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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} $
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