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By combining the discrete energy method and the mathematical induction method, the proposed methods proved to be unconditional stable and convergent. In order to overcome the possible singularity of the solution near the initial stage, a difference scheme based on non-uniform mesh is also given. Some numerical experiments are carried out to support our theoretical results. The results indicate that the our two main schemes has the almost same accuracy and the fast scheme can reduce the storage and computational cost significantly.&lt;\/p&gt;&lt;\/abstract&gt;<\/jats:p>","DOI":"10.3934\/nhm.2023011","type":"journal-article","created":{"date-parts":[[2022,12,22]],"date-time":"2022-12-22T06:42:45Z","timestamp":1671691365000},"page":"291-309","source":"Crossref","is-referenced-by-count":0,"title":["Effective difference methods for solving the variable coefficient fourth-order fractional sub-diffusion equations"],"prefix":"10.3934","volume":"18","author":[{"given":"Zhe","family":"Pu","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences and V.C. and V.R. Key Lab, Sichuan Normal University, Chengdu 610068, China"},{"name":"School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China"}]},{"given":"Maohua","family":"Ran","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences and V.C. and V.R. Key Lab, Sichuan Normal University, Chengdu 610068, China"},{"name":"School of Mathematics, Aba Teachers University, Aba 623002, China"}]},{"given":"Hong","family":"Luo","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences and V.C. and V.R. Key Lab, Sichuan Normal University, Chengdu 610068, China"}]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2023011-1","doi-asserted-by":"publisher","unstructured":"V. Srivastava, K. N. Rai, A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues, <i>Math. Comput. Model.<\/i>, <b>51<\/b> (2010), 616\u2013624. https:\/\/doi.org\/10.1016\/j.mcm.2009.11.002","DOI":"10.1016\/j.mcm.2009.11.002"},{"key":"key-10.3934\/nhm.2023011-2","doi-asserted-by":"publisher","unstructured":"Y. Z. Povstenko, Fractional Cattaneo-type equations and generalized thermoelasticity, <i>J. Therm. 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