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The theoretic properties of the fully discrete LDG scheme are proved in detail by mathematical induction, and the method is proved to be unconditionally stable and convergent with $ {\\rm O}(\\tau+{h^{k+1}}) $, where $ h $ is the spatial step, $ \\tau $ is the temporal step and $ k $ is the degree of the piecewise $ P^k $ polynomial. In order to show the efficiency of our method, some numerical examples are carried out by Matlab.&lt;\/p&gt;&lt;\/abstract&gt;<\/jats:p>","DOI":"10.3934\/nhm.2023022","type":"journal-article","created":{"date-parts":[[2023,1,18]],"date-time":"2023-01-18T11:57:47Z","timestamp":1674043067000},"page":"532-546","source":"Crossref","is-referenced-by-count":1,"title":["A fully discrete local discontinuous Galerkin method for variable-order fourth-order equation with Caputo-Fabrizio derivative based on generalized numerical fluxes"],"prefix":"10.3934","volume":"18","author":[{"given":"Liuchao","family":"Xiao","sequence":"first","affiliation":[{"name":"College of Science, Henan University of Technology, Zhengzhou 450001, China"}]},{"given":"Wenbo","family":"Li","sequence":"additional","affiliation":[{"name":"College of Science, Henan University of Technology, Zhengzhou 450001, China"}]},{"given":"Leilei","family":"Wei","sequence":"additional","affiliation":[{"name":"College of Science, Henan University of Technology, Zhengzhou 450001, China"}]},{"given":"Xindong","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Xinjiang Normal University, Urumqi 830017, China"}]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2023022-1","doi-asserted-by":"publisher","unstructured":"Y. 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