{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T10:56:37Z","timestamp":1760784997483,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T00:00:00Z","timestamp":1631836800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, the time-fractional wave equation associated with the space-fractional Fokker\u2013Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the Green function G3(t) of the three-term time-fractional wave equation with constant coefficients is also studied for two physical and biological models. The explicit analytic solutions, for the two studied models, are expressed in terms of the Weber, hypergeometric, exponential, and Mittag\u2013Leffler functions. The relation to the diffusion equation is given. The asymptotic behaviors of the Mittag\u2013Leffler function, the hypergeometric function 1F1, and the exponential functions are compared numerically. The Gr\u00fcnwald\u2013Letnikov scheme is used to derive the approximate difference schemes of the Caputo time-fractional operator and the Feller\u2013Riesz space-fractional operator. The explicit difference scheme is numerically studied, and the simulations of the approximate solutions are plotted for different values of the fractional orders.<\/jats:p>","DOI":"10.3390\/axioms10030230","type":"journal-article","created":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T21:23:29Z","timestamp":1631913809000},"page":"230","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["The Approximate and Analytic Solutions of the Time-Fractional Intermediate Diffusion Wave Equation Associated with the Fokker\u2013Planck Operator and Applications"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1970-4847","authenticated-orcid":false,"given":"Entsar A.","family":"Abdel-Rehim","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Suez Canal University, Eldaree Street, Ismailia 41522, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,17]]},"reference":[{"key":"ref_1","unstructured":"Risken, H. 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