Research Article
BibTex RIS Cite

Year 2021, , 159 - 179, 04.02.2021

Ömer Oruç , Alaattin Esen , Fatih Bulut

https://doi.org/10.15672/hujms.610834
Cited By: 5

Abstract

References

  • [1] C. Andrade and S. McKee, High accuracy A.D.I. methods for fourth-order parabolic equations with variable coefficients, J. Comput. Appl. Math. 3 (1), 11–14, 1977.
  • [2] T. Aziz, A. Khan and J. Rashidinia, Spline methods for the solution of fourth-order parabolic partial differential equations, Appl. Math. Comput. 167, 153–166, 2005.
  • [3] H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Sys- tems, Birkhauser, Boston, 1989.
  • [4] H. Caglar and N. Caglar, Fifth-degree B-spline solution for a fourth-order parabolic partial differential equations, Appl. Math. Comput. 201, 597–603, 2008.
  • [5] C. Chen and C.H. Hsiao, Haar wavelet method for solving lumped and distributed parameter systems, IEE Proc. Control Theory Appl. 144, 87–94, 1997.
  • [6] C. Chen and C.H. Hsiao, Wavelet approach to optimising dynamic systems, IEE Proc. Control Theory Appl. 146, 213–219, 1997.
  • [7] L. Collatz, Hermitian methods for initial value problems in partial differential equa- tions, in: J.J.H. Miller (Ed.), Topics in Numerical Analysis, Academic Press, New York, 41–61, 1973.
  • [8] S.D. Conte, A stable implicit finite difference approximation to a fourth order parabolic equation, J. Assoc. Comput. Mech. 4, 18–23, 1957.
  • [9] S.H. Crandall, Numerical treatment of a fourth order partial differential equations, J. Assoc. Comput. Mech. 1, 111–118, 1954.
  • [10] A. Danaee, Arshad Khan, Islam Khan, Tariq Aziz and D.J. Evans, Hopscotch proce- dure for a fourth-order parabolic partial differential equation, Math. Comput. Simulat. XXIV, 326–329, 1982.
  • [11] D.J. Evans, A stable explicit method for the finite difference solution of a fourth order parabolic partial differential equation, Comput. J. 8, 280–287, 1965.
  • [12] D.J. Evans and W.S.Yousif, A note on solving the fourth order parabolic equation by the age method, Int. J. Comput. Math. 40, 93–97, 1991.
  • [13] G. Fairweather and A.R. Gourlay, Some stable difference approximations to a fourth order parabolic partial differential equation, Math. Comput. 21, 1–11, 1967.
  • [14] H. Haddadpour, An exact solution for variable coefficients fourth-order wave equation using the Adomian method, Math. Comput. Model. 44, 144–1152, 2006.
  • [15] C.H. Hsiao, Haar wavelet direct method for solving variational problems, Math. Com- put. Simul. 64, 569–585, 2004.
  • [16] C.H. Hsiao and W.J. Wang, State analysis of time-varying singular nonlinear systems via Haar wavelets Math. Comput. Simul. 51, 91–100, 1999.
  • [17] C.H. Hsiao and W.J. Wang, State analysis of time-varying singular bilinear systems via Haar wavelets, Math. Comput. Simul. 52, 11–20, 2000.
  • [18] C.H. Hsiao and W.J. Wang, Haar wavelet approach to nonlinear stiff systems, Math. Comput. Simul. 57, 347–353, 2001.
  • [19] J.D. Hunter, Matplotlib: A 2D graphics environment, Comput Sci Eng, 9 (3), 90–95, 2007.
  • [20] M.K. Jain, S.R.K. Iyengar and A.G. Lone, Higher order difference formulas for a fourth order parabolic partial differential equation, Int. J. Numer. Methods Eng. 10, 1357–1367, 1976.
  • [21] R. Jiwari, Haar wavelet quasilinearization approach for numerical simulation of Burg- ers’ equation, Comput. Phys. Commun. 183, 2413–2423, 2012.
  • [22] R. Jiwari, A hybrid numerical scheme for the numerical solution of the Burgers’ equation, Comput. Phys. Commun. 188, 59–67 2015.
  • [23] R. Jiwari, V. Kumar, R. Karan and A. S. Alshomrani, Haar wavelet quasilinearization approach for MHD Falkner–Skan flow over permeable wall via Lie group method, Int. J. Numer. Method H. 27 (6), 1332–1350, 2017.
  • [24] H. Kaur, R.C. Mittal and V. Mishra, Haar wavelet approximate solutions for the generalized Lane–Emden equations arising in astrophysics, Comput. Phys. Commun. 184, 2169–2177, 2013.
  • [25] A.Q.M. Khaliq and E.H. Twizell, A family of second order methods for variable coef- ficient fourth order parabolic partial differential equations, Int. J. Comput. Math. 23, 63–76, 1987.
