{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T07:51:32Z","timestamp":1776930692790,"version":"3.51.2"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2018,10,15]],"date-time":"2018-10-15T00:00:00Z","timestamp":1539561600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100002383","name":"King Saud University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002383","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["RACSAM"],"published-print":{"date-parts":[[2019,7]]},"DOI":"10.1007\/s13398-018-0590-0","type":"journal-article","created":{"date-parts":[[2018,10,15]],"date-time":"2018-10-15T01:41:41Z","timestamp":1539567701000},"page":"1873-1891","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":69,"title":["A numerical study of fractional relaxation\u2013oscillation equations involving $$\\psi $$ \u03c8 -Caputo fractional derivative"],"prefix":"10.1007","volume":"113","author":[{"given":"Ricardo","family":"Almeida","sequence":"first","affiliation":[]},{"given":"Mohamed","family":"Jleli","sequence":"additional","affiliation":[]},{"given":"Bessem","family":"Samet","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,10,15]]},"reference":[{"key":"590_CR1","doi-asserted-by":"publisher","first-page":"460","DOI":"10.1016\/j.cnsns.2016.09.006","volume":"44","author":"R Almeida","year":"2017","unstructured":"Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear. Sci. Numer. Simulat. 44, 460\u2013481 (2017)","journal-title":"Commun. Nonlinear. Sci. Numer. Simulat."},{"key":"590_CR2","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1002\/mma.4617","volume":"41","author":"R Almeida","year":"2018","unstructured":"Almeida, R., Malinowska, A.B., Monteiro, M.T.T.: Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Methods. Appl. Sci. 41, 336\u2013352 (2018)","journal-title":"Math. Methods. Appl. Sci."},{"key":"590_CR3","doi-asserted-by":"publisher","unstructured":"Baleanu, D., Bhrawy, A.H., Taha, T.M.: Two efficient generalized Laguerre spectral algorithms for fractional initial value problems. Abstr. Appl. Anal. 2013. https:\/\/doi.org\/10.1155\/2013\/546502.","DOI":"10.1155\/2013\/546502."},{"key":"590_CR4","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1016\/j.jcp.2014.03.039","volume":"293","author":"AH Bhrawy","year":"2015","unstructured":"Bhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S.: A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J. Comput. Phys. 293, 142\u2013156 (2015)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"590_CR5","first-page":"3","volume":"11","author":"W Chen","year":"2010","unstructured":"Chen, W., Zhang, X.D., Korosak, D.: Investigation on fractional and fractal derivative relaxation\u2013oscillation models. Int. J. Nonlinear. Sci. Numer. Simulat. 11(1), 3\u20139 (2010)","journal-title":"Int. J. Nonlinear. Sci. Numer. Simulat."},{"key":"590_CR6","volume-title":"Introduction to Hilbert Spaces with Applications","author":"L Debnath","year":"1999","unstructured":"Debnath, L., Mikusi\u0144ski, P.: Introduction to Hilbert Spaces with Applications, 2nd edn. Academic Press, Cambridge (1999)","edition":"2"},{"key":"590_CR7","unstructured":"Dimitrov, Y.: Higher-order numerical solutions of the fractional relaxation\u2013oscillation equation using fractional integration (2016). arXiv:1603.08733"},{"key":"590_CR8","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1080\/10652460802030672","volume":"20","author":"EH Doha","year":"2009","unstructured":"Doha, E.H., Ahmed, H.M., El-Soubhy, S.I.: Explicit formulae for the coefficients of integrated expansions of Laguerre and Hermite polynomials and their integrals. Integral Transform. Spec. Funct. 20, 491\u2013503 (2009)","journal-title":"Integral Transform. Spec. Funct."},{"key":"590_CR9","doi-asserted-by":"publisher","first-page":"2364","DOI":"10.1016\/j.camwa.2011.07.024","volume":"62","author":"EH Doha","year":"2011","unstructured":"Doha, E.H., Bhrawy, A.H., Ezz-Eliden, S.S.: A Chebeshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 62, 2364\u20132373 (2011)","journal-title":"Comput. Math. Appl."},{"key":"590_CR10","doi-asserted-by":"publisher","first-page":"4931","DOI":"10.1016\/j.apm.2011.12.031","volume":"36","author":"EH Doha","year":"2012","unstructured":"Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 36, 4931\u20134943 (2012)","journal-title":"Appl. Math. Model."},{"issue":"8","key":"590_CR11","doi-asserted-by":"publisher","first-page":"5927","DOI":"10.1016\/j.apm.2012.12.015","volume":"37","author":"M Gulsu","year":"2013","unstructured":"Gulsu, M., Ozturk, Y., Anapali, A.: Numerical approach for solving fractional relaxation\u2013oscillation equation. Appl. Math. Model. 37(8), 5927\u20135937 (2013)","journal-title":"Appl. Math. Model."},{"key":"590_CR12","first-page":"98","volume":"2","author":"M Hamarsheh","year":"2015","unstructured":"Hamarsheh, M., Ismail, A., Odibat, Z.: Optimal homotopy asymptotic method for solving fractional relaxation\u2013oscillation equation. J. Interpol. Appl. Sci. Comput. 2, 98\u2013111 (2015)","journal-title":"J. Interpol. Appl. Sci. Comput."},{"key":"590_CR13","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1080\/00207178108922549","volume":"34","author":"C Hwang","year":"1981","unstructured":"Hwang, C., Shih, Y.: Laguerre operational matrices for fractional calculus and applications. Int. J. Control. 