Direct and inverse problems of ROD equation using finite element method and a correction technique

Mirzaei, Hanif; Ghanbari, Kazem; Abbasnavaz, Vahid; Mingarelli, Angelo

doi:10.22034/cmde.2024.57676.2417

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	Direct and inverse problems of ROD equation using finite element method and a correction technique
Computational Methods for Differential Equations
مقاله 2، دوره 12، شماره 4، دی 2024، صفحه 651-668 اصل مقاله (625.48 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2024.57676.2417
نویسندگان
Hanif Mirzaei¹؛ Kazem Ghanbari^* ¹؛ Vahid Abbasnavaz¹؛ Angelo Mingarelli²
¹Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran.
²School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6, Canada.
چکیده
The free vibrations of a rod are governed by a differential equation of the form $(a(x)y^\prime)^\prime+\lambda a(x)y(x)=0$, where $a(x)$ is the cross sectional area and $\lambda$ is an eigenvalue parameter. Using the finite element method (FEM) we transform this equation to a generalized matrix eigenvalue problem of the form $(K-\Lambda M)u=0$ and, for given $a(x)$, we correct the eigenvalues $\Lambda$ of the matrix pair $(K,M)$ to approximate the eigenvalues of the rod equation. The results show that with step size $h$ the correction technique reduces the error from $O(h^2i^4)$ to $O(h^2i^2)$ for the $i$-th eigenvalue. We then solve the inverse spectral problem by imposing numerical algorithms that approximate the unknown coefficient $a(x)$ from the given spectral data. The cross section is obtained by solving a nonlinear system using Newton's method along with a regularization technique. Finally, we give numerical examples to illustrate the efficiency of the proposed algorithms.
کلیدواژه‌ها
Rod equation؛ Eigenvalue؛ Finite element method؛ Direct problem؛ Inverse problem؛ Sturm-Liouville