  • [26] A. Khan, I. Khan, and T. Aziz, Sextic spline solution for solving a fourth-order parabolic partial differential equation, Int. J. Comput. Math. 82 (7), 871–879, 2005.
  • [27] M. Kirs, M. Mikola, A. Haavajõe, E. Õunapuu, B. Shvartsman and J. Majak, Haar wavelet method for vibration analysis of nanobeams, Waves Wavelets Fractals, 2 (1), 2016.
  • [28] K. Kunisch and E. Graif, Parameter estimation for the Euler–Bernoulli beam, Mat. Apficada Comput. 4, 95–124, 1985.
  • [29] U. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul. 68, 127–143, 2005.
  • [30] U. Lepik, Numerical solution of evolution equations by the Haar wavelet method, Appl. Math. Comput. 185, 695–704, 2007.
  • [31] U. Lepik, Solving PDEs with the aid of two-dimensional Haar wavelets, Comput. Math. with Appl. 61, 1873–1879, 2011.
  • [32] Y. Liu and C.S. Gurram, The use of He’s variational iteration method for obtaining the free vibration of an Euler–Bernoulli beam, Math. Comput. Model. 50, 1545–1552, 2009.
  • [33] J. Majak, B. Shvartsman, M. Kirs, M. Pohlak and H. Herranen, Convergence theorem for the Haar wavelet based discretization method, Compos. Struct. 126, 227–232, 2015.
  • [34] J. Majak, M. Pohlak, K. Karjust, M. Eerme, J. Kurnitski and B.S. Shvartsman, New higher order Haar wavelet method: Application to FGM structures, 201, 72–78, 2018. https://doi.org/10.1016/j.compstruct.2018.06.013.
  • [35] J. Majak, B. Shvartsman, K. Karjust, M. Mikola, A. Haavajõe and M. Pohlak, On the accuracy of the Haar wavelet discretization method, Compos. B. Eng. 80, 321–327, 2015.
  • [36] R.C. Mittal and R.K. Jain, B-splines methods with redefined basis functions for solving fourth order parabolic partial differential equations, Appl. Math. Comput. 217, 9741– 9755, 2011.
  • [37] R.C. Mittal and S. Pandit, Sensitivity analysis of shock wave Burgers’ equation via a novel algorithm based on scale-3 Haar wavelets, Int. J. Comput. Math. 95 (3), 601–625, 2017.
  • [38] R.C. Mittal and S. Pandit, A Numerical Algorithm to Capture Spin Patterns of Frac- tional Bloch NMR Flow Models, J. Comput. Nonlinear Dynam. 14 (8), 2019.
  • [39] R.C. Mittal and S. Pandit, New Scale-3 Haar Wavelets Algorithm for Numerical Simulation of Second Order Ordinary Differential Equations, P. Natl. A. Sci. India A, 89, 799–808, 2019.
  • [40] R.C. Mittal and S. Pandit, Quasilinearized Scale-3 Haar Wavelets based Algorithm for Numerical Simulation of Fractional Dynamical System, Eng. Computations. 35 (5), 1907–1931, 2018.
  • [41] R. Mohammadi, Sextic B-spline collocation method for solving Euler–Bernoulli Beam Models, Appl. Math. Comput. 241, 151–166, 2014.
  • [42] Ö. Oruç, F. Bulut and A. Esen, A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers equation, J. Math. Chem. 53 (7), 1592–1607, 2015.
  • [43] Ö. Oruç, F. Bulut and A. Esen, Numerical Solutions of Regularized Long Wave Equa- tion By Haar Wavelet Method, Mediterr. J. Math. 13 (5), 3235–3253, 2016.
  • [44] Ö. Oruç, A. Esen and F. Bulut, A Haar wavelet collocation method for coupled non- linear Schrödinger–KdV equations, Int. J. Mod. Phys. C, 27 (9), 2016.
  • [45] S. Pandit, M. Kumar and S. Tiwari, Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients, Comput. Phys. Commun. 187, 83– 90, 2015.
  • [46] J. Rashidinia and R. Mohammadi, Sextic spline solution of variable coefficient fourth- order parabolic equations, Int. J. Comput. Math. 87 (15), 3443–3454, 2010.
  • [47] R.D. Richtmyer and K.W. Morton, Difference methods for Initial value Problems, second ed., John Wiley & Sons, 1967.
  • [48] Z. Shi, Y. Cao and Q.J. Chen, Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method, Appl. Math. Model. 36, 5143–5161, 2012.
  • [49] R.C. Smith, K.L. Bowers and J. Lund, A fully Sinc–Galerkin method for Euler– Bernoulli Beam Models, Numer. Methods Partial Diff. Equ. 8, 171–202, 1992.
  • [50] S.P. Timoshenko and J.M. Gere, Theory of Elastic Stability, McGraw-Hill, New York, 1961.
  • [51] J. Todd, A direct approach to the problem of stability in the numerical solution of partial differential equations, Commun. Pure Appl. Math. 9, 597–612, 1956.
  • [52] A.M.Wazwaz, Analytic treatment for variable coefficient fourth-order parabolic partial differential equations, Appl. Math. Comput. 123, 219–227, 2001.

Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method

Year 2021, , 159 - 179, 04.02.2021

Ömer Oruç , Alaattin Esen , Fatih Bulut

https://doi.org/10.15672/hujms.610834
Cited By: 5

Abstract

In this study, we analyze the performance of a numerical scheme based on 3-scale Haar wavelets for dynamic Euler-Bernoulli equation, which is a fourth order time dependent partial differential equation. This type of equations governs the behaviour of a vibrating beam and have many applications in elasticity. For its solution, we first rewrite the fourth order time dependent partial differential equation as a system of partial differential equations by introducing a new variable, and then use finite difference approximations to discretize in time, as well as 3-scale Haar wavelets to discretize in space. By doing so, we obtain a system of algebraic equations whose solution gives wavelet coefficients for constructing the numerical solution of the partial differential equation. To test the accuracy and reliability of the numerical scheme based on 3-scale Haar wavelets, we apply it to five test problems including variable and constant coefficient, as well as homogeneous and non-homogeneous partial differential equations. The obtained results are compared wherever possible with those from previous studies. Numerical results are tabulated and depicted graphically. In the applications of the proposed method, we achieve high accuracy even with small number of collocation points. 

Keywords

3-Scale Haar wavelets Vibrating beam Dynamic Euler-Bernoulli equation

References

  • [1] C. Andrade and S. McKee, High accuracy A.D.I. methods for fourth-order parabolic equations with variable coefficients, J. Comput. Appl. Math. 3 (1), 11–14, 1977.
  • [2] T. Aziz, A. Khan and J. Rashidinia, Spline methods for the solution of fourth-order parabolic partial differential equations, Appl. Math. Comput. 167, 153–166, 2005.
  • [3] H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Sys- tems, Birkhauser, Boston, 1989.
  • [4] H. Caglar and N. Caglar, Fifth-degree B-spline solution for a fourth-order parabolic partial differential equations, Appl. Math. Comput. 201, 597–603, 2008.
  • [5] C. Chen and C.H. Hsiao, Haar wavelet method for solving lumped and distributed parameter systems, IEE Proc. Control Theory Appl. 144, 87–94, 1997.
  • [6] C. Chen and C.H. Hsiao, Wavelet approach to optimising dynamic systems, IEE Proc. Control Theory Appl. 146, 213–219, 1997.
  • [7] L. Collatz, Hermitian methods for initial value problems in partial differential equa- tions, in: J.J.H. Miller (Ed.), Topics in Numerical Analysis, Academic Press, New York, 41–61, 1973.
  • [8] S.D. Conte, A stable implicit finite difference approximation to a fourth order parabolic equation, J. Assoc. Comput. Mech. 4, 18–23, 1957.
  • [9] S.H. Crandall, Numerical treatment of a fourth order partial differential equations, J. Assoc. Comput. Mech. 1, 111–118, 1954.
  • [10] A. Danaee, Arshad Khan, Islam Khan, Tariq Aziz and D.J. Evans, Hopscotch proce- dure for a fourth-order parabolic partial differential equation, Math. Comput. Simulat. XXIV, 326–329, 1982.
  • [11] D.J. Evans, A stable explicit method for the finite difference solution of a fourth order parabolic partial differential equation, Comput. J. 8, 280–287, 1965.
