak nonlinear dissipation, Math. Meth. Appl. Sci., 32 (2009), 454-469.  doi: 10.1002/mma.1047.  Google Scholar

[13]

S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear damped Timoshenko systems with second sound–global existence and exponential stability, Math. Meth. Appl. Sci., 32 (2009), 505-534.  doi: 10.1002/mma.1049.  Google Scholar

[14]

S. A. Messaoudi and A. Soufyane, Boundary stabilization of solutions of a nonlinear system of Timoshenko type, Nonlinear Anal., 67 (2007), 2107-2121.  doi: 10.1016/j.na.2006.08.039.  Google Scholar

[15]

J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278.  doi: 10.1016/S0022-247X(02)00436-5.  Google Scholar

[16]

B. V. Numerov, A method of extrapolation of perturbations, Monthly Notices of the Royal Astronomical Society., 84 (1924), 592-601.  doi: 10.1093/mnras/84.8.592.  Google Scholar

[17]

B. V. Numerov, Note on the numerical integration of $d^2x/dt^2 = f(x,t)$, Astronomische Nachrichten, 230 (1927), 359-364.  doi: 10.1002/asna.19272301903.  Google Scholar

[18]

C. A. Raposo, J. A. D. Chuquipoma, J. A. J. Avila and M. L. Santos, Exponential decay and numerical solution for a Timoshenko system with delay term in the internal feedback, International Journal of Analysis and Applications., 3 (2013), 1-13.   Google Scholar

[19]

D. M. Serre, Theory and applications, Translated from the 2001 French original. Graduate Texts in Mathematics., 216. Springer-Verlag, New York, 2002.  Google Scholar

[20]

A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differential Equations., (2003), No. 29, 14 pp.  Google Scholar

[21]

B. Wang, T. Sun and D. Liang, The conservative and fourth-order compact finite difference schemes for regularized long wave equation, J. Comput. Appl. Math., 356 (2019), 98-117.  doi: 10.1016/j.cam.2019.01.036.  Google Scholar

[22]

E. Zauderer, Partial Differential Equations of Applied Mathematics, Pure and Applied Mathematics (New York), Wiley-Interscience, John Wiley & Sons, Hoboken, NJ., 2006. doi: 10.1002/9781118033302.  Google Scholar

Figure 1.  Mesh of the domain $ \Omega_{h} \times T_{n} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.   
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.   
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.   
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423
[2]

Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago. Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 343-362. doi: 10.3934/dcdss.2016.9.343
[3]

Qingguang Guan, Max Gunzburger. Stability and convergence of time-stepping methods for a nonlocal model for diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1315-1335. doi: 10.3934/dcdsb.2015.20.1315
[4]

Jitraj Saha, Nilima Das, Jitendra Kumar, Andreas Bück. Numerical solutions for multidimensional fragmentation problems using finite volume methods. Kinetic & Related Models, 2019, 12 (1) : 79-103. doi: 10.3934/krm.2019004
[5]

Houda Hani, Moez Khenissi. Asymptotic behaviors of solutions for finite difference analogue of the Chipot-Weissler equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1421-1445. doi: 10.3934/dcdss.2016057
[6]

Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269
[7]

Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561
[8]

Yong Liu. Even solutions of the Toda system with prescribed asymptotic behavior. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1779-1790. doi: 10.3934/cpaa.2011.10.1779
[9]

Yongqin Liu, Shuichi Kawashima. Asymptotic behavior of solutions to a model system of a radiating gas. Communications on Pure & Applied Analysis, 2011, 10 (1) : 209-223. doi: 10.3934/cpaa.2011.10.209
[10]

Zhong Tan, Leilei Tong. Asymptotic stability of stationary solutions for magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3435-3465. doi: 10.3934/dcds.2017146
[11]

Kun Wang, Yinnian He, Yanping Lin. Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1551-1573. doi: 10.3934/dcdsb.2012.17.1551
[12]

Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods. Kinetic & Related Models, 2017, 10 (1) : 61-91. doi: 10.3934/krm.2017003
[13]

Jeffrey R. Haack, Cory D. Hauck. Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation. Kinetic & Related Models, 2008, 1 (4) : 573-590. doi: 10.3934/krm.2008.1.573
[14]

Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105
[15]

Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619
[16]

Ramon Quintanilla, Reinhard Racke. Stability in thermoelasticity of type III. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 383-400. doi: 10.3934/dcdsb.2003.3.383
[17]

Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155
[18]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[19]

Jiangshan Wang, Lingxiong Meng, Hongen Jia. Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model. Electronic Research Archive, 2020, 28 (3) : 1191-1205. doi: 10.3934/era.2020065
[20]

Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (37)
  • HTML views (183)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences