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

* Corresponding author: Makram Hamouda

Dedicated to the memory of Professor Ezzedine Zahrouni

Received  June 2020 Revised  November 2020 Published  August 2021 Early access  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.

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
References:
[1]

K. AmmariA. 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. AyadiA. BchatniaM. 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. BchatniaS. ChebbiM. 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. MessaoudiM. 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. RaposoJ. A. D. ChuquipomaJ. 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. WangT. 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. AmmariA. 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. AyadiA. BchatniaM. 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. BchatniaS. ChebbiM. 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. MessaoudiM. 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. RaposoJ. A. D. ChuquipomaJ. 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. WangT. 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.

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