  • [12] D.J. Evans and W.S.Yousif, A note on solving the fourth order parabolic equation by the age method, Int. J. Comput. Math. 40, 93–97, 1991.
  • [13] G. Fairweather and A.R. Gourlay, Some stable difference approximations to a fourth order parabolic partial differential equation, Math. Comput. 21, 1–11, 1967.
  • [14] H. Haddadpour, An exact solution for variable coefficients fourth-order wave equation using the Adomian method, Math. Comput. Model. 44, 144–1152, 2006.
  • [15] C.H. Hsiao, Haar wavelet direct method for solving variational problems, Math. Com- put. Simul. 64, 569–585, 2004.
  • [16] C.H. Hsiao and W.J. Wang, State analysis of time-varying singular nonlinear systems via Haar wavelets Math. Comput. Simul. 51, 91–100, 1999.
  • [17] C.H. Hsiao and W.J. Wang, State analysis of time-varying singular bilinear systems via Haar wavelets, Math. Comput. Simul. 52, 11–20, 2000.
  • [18] C.H. Hsiao and W.J. Wang, Haar wavelet approach to nonlinear stiff systems, Math. Comput. Simul. 57, 347–353, 2001.
  • [19] J.D. Hunter, Matplotlib: A 2D graphics environment, Comput Sci Eng, 9 (3), 90–95, 2007.
  • [20] M.K. Jain, S.R.K. Iyengar and A.G. Lone, Higher order difference formulas for a fourth order parabolic partial differential equation, Int. J. Numer. Methods Eng. 10, 1357–1367, 1976.
  • [21] R. Jiwari, Haar wavelet quasilinearization approach for numerical simulation of Burg- ers’ equation, Comput. Phys. Commun. 183, 2413–2423, 2012.
  • [22] R. Jiwari, A hybrid numerical scheme for the numerical solution of the Burgers’ equation, Comput. Phys. Commun. 188, 59–67 2015.
  • [23] R. Jiwari, V. Kumar, R. Karan and A. S. Alshomrani, Haar wavelet quasilinearization approach for MHD Falkner–Skan flow over permeable wall via Lie group method, Int. J. Numer. Method H. 27 (6), 1332–1350, 2017.
  • [24] H. Kaur, R.C. Mittal and V. Mishra, Haar wavelet approximate solutions for the generalized Lane–Emden equations arising in astrophysics, Comput. Phys. Commun. 184, 2169–2177, 2013.
  • [25] A.Q.M. Khaliq and E.H. Twizell, A family of second order methods for variable coef- ficient fourth order parabolic partial differential equations, Int. J. Comput. Math. 23, 63–76, 1987.
  • [26] A. Khan, I. Khan, and T. Aziz, Sextic spline solution for solving a fourth-order parabolic partial differential equation, Int. J. Comput. Math. 82 (7), 871–879, 2005.
  • [27] M. Kirs, M. Mikola, A. Haavajõe, E. Õunapuu, B. Shvartsman and J. Majak, Haar wavelet method for vibration analysis of nanobeams, Waves Wavelets Fractals, 2 (1), 2016.
  • [28] K. Kunisch and E. Graif, Parameter estimation for the Euler–Bernoulli beam, Mat. Apficada Comput. 4, 95–124, 1985.
  • [29] U. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul. 68, 127–143, 2005.
  • [30] U. Lepik, Numerical solution of evolution equations by the Haar wavelet method, Appl. Math. Comput. 185, 695–704, 2007.
  • [31] U. Lepik, Solving PDEs with the aid of two-dimensional Haar wavelets, Comput. Math. with Appl. 61, 1873–1879, 2011.
  • [32] Y. Liu and C.S. Gurram, The use of He’s variational iteration method for obtaining the free vibration of an Euler–Bernoulli beam, Math. Comput. Model. 50, 1545–1552, 2009.
