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

فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری

تعداد نشریات	45
تعداد شماره‌ها	1,560
تعداد مقالات	19,049
تعداد مشاهده مقاله	65,209,199
تعداد دریافت فایل اصل مقاله	23,842,347

	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.
آمار تعداد مشاهده مقاله: 700 تعداد دریافت فایل اصل مقاله: 658

سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب

پیوندهای مفید

آمار

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