{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:15:42Z","timestamp":1760148942042,"version":"build-2065373602"},"reference-count":50,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,6,21]],"date-time":"2023-06-21T00:00:00Z","timestamp":1687305600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m\u2208Z+, where the fractional derivative of order \u03b1=q\u03b3, for which \u03b3\u2208Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm\u2019s accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of fractional problem. Under Caputo meaning of the symmetry order, the obtained results are illustrated numerically and graphically. Geometrically, the behavior of the obtained solutions declares that the changing of the fractional derivative parameter values in their domain alters the style of these solutions in a symmetric meaning, as well as indicates harmony and symmetry, which leads them to fully coincide at the value of the ordinary derivative. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for integral and integro-differential equations of fractional order.<\/jats:p>","DOI":"10.3390\/sym15071296","type":"journal-article","created":{"date-parts":[[2023,6,22]],"date-time":"2023-06-22T01:49:32Z","timestamp":1687398572000},"page":"1296","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5374-0916","authenticated-orcid":false,"given":"Abedel-Karrem","family":"Alomari","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Yarmouk University, Irbid 21163, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9099-5619","authenticated-orcid":false,"given":"Mohammad","family":"Alaroud","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman 11953, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nedal","family":"Tahat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa 13133, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1001-4433","authenticated-orcid":false,"given":"Adel","family":"Almalki","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al-Qunfudah University College, Umm Al-Qura University, Mecca 21955, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,21]]},"reference":[{"unstructured":"Loverro, A. 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