{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T03:39:54Z","timestamp":1768880394195,"version":"3.49.0"},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T00:00:00Z","timestamp":1694736000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T00:00:00Z","timestamp":1694736000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2023,10]]},"DOI":"10.1007\/s40314-023-02444-1","type":"journal-article","created":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T12:01:54Z","timestamp":1694779314000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Review of recursive and operational approaches of the Tau method with a new extension"],"prefix":"10.1007","volume":"42","author":[{"given":"Sedaghat","family":"Shahmorad","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Younes","family":"Talaei","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2909-8753","authenticated-orcid":false,"given":"Cemil","family":"Tun\u00e7","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,9,15]]},"reference":[{"key":"2444_CR1","doi-asserted-by":"crossref","unstructured":"Chauhan HVS, Singh BS, Tun\u00e7 C, Tun\u00e7 O (2022) On the existence of solutions of non-linear 2D Volterra integral equations in a Banach Space. Rev. R. Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Mat. RACSAM 116(3): Paper No. 101. 45. https:\/\/doi.org\/10.1007\/s13398-022-01246-0","DOI":"10.1007\/s13398-022-01246-0"},{"key":"2444_CR2","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1002\/zamm.19680480607","volume":"48","author":"T Chaves","year":"1968","unstructured":"Chaves T, Ortiz EL (1968) On the numerical solution of two point boundary value problems for linear differential equations. Z Angew Math Mech 48:415\u2013418","journal-title":"Z Angew Math Mech"},{"issue":"2","key":"2444_CR3","first-page":"1580","volume":"188","author":"G Ebadi","year":"2007","unstructured":"Ebadi G, Rahimi-Ardabili MY, Shahmorad S (2007) Numerical solution of the nonlinear Volterra integro-differential equations by the Tau method. Appl Math Comput 188(2):1580\u20131586","journal-title":"Appl Math Comput"},{"issue":"3","key":"2444_CR4","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0898-1221(93)90145-L","volume":"25","author":"MK El-Daou","year":"1993","unstructured":"El-Daou MK, Ortiz EL, Samara H (1993) A unified approach to the Tau method and Chebyshev series expansion techniques. Comput Math Appl 25(3):73\u201382","journal-title":"Comput Math Appl"},{"issue":"2","key":"2444_CR5","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1007\/BF03021557","volume":"9","author":"SM Hosseini","year":"2002","unstructured":"Hosseini SM, Shahmorad S (2002) A matrix formulation of the Tau method for Fredholm and Volterra linear integro-differential equations. Korean J Comput Appl Math 9(2):497\u2013507","journal-title":"Korean J Comput Appl Math"},{"key":"2444_CR6","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/S0307-904X(02)00099-9","volume":"27","author":"SM Hosseini","year":"2003","unstructured":"Hosseini SM, Shahmorad S (2003) Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases. Appl Math Model 27:145\u2013154","journal-title":"Appl Math Model"},{"issue":"3","key":"2444_CR7","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1007\/s11075-014-9858-4","volume":"68","author":"SA Hosseini","year":"2015","unstructured":"Hosseini SA, Shahmorad S, Talati F (2015) A matrix based method for two dimensional nonlinear Volterra\u2013Fredholm integral equations. Numer Algorithms 68(3):511\u2013529","journal-title":"Numer Algorithms"},{"key":"2444_CR8","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1016\/j.joems.2014.02.004","volume":"23","author":"S Kumar","year":"2015","unstructured":"Kumar S, Kumar A et al (2015) Analytical solution of Abel integral equation arising in astrophysics via Laplace transform. J Egypt Math Soc 23:102\u2013107","journal-title":"J Egypt Math Soc"},{"key":"2444_CR9","unstructured":"Lanczos C (1988) Applied analysis. Reprint of the (1956) original Dover Books Adv. Math. Dover Publications Inc, New York"},{"key":"2444_CR10","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1002\/cnm.1630030305","volume":"3","author":"K Liu","year":"1987","unstructured":"Liu K, Ortiz EL (1987) Tau method approximate solution of high-order differential eigenvalue problems defined in the complex plane, with an application to Orr\u2013Sommerfeld stability equation. Commun Appl Numer Methods 3:187\u2013194","journal-title":"Commun Appl Numer Methods"},{"key":"2444_CR11","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1090\/S0025-5718-1984-0744930-9","volume":"43","author":"P Onumanyi","year":"1984","unstructured":"Onumanyi P, Ortiz EL (1984) Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the Tau method. Math Comput 43:189\u2013203","journal-title":"Math Comput"},{"key":"2444_CR12","doi-asserted-by":"publisher","first-page":"480","DOI":"10.1137\/0706044","volume":"6","author":"EL Ortiz","year":"1969","unstructured":"Ortiz EL (1969) The Tau method. SIAM J Numer Anal 6:480\u2013492","journal-title":"SIAM J Numer Anal"},{"issue":"13","key":"2444_CR13","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1016\/0377-0427(85)90044-5","volume":"12","author":"EL Ortiz","year":"1985","unstructured":"Ortiz EL, Pun KS (1985) Numerical solution of nonlinear partial differential equations with Tau method. J Comput Appl Math 12(13):511\u2013516","journal-title":"J Comput Appl Math"},{"key":"2444_CR14","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1007\/BF02243435","volume":"27","author":"EL Ortiz","year":"1981","unstructured":"Ortiz EL, Samara H (1981) An operational approach to the Tau Method for the numerical solution of nonlinear differential equations. Computing 27:15\u201325","journal-title":"Computing"},{"issue":"1","key":"2444_CR15","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/0898-1221(84)90081-6","volume":"10","author":"EL Ortiz","year":"1984","unstructured":"Ortiz EL, Samara H (1984) Numerical solution of partial differential equations with variable coefficients with an operational approach to the Tau method. Comput Math Appl 10(1):5\u201313","journal-title":"Comput Math Appl"},{"key":"2444_CR16","first-page":"465","volume":"168","author":"J Pour-Mahmoud","year":"2005","unstructured":"Pour-Mahmoud J, Rahimi-Ardabili MY, Shahmorad S (2005) Numerical solution of the system of Fredholm integro-differential equations by the Tau method. Appl Math Comput 168:465\u2013478","journal-title":"Appl Math Comput"},{"key":"2444_CR17","doi-asserted-by":"crossref","unstructured":"Tari A, Rahimi-Ardabili MY, Shahmorad S, Talati F (2009) Development of the Tau method for the numerical solution of two-dimensional linear Volterra integro-differential equations. Comput Appl Math 9(4):421\u2013435","DOI":"10.2478\/cmam-2009-0027"},{"key":"2444_CR18","doi-asserted-by":"crossref","unstructured":"Tun\u00e7 O, Tun\u00e7 C (2023) On Ulam stabilities of delay Hammerstein integral equation. Symmetry 15(9):1736. https:\/\/doi.org\/10.3390\/sym15091736","DOI":"10.3390\/sym15091736"},{"key":"2444_CR19","doi-asserted-by":"publisher","first-page":"494","DOI":"10.1016\/j.mcm.2012.07.004","volume":"57","author":"SK Vanani","year":"2013","unstructured":"Vanani SK, Soleyman F (2013) Tau approximate solution of weakly singular Volterra integral equations. Math Comput Model 57:494\u2013502","journal-title":"Math Comput Model"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02444-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-023-02444-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02444-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T21:31:26Z","timestamp":1703194286000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-023-02444-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,15]]},"references-count":19,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["2444"],"URL":"https:\/\/doi.org\/10.1007\/s40314-023-02444-1","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,15]]},"assertion":[{"value":"17 August 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 August 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 August 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 September 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"307"}}