{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:12:17Z","timestamp":1760238737822,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,2]],"date-time":"2020-09-02T00:00:00Z","timestamp":1599004800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51706019"],"award-info":[{"award-number":["51706019"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we develop a new time-space fractional constitution relation to study the unsteady boundary layer flow over a stretching sheet. For the convenience of calculation, the boundary layer flow is simulated as a symmetrical rectangular area. The implicit difference method combined with an L1-algorithm and shift Gr\u00fcnwald scheme is used to obtain the numerical solutions of the fractional governing equation. The validity and solvability of the present numerical method are analyzed systematically. The numerical results show that the thickness of the velocity boundary layer increases with an increase in the space fractional parameter \u03b3. For a different stress fractional parameter \u03b1, the viscoelastic fluid will exhibit viscous or elastic behavior, respectively. Furthermore, the numerical method in this study is validated and can be extended to other time-space fractional boundary layer models.<\/jats:p>","DOI":"10.3390\/sym12091446","type":"journal-article","created":{"date-parts":[[2020,9,2]],"date-time":"2020-09-02T11:07:14Z","timestamp":1599044834000},"page":"1446","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Numerical Solutions of Unsteady Boundary Layer Flow with a Time-Space Fractional Constitutive Relationship"],"prefix":"10.3390","volume":"12","author":[{"given":"Weidong","family":"Yang","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"},{"name":"School of Energy and Environmental Engineering, University of Science and Technology Beijing, Beijing 100083, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5909-1301","authenticated-orcid":false,"given":"Xuehui","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"}]},{"given":"Yuan","family":"Meng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0643-390X","authenticated-orcid":false,"given":"Xinru","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Energy and Environmental Engineering, University of Science and Technology Beijing, Beijing 100083, China"}]},{"given":"Shiyun","family":"Mi","sequence":"additional","affiliation":[{"name":"Research Institute of Petroleum Exploration and Development, Xueyuan Road No.20, Haidian District, Beijing 100083, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1007\/BF00879562","article-title":"A new dissipation model based on memory mechanism","volume":"91","author":"Caputo","year":"1971","journal-title":"Pure Appl. 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