مراجع
[1] M. O. Adewole, Finite element solution of a class of parabolic integro-differential equations with nonhomogeneous jump condition using FreeFEM++, Computational Methods for Differential Equations, 12(2) (2023), 314–328. [2] S. Akbarpoor and A. H. Dabbaghian, Approximate solution of inverse Nodal problem with conformable derivative, Computational Methods for Differential Equations, 12(2) (2024), 610–623. [3] H. Altundag and H. Taseli, Singular inverse Sturm-Liouville problems with Hermite pseudospectral methods, Eur. Phys. J. Plus., 136 (2021), 1–13. [4] H. Altundag, C. Bockmann, and H. Ta¸seli, Inverse Sturm-Liouville problems with pseudospectral methods, International journal of computer mathematics., 92(7) (2015), 1373–1384. [5] R. Amirov and I. Adalar, Eigenvalues of Sturm-Liouville operators and prime numbers, Electronic journal of differential equations., (2017), 1–3. [6] L. E. Andersson, Inverse eigenvalue problems for a Sturm-Liouville equation in impedance form, Inverse Probl., 4 (1988), 929–971. [7] A. L. Andrew and J. W. Paine, Correction of finite element estimates for Sturm-Liouville eigenvalues, Numer. Math., 50 (1986), 205–215. [8] A. L. Andrew,Computing SturmLiouville potentials from two spectra, Inverse problems., 22 (2006), 1–12. [9] A. Bilotta, A. Morassi, and E. Turco, Reconstructing blockages in a symmetric duct via quasi-isospectral horn operators Journal of sound and vibration., 366 (2016), 149–172. [10] Z. Chen, Finite element method and their applications, Springer-Verlag, Berlin., 2005. [11] S. S. Ezhak and M. Y. Telnova, Estimates for The first eigenvalue of the Sturm-Liouville problem with potentials in weighted spaces, Journal of mathematical sciences., 244 (2020), 216-234. [12] G. Freiling and V. A. Yurko, Inverse Sturm-Liouville problems and their applications, NOVA science publications, New York., 2001. [13] Q. Gao, Z. Huang, and X. Cheng, A finite difference method for an inverse Sturm-Liouville problem in impedance form, Numer. Algor., 70 (2015), 669–690. [14] Q. Gao, Q. Zhao, X. Zheng, and Y. Ling, Convergence of Numerov’s method for inverse Sturm-Liouville problems, Applied mathematics and computation., 293 (2017), 1–17. [15] Q. Gao, Q. Zhao and M. Chen, On a modified Numerov’s method for inverse Sturm-Liouville problems, International journal of computer mathematics., 95(2) (2018), 412–426. [16] J. C. Gilbert, On the realization of the Wolf conditions in reduced Quasi-Newton for equality constrained optimization, SIAM J. Optim., 7(3) (1997), 780–813. [17] G. M. L. Gladwell, Inverse problem in vibration, Kluwer academic publishers, New York., 2004. [18] G. M. L. Gladwell and A. Morassi, On isospectral rods, horns and strings, Inverse problems., 11 (1995), 533–554. [19] G. M. L. Gladwell and A. Morassi, A family of isospectral EulerBernoulli beams, Inverse problems., 26 (2010), 1–12. [20] K. Ghanbari, H. Mirzaei, and G. L. M. Gladwell, Reconstructing of a rod using one spectrum and minimal mass condition, Inverse problem in science and engineering., 22 (2014), 325–333. [21] A. Kirsch, An introduction to the mathematical theory of inverse problems, Springer-Verlag, New York., 1996. [22] V. Ledoux, M. V. Daele, and G. V. Berghe, Matslise: A Matlab package for the numerical solution of SturmLiouville and Schrodinger equations, ACM translations on mathematical software., 31 (2005), 532–554. [23] C. S. Liu, Computing the eigenvalues of the generalized Sturm-Liouville problems based on the Lie-group SL(2,R), Journal of computational and applied mathematics., 236(17) (2012), 4547–4560. [24] L. Losonczi, Eigenvalues and eigenvactors of some tridiagonal matrices, Acta Math.Hung., 60(1-2) (1992), 309– 323. [25] G. T. Lubo and G. F. Duressa,Linear B-spline finite element Method for solving delay reaction diffusion equation, Computational Methods for Differential Equations., 11 (2023), 161–174. [26] C. Magherini, A corrected spectral method for Sturm-Liouville problems with unbounded potential at one endpoint, Journal of computational and applied mathematics., 364 (2020), 1–15. [27] H. Mirzaei, Computing the eigenvalues of fourth order Sturm–Liouville problems with Lie Group method, Iranian journal of numerical analysis and optimization., 7 (2017), 1–12. [28] H. Mirzaei, A family of isospectral fourth order Sturm–Liouville problems and equivalent beam equations, Mathematical communications., 23 (2018), 15–27. [29] H. Mirzaei, Higher-order Sturm–Liouville problems with the same eigenvalues, Turkish journal of mathematics., 44(2) (2020), 409–417. [30] H. Mirzaei, K. Ghanbari, and M. Emami, Direct and Inverse Problems of String Equation by Numerovs Method, Iranian Journal of Science., 47 (2023), 871–884. [31] A. B. Mingarelli, A note on Sturm-Liouville problems whose spectrum is the set of prime numbers, Electronic journal of differential equations., 44(123) (2011), 1–4. [32] A. Morassi, Quasi-isospectral vibrating systems, Technical report, University of Udine., 2012. [33] A. Morassi, Constructing rods with given natural frequencies, Mechanical systems and signal processing., 40 (2013), 288–300. [34] S. Mosazadeh, A new approach to uniqueness for inverse Sturm-Liouville problems on finite intervals, Turkish journal of mathematics., 41 (2017), 1224–1234. [35] A. S. Ozkan and A. Adalar, Inverse nodal problems for Sturm-Liouville equation with nonlocal boundary conditions, Journal of Mathematical Analysis and Applications., 520(1) (2023), 126904. [36] J. W. Paine, A numerical method for the inverse Sturm-Liouville problem, Siam J. Sci. Stat. Comput., 5 (1984), 149–156. [37] J. W. Paine, F. R. Hoog, and R. S. Anderssen, On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems, Computing., 26 (1981), 123–139. [38] M. Shahriari, A. A. Jodayree, and G. Teschl, Uniqueness for inverse Sturm-Liouville problems with a finite number of transmission conditions Journal of mathematical analysis and applications., 365 (2012), 19–29. [39] C. W. Soh, Isospectral Euler-Bernoulli beams via factorization and the Lie method, Int. J. Non-Linear Mech., 44 (2009), 396-403. [40] J. Stoer and R. Bulirsch, Introduction to numerical analysis, Third edition, Springer-Verlag, Wurzburg., 2002. [41] J.G. Sun, Multiple eigenvalue sensitivity analysis, Linear algebra appl., 137–138 (1999), 183–211. [42] X. Wu, P. Niu and G. Wei, An inverse eigenvalue problem for a nonlocal Sturm-Liouville operator, Journal of Mathematical Analysis and Applications., 494(2) (2021), 124661. [43] M. Zhang and K. Li, Dependence of eigenvalues of Sturm-Liouville problems with eigenparameter dependent boundary conditions, Applied mathematics and computation., 378 (2020), 1–14.
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Direct and inverse problems of ROD equation using finite element method and a correction technique