{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T09:46:00Z","timestamp":1778319960013,"version":"3.51.4"},"reference-count":25,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,19]],"date-time":"2021-11-19T00:00:00Z","timestamp":1637280000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007345","name":"King Mongkut's University of Technology North Bangkok","doi-asserted-by":"publisher","award":["KMUTNB-64-KNOW-42"],"award-info":[{"award-number":["KMUTNB-64-KNOW-42"]}],"id":[{"id":"10.13039\/501100007345","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit.<\/jats:p>","DOI":"10.3390\/sym13112215","type":"journal-article","created":{"date-parts":[[2021,11,21]],"date-time":"2021-11-21T21:00:50Z","timestamp":1637528450000},"page":"2215","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation"],"prefix":"10.3390","volume":"13","author":[{"given":"Haji","family":"Gul","sequence":"first","affiliation":[{"name":"Department of Mathematics, Abdulwali Khan University, Mardan 23200, Khyber Pakhtunkhwa, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5507-8513","authenticated-orcid":false,"given":"Sajjad","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shaheed Benazir Bhutto University Sheringal Dir (Upper), Sheringal Dir (Upper) 18050, Khyber Pakhtunkhwa, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8851-4844","authenticated-orcid":false,"given":"Kamal","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Malakand, Chakadara Dir (Lower), Lower Dir 18800, Khyber Pakhtunkhwa, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5808-4951","authenticated-orcid":false,"given":"Shakoor","family":"Muhammad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdulwali Khan University, Mardan 23200, Khyber Pakhtunkhwa, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8455-1402","authenticated-orcid":false,"given":"Thanin","family":"Sitthiwirattham","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand"}]},{"given":"Saowaluck","family":"Chasreechai","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1186\/s13662-020-03167-x","article-title":"An efficient approach for solution of fractional-order Helmholtz equations","volume":"2021","author":"Shah","year":"2021","journal-title":"Adv. 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