34, 577\u2013584 (1981)","journal-title":"Int. J. Control."},{"key":"590_CR14","doi-asserted-by":"crossref","unstructured":"Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012.: Article ID 142 (2012)","DOI":"10.1186\/1687-1847-2012-142"},{"issue":"21","key":"590_CR15","doi-asserted-by":"publisher","first-page":"6461","DOI":"10.1016\/j.apm.2015.01.060","volume":"39","author":"H Khosravian-Arab","year":"2015","unstructured":"Khosravian-Arab, H., Almeida, R.: Numerical solution for fractional variational problems using the Jacobi polynomials. Appl. Math. Model. 39(21), 6461\u20136470 (2015)","journal-title":"Appl. Math. Model."},{"key":"590_CR16","volume-title":"Theory and Applications of Fractional Differential Equations","author":"AA Kilbas","year":"2006","unstructured":"Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, San Diego (2006)"},{"issue":"2","key":"590_CR17","doi-asserted-by":"publisher","first-page":"665","DOI":"10.1137\/15M1028583","volume":"54","author":"RR Lederman","year":"2016","unstructured":"Lederman, R.R., Rokhlin, V.: On the analytical and numerical properties of the truncated Laplace transform. Part II. SIAM J. Numer. Anal 54(2), 665\u2013687 (2016)","journal-title":"SIAM J. Numer. Anal"},{"key":"590_CR18","volume-title":"The Special Functions and Their Approximations","author":"Y Luke","year":"1969","unstructured":"Luke, Y.: The Special Functions and Their Approximations, vol. 2. Academic Press, New York (1969)"},{"key":"590_CR19","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1016\/j.sigpro.2010.08.003","volume":"91","author":"R Magin","year":"2011","unstructured":"Magin, R., Ortigueira, M.D., Podlubny, I., Trujillo, J.: On the fractional signals and systems. Signal Process. 91, 350\u2013371 (2011)","journal-title":"Signal Process."},{"key":"590_CR20","doi-asserted-by":"publisher","first-page":"1461","DOI":"10.1016\/0960-0779(95)00125-5","volume":"7","author":"F Mainardi","year":"1996","unstructured":"Mainardi, F.: Fractional relaxation oscillation and fractional diffusion-wave phenomena. Chaos Solit. Fract. 7, 1461\u20131477 (1996)","journal-title":"Chaos Solit. Fract."},{"issue":"1","key":"590_CR21","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1016\/j.amc.2006.07.102","volume":"186","author":"Z Odibat","year":"2007","unstructured":"Odibat, Z., Shawagfeh, N.T.: Generalized Taylor\u2019s formula. Appl. Math. Comput. 186(1), 286\u2013293 (2007)","journal-title":"Appl. Math. Comput."},{"key":"590_CR22","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1016\/0016-0032(83)90082-0","volume":"316","author":"PN Paraskevopoulos","year":"1983","unstructured":"Paraskevopoulos, P.N.: Chebyshev series approach to system identification, analysis and control. J. Frankl. Inst. 316, 135\u2013157 (1983)","journal-title":"J. Frankl. Inst."},{"key":"590_CR23","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1109\/TAC.1985.1103992","volume":"30","author":"PN Paraskevopoulos","year":"1985","unstructured":"Paraskevopoulos, P.N.: Legendre series approach to identification and analysis of linear systems. IEEE Trans. Autom. Control 30, 585\u2013589 (1985)","journal-title":"IEEE Trans. Autom. Control"},{"key":"590_CR24","doi-asserted-by":"publisher","first-page":"1326","DOI":"10.1016\/j.camwa.2009.07.006","volume":"59","author":"A Saadatmandi","year":"2010","unstructured":"Saadatmandi, A., Deghan, M.: A new operational matrix for solving fractional-order differential equation. Comput. Math. Appl. 59, 1326\u20131336 (2010)","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"590_CR25","first-page":"1","volume":"6","author":"FA Shah","year":"2017","unstructured":"Shah, F.A., Abass, R.: Generalized wavelet collocation method for solving fractional relaxation\u2013oscillation equation arising in fluid mechanics. Int. J. Comput. Math. Sci. Eng. 6(2), 1\u201317 (2017)","journal-title":"Int. J. Comput. Math. Sci. Eng."},{"key":"590_CR26","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/S0378-4371(03)00598-3","volume":"329","author":"A Tofighi","year":"2003","unstructured":"Tofighi, A.: The intrinsic damping of the fractional oscillator. Phys. A 329, 29\u201334 (2003)","journal-title":"Phys. A"}],"container-title":["Revista de la Real Academia de Ciencias Exactas, F\u00edsicas y Naturales. Serie A. Matem\u00e1ticas"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s13398-018-0590-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s13398-018-0590-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s13398-018-0590-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,4]],"date-time":"2022-09-04T00:09:48Z","timestamp":1662250188000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s13398-018-0590-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,15]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,7]]}},"alternative-id":["590"],"URL":"https:\/\/doi.org\/10.1007\/s13398-018-0590-0","relation":{},"ISSN":["1578-7303","1579-1505"],"issn-type":[{"value":"1578-7303","type":"print"},{"value":"1579-1505","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,10,15]]},"assertion":[{"value":"10 April 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 October 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 October 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}