{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,22]],"date-time":"2026-03-22T18:54:48Z","timestamp":1774205688332,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T00:00:00Z","timestamp":1667952000000},"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>Wavelet transforms or wavelet analysis represent a recently created mathematical tool for assistance in resolving various issues. Wavelets can also be used in numerical analysis. In this study, we solve pantograph delay differential equations using the Modified Laguerre Wavelet method (MLWM), an effective numerical technique. Fractional derivatives are defined using the Caputo operator. The convergence of the suggested strategy is carefully examined. The suggested strategy is straightforward, effective, and simple in comparison with previous approaches. Specific examples are provided to demonstrate the current scenario\u2019s reliability and accuracy. Compared with other methodologies, our results show a higher accuracy level. With the aid of tables and graphs, we demonstrate the effectiveness of the proposed approach by comparing results of the actual and suggested methods and demonstrating their strong agreement. For better understanding of the proposed method, we show the pointwise solution in the tables provided which confirm the accuracy at each point of the proposed method. Additionally, the results of employing the suggested method to various fractional-orders are compared, which demonstrates that when a value shifts from fractional-order to integer-order, the result approaches the exact solution. Owing to its novelty and scientific significance, the suggested technique can also be used to solve additional nonlinear delay differential equations of fractional-order.<\/jats:p>","DOI":"10.3390\/sym14112356","type":"journal-article","created":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T02:44:32Z","timestamp":1667961872000},"page":"2356","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Evaluation of Fractional-Order Pantograph Delay Differential Equation via Modified Laguerre Wavelet Method"],"prefix":"10.3390","volume":"14","author":[{"given":"Aisha Abdullah","family":"Alderremy","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Khalid University, Abha 61413, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4306-8489","authenticated-orcid":false,"given":"Rasool","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali khan University, Mardan 23200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nehad Ali","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Sejong University, Seoul 05006, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shaban","family":"Aly","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, AL-Azhar University, Assiut 11884, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,9]]},"reference":[{"key":"ref_1","unstructured":"Miller, K.S., and Ross, B. 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