Figure 1.
Mesh of the domain $ \Omega_{h} \times T_{n} $
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August
2021, 14(8): 2975-2992.
doi: 10.3934/dcdss.2021001
Numerical solutions for a Timoshenko-type system with thermoelasticity with second sound
| 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 |
We consider in this article a nonlinear vibrating Timoshenko system with thermoelasticity with second sound. We first recall the results obtained in [
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 and Continuous Dynamical Systems - S,
2021, 14
(8)
: 2975-2992.
doi: 10.3934/dcdss.2021001
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [19] |
D. M. Serre, Theory and applications, Translated from the 2001 French original. Graduate Texts in Mathematics., 216. Springer-Verlag, New York, 2002. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [19] |
D. M. Serre, Theory and applications, Translated from the 2001 French original. Graduate Texts in Mathematics., 216. Springer-Verlag, New York, 2002. |
| [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. |
| [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. |
| [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. |
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