  • [33] J. Majak, B. Shvartsman, M. Kirs, M. Pohlak and H. Herranen, Convergence theorem for the Haar wavelet based discretization method, Compos. Struct. 126, 227–232, 2015.
  • [34] J. Majak, M. Pohlak, K. Karjust, M. Eerme, J. Kurnitski and B.S. Shvartsman, New higher order Haar wavelet method: Application to FGM structures, 201, 72–78, 2018. https://doi.org/10.1016/j.compstruct.2018.06.013.
  • [35] J. Majak, B. Shvartsman, K. Karjust, M. Mikola, A. Haavajõe and M. Pohlak, On the accuracy of the Haar wavelet discretization method, Compos. B. Eng. 80, 321–327, 2015.
  • [36] R.C. Mittal and R.K. Jain, B-splines methods with redefined basis functions for solving fourth order parabolic partial differential equations, Appl. Math. Comput. 217, 9741– 9755, 2011.
  • [37] R.C. Mittal and S. Pandit, Sensitivity analysis of shock wave Burgers’ equation via a novel algorithm based on scale-3 Haar wavelets, Int. J. Comput. Math. 95 (3), 601–625, 2017.
  • [38] R.C. Mittal and S. Pandit, A Numerical Algorithm to Capture Spin Patterns of Frac- tional Bloch NMR Flow Models, J. Comput. Nonlinear Dynam. 14 (8), 2019.
  • [39] R.C. Mittal and S. Pandit, New Scale-3 Haar Wavelets Algorithm for Numerical Simulation of Second Order Ordinary Differential Equations, P. Natl. A. Sci. India A, 89, 799–808, 2019.
  • [40] R.C. Mittal and S. Pandit, Quasilinearized Scale-3 Haar Wavelets based Algorithm for Numerical Simulation of Fractional Dynamical System, Eng. Computations. 35 (5), 1907–1931, 2018.
  • [41] R. Mohammadi, Sextic B-spline collocation method for solving Euler–Bernoulli Beam Models, Appl. Math. Comput. 241, 151–166, 2014.
  • [42] Ö. Oruç, F. Bulut and A. Esen, A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers equation, J. Math. Chem. 53 (7), 1592–1607, 2015.
  • [43] Ö. Oruç, F. Bulut and A. Esen, Numerical Solutions of Regularized Long Wave Equa- tion By Haar Wavelet Method, Mediterr. J. Math. 13 (5), 3235–3253, 2016.
  • [44] Ö. Oruç, A. Esen and F. Bulut, A Haar wavelet collocation method for coupled non- linear Schrödinger–KdV equations, Int. J. Mod. Phys. C, 27 (9), 2016.
  • [45] S. Pandit, M. Kumar and S. Tiwari, Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients, Comput. Phys. Commun. 187, 83– 90, 2015.
  • [46] J. Rashidinia and R. Mohammadi, Sextic spline solution of variable coefficient fourth- order parabolic equations, Int. J. Comput. Math. 87 (15), 3443–3454, 2010.
  • [47] R.D. Richtmyer and K.W. Morton, Difference methods for Initial value Problems, second ed., John Wiley & Sons, 1967.
  • [48] Z. Shi, Y. Cao and Q.J. Chen, Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method, Appl. Math. Model. 36, 5143–5161, 2012.
  • [49] R.C. Smith, K.L. Bowers and J. Lund, A fully Sinc–Galerkin method for Euler– Bernoulli Beam Models, Numer. Methods Partial Diff. Equ. 8, 171–202, 1992.
  • [50] S.P. Timoshenko and J.M. Gere, Theory of Elastic Stability, McGraw-Hill, New York, 1961.
  • [51] J. Todd, A direct approach to the problem of stability in the numerical solution of partial differential equations, Commun. Pure Appl. Math. 9, 597–612, 1956.
  • [52] A.M.Wazwaz, Analytic treatment for variable coefficient fourth-order parabolic partial differential equations, Appl. Math. Comput. 123, 219–227, 2001.
There are 52 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Ömer Oruç 0000-0002-6655-3543

