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The technique is a mixture of the Laplace transformation and homotopy perturbation technique. Fractional derivatives with Mittag\u2010Leffler and exponential laws in sense of Caputo are considered. Moreover, this paper aims to show the Whitham\u2013Broer\u2013Kaup equations with both derivatives to see their difference in a real\u2010world problem. The efficiency of both operators is confirmed by the outcome of the actual results of the Whitham\u2013Broer\u2013Kaup equations. Some problems have been presented to compare the solutions achieved with both fractional\u2010order derivatives.<\/jats:p>","DOI":"10.1155\/2021\/7979365","type":"journal-article","created":{"date-parts":[[2021,7,22]],"date-time":"2021-07-22T23:05:07Z","timestamp":1626995107000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["Numerical Investigation of the Time\u2010Fractional Whitham\u2013Broer\u2013Kaup Equation Involving without Singular Kernel Operators"],"prefix":"10.1155","volume":"2021","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4755-9381","authenticated-orcid":false,"given":"Muhammad","family":"Naeem","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3270-9124","authenticated-orcid":false,"given":"Ahmed M.","family":"Zidan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4306-8489","authenticated-orcid":false,"given":"Rasool","family":"Shah","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1702-8643","authenticated-orcid":false,"given":"Ahmed","family":"Alsanad","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8512-9687","authenticated-orcid":false,"given":"Abdu","family":"Gumaei","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2021,7,22]]},"reference":[{"key":"e_1_2_11_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0375-9601(01)00333-4"},{"key":"e_1_2_11_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cjph.2020.05.026"},{"key":"e_1_2_11_3_2","article-title":"New models of fractional blood ethanol and two cell cubic autocatalator reaction equations","volume":"43","author":"Alderremy A. 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