{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T21:07:11Z","timestamp":1774386431987,"version":"3.50.1"},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,4,1]]},"abstract":"Abstract<\/jats:title>\n \n The free longitudinal vibrations of a rod are described by a differential equation of the form\n \n \n \n \n \n \n (<\/m:mo>\n P<\/m:mi>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n y<\/m:mi>\n \u2032<\/m:mo>\n )<\/m:mo>\n <\/m:mrow>\n \u2032<\/m:mo>\n <\/m:msup>\n +<\/m:mo>\n \u03bb<\/m:mi>\n P<\/m:mi>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n y<\/m:mi>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n 0<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {(P(x)y\\prime)^{\\prime}+\\lambda P(x)y(x)=0}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n \n \n P<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {P(x)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is the cross section area at point\n x<\/jats:italic>\n and \u03bb is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form\n \n \n \n \n \n \ud835\udc00<\/m:mi>\n \u2062<\/m:mo>\n Y<\/m:mi>\n <\/m:mrow>\n =<\/m:mo>\n \n \u039b<\/m:mi>\n \u2062<\/m:mo>\n \ud835\udc01<\/m:mi>\n \u2062<\/m:mo>\n Y<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathbf{A}Y=\\Lambda\\mathbf{B}Y}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n \n \ud835\udc00<\/m:mi>\n <\/m:math>\n \n {\\mathbf{A}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n \n \ud835\udc01<\/m:mi>\n <\/m:math>\n \n {\\mathbf{B}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n are Jacobi and diagonal matrices dependent to cross section\n \n \n \n \n P<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {P(x)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , respectively. Then we estimate the eigenvalues of the rod equation by correcting the eigenvalues of the resulting matrix eigenvalue problem. We give a method based on a correction idea to construct the cross section\n \n \n \n \n P<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {P(x)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n by solving an inverse matrix eigenvalue problem. We give some numerical examples to illustrate the efficiency of the proposed method. The results show that the convergence order of the method is\n \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n h<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {O(h^{2})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1515\/cmam-2024-0001","type":"journal-article","created":{"date-parts":[[2024,8,2]],"date-time":"2024-08-02T13:23:00Z","timestamp":1722604980000},"page":"475-485","source":"Crossref","is-referenced-by-count":0,"title":["An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod"],"prefix":"10.1515","volume":"25","author":[{"given":"Hanif","family":"Mirzaei","sequence":"first","affiliation":[{"name":"Faculty of Basic Sciences , 114739 Sahand University of Technology , Tabriz , Iran"}]},{"given":"Vahid","family":"Abbasnavaz","sequence":"additional","affiliation":[{"name":"Faculty of Basic Sciences , 114739 Sahand University of Technology , Tabriz , Iran"}]},{"given":"Kazem","family":"Ghanbari","sequence":"additional","affiliation":[{"name":"Faculty of Basic Sciences , 114739 Sahand University of Technology , Tabriz , Iran"}]}],"member":"374","published-online":{"date-parts":[[2024,8,2]]},"reference":[{"key":"2026032417390185058_j_cmam-2024-0001_ref_001","doi-asserted-by":"crossref","unstructured":"H. Altunda\u01e7 and H. Ta\u015feli,\nSingular inverse Sturm\u2013Liouville problems with Hermite pseudospectral methods,\nEur. Phys. J. Plus. 136 (2021), 1\u201313.","DOI":"10.1140\/epjp\/s13360-021-02000-y"},{"key":"2026032417390185058_j_cmam-2024-0001_ref_002","doi-asserted-by":"crossref","unstructured":"A. L. Andrew,\nAsymptotic correction of more Sturm\u2013Liouville eigenvalue estimates,\nBIT 43 (2003), no. 3, 485\u2013503.","DOI":"10.1023\/B:BITN.0000007052.66222.6d"},{"key":"2026032417390185058_j_cmam-2024-0001_ref_003","doi-asserted-by":"crossref","unstructured":"A. L. Andrew,\nComputing Sturm\u2013Liouville potentials from two spectra,\nInverse Problems 22 (2006), no. 6, 2069\u20132081.","DOI":"10.1088\/0266-5611\/22\/6\/010"},{"key":"2026032417390185058_j_cmam-2024-0001_ref_004","doi-asserted-by":"crossref","unstructured":"N. Bebiano and J. da Provid\u00eancia,\nInverse spectral problems for structured pseudo-symmetric matrices,\nLinear Algebra Appl. 438 (2013), no. 10, 4062\u20134074.","DOI":"10.1016\/j.laa.2012.07.023"},{"key":"2026032417390185058_j_cmam-2024-0001_ref_005","doi-asserted-by":"crossref","unstructured":"B. G. S. Doman,\nThe Classical Orthogonal Polynomials,\nWorld Scientific, Hackensack, 2015.","DOI":"10.1142\/9700"},{"key":"2026032417390185058_j_cmam-2024-0001_ref_006","doi-asserted-by":"crossref","unstructured":"Q. Gao, Z. Huang and X. Cheng,\nA finite difference method for an inverse Sturm\u2013Liouville problem in impedance form,\nNumer. Algorithms 70 (2015), no. 3, 669\u2013690.","DOI":"10.1007\/s11075-015-9968-7"},{"key":"2026032417390185058_j_cmam-2024-0001_ref_007","doi-asserted-by":"crossref","unstructured":"K. Ghanbari, H. Mirzaei and G. M. L. Gladwell,\nReconstruction of a rod using one spectrum and minimal mass condition,\nInverse Probl. Sci. 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