Alaattin Esen 0000-0002-7927-5941

Fatih Bulut 0000-0001-6603-2468

Publication Date February 4, 2021
Published in Issue Year 2021

Cite

APA Oruç, Ö., Esen, A., & Bulut, F. (2021). Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method. Hacettepe Journal of Mathematics and Statistics, 50(1), 159-179. https://doi.org/10.15672/hujms.610834
AMA Oruç Ö, Esen A, Bulut F. Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method. Hacettepe Journal of Mathematics and Statistics. February 2021;50(1):159-179. doi:10.15672/hujms.610834
Chicago Oruç, Ömer, Alaattin Esen, and Fatih Bulut. “Numerical Investigation of Dynamic Euler-Bernoulli Equation via 3-Scale Haar Wavelet Collocation Method”. Hacettepe Journal of Mathematics and Statistics 50, no. 1 (February 2021): 159-79. https://doi.org/10.15672/hujms.610834.
EndNote Oruç Ö, Esen A, Bulut F (February 1, 2021) Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method. Hacettepe Journal of Mathematics and Statistics 50 1 159–179.
IEEE Ö. Oruç, A. Esen, and F. Bulut, “Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 1, pp. 159–179, 2021, doi: 10.15672/hujms.610834.
ISNAD Oruç, Ömer et al. “Numerical Investigation of Dynamic Euler-Bernoulli Equation via 3-Scale Haar Wavelet Collocation Method”. Hacettepe Journal of Mathematics and Statistics 50/1 (February 2021), 159-179. https://doi.org/10.15672/hujms.610834.
JAMA Oruç Ö, Esen A, Bulut F. Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method. Hacettepe Journal of Mathematics and Statistics. 2021;50:159–179.
MLA Oruç, Ömer et al. “Numerical Investigation of Dynamic Euler-Bernoulli Equation via 3-Scale Haar Wavelet Collocation Method”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 1, 2021, pp. 159-7, doi:10.15672/hujms.610834.
Vancouver Oruç Ö, Esen A, Bulut F. Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method. Hacettepe Journal of Mathematics and Statistics. 2021;50(1):159-7.

Cited By

Multiresolution method for bending of plates with complex shapes
Applied Mathematics and Mechanics
https://doi.org/10.1007/s10483-023-2972-8
A finite-difference and Haar wavelets hybrid collocation technique for non-linear inverse Cauchy problems
Applied Mathematics in Science and Engineering
https://doi.org/10.1080/17415977.2022.2026350
Collocation approaches to the mathematical model of an Euler–Bernoulli beam vibrations
Mathematics and Computers in Simulation
https://doi.org/10.1016/j.matcom.2022.01.027
A Modified Algorithm Based on Haar Wavelets for the Numerical Simulation of Interface Models
Journal of Function Spaces
https://doi.org/10.1155/2022/1541486
Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam ∗
Journal of Mathematics
https://doi.org/10.1155/2021/3920937

https://dergipark.org.tr/en/pub/hujms