{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T23:31:49Z","timestamp":1768260709900,"version":"3.49.0"},"reference-count":29,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2016,4,19]],"date-time":"2016-04-19T00:00:00Z","timestamp":1461024000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we consider the local fractional decomposition method, variational iteration method, and differential transform method for analytic treatment of linear and nonlinear local fractional differential equations, homogeneous or nonhomogeneous. The operators are taken in the local fractional sense. Some examples are given to demonstrate the simplicity and the efficiency of the presented methods.<\/jats:p>","DOI":"10.3390\/e18040150","type":"journal-article","created":{"date-parts":[[2016,4,19]],"date-time":"2016-04-19T06:32:53Z","timestamp":1461047573000},"page":"150","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["On the Approximate Solutions of Local Fractional Differential Equations with Local Fractional Operators"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6807-6675","authenticated-orcid":false,"given":"Hossein","family":"Jafari","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, University of South Africa (UNISA), Pretoria 0003, South Africa"},{"name":"Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar 47416, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hassan","family":"Jassim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar 47416, Iran"},{"name":"Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah 64001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7855-508X","authenticated-orcid":false,"given":"Fairouz","family":"Tchier","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Saud University, P.O. 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Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/S0045-7825(98)00108-X","article-title":"Approximate analytical solution for seepage flow with fractional derivatives in porous media","volume":"167","author":"He","year":"1998","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_3","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/9789812817747"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.chaos.2004.12.030","article-title":"Synchronization of two coupled fractional order chaotic oscillators","volume":"26","author":"Gao","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1125","DOI":"10.1016\/j.chaos.2005.02.023","article-title":"Chaotic dynamics and synchronization of fractional order Arneodo\u2019s systems","volume":"26","author":"Lu","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1016\/j.chaos.2005.04.037","article-title":"A note on the fractional-order Chen system","volume":"27","author":"Lu","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_8","first-page":"1","article-title":"Approximate controllability of abstract impulsive fractional neutral evolution equations with infinite delay in Banach spaces","volume":"2013","author":"Chalishajar","year":"2013","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"33","DOI":"10.5890\/DNC.2014.03.003","article-title":"Boundary value problems for impulsive fractional evolution integrodifferential equations with Gronwall\u2019s inequality in Banach spaces","volume":"3","author":"Chalishajar","year":"2014","journal-title":"Discontin. Nonlinearity Complex."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"758","DOI":"10.1016\/S0252-9602(13)60036-3","article-title":"Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving Gronwall\u2019s inequality in Banach spaces","volume":"33","author":"Chalishajar","year":"2013","journal-title":"Acta Math. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Yan, S.P., Jafari, H., and Jassim, H.K. (2014). Local fractional Adomian decomposition and function decomposition methods for solving Laplace equation within local fractional operators. Adv. Math. Phys., 2014.","DOI":"10.1155\/2014\/161580"},{"key":"ref_12","first-page":"2574","article-title":"Local fractional Adomian decomposition method for solving two dimensional heat conduction equations within local fractional operators","volume":"9","author":"Jafari","year":"2014","journal-title":"J. Adv. Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Machado, J.A.T., Cattani, C., Baleanu, C.M., and Yang, X.J. (2014). Local fractional variational iteration and decomposition methods for wave equation on Cantor sets within local fractional operators. Abstr. Appl. Anal., 2014.","DOI":"10.1155\/2014\/535048"},{"key":"ref_14","first-page":"1055","article-title":"Local fractional variational iteration method for nonlinear partial differential equations within local fractional operators","volume":"10","author":"Jafari","year":"2015","journal-title":"Appl. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/j.amc.2015.10.072","article-title":"A new numerical technique for solving the local fractional diffusion equation: Two-dimensional extended differential transform approach","volume":"274","author":"Yang","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Yang, A.M., Yang, X.J., and Li, Z.B. (2013). Local fractional series expansion method for solving wave and diffusion equations Cantor sets. Abstr. Appl. Anal., 2013.","DOI":"10.1155\/2013\/351057"},{"key":"ref_17","first-page":"736","article-title":"Local fractional series expansion method for solving Laplace and Schrodinger equations on Cantor sets within local fractional operators","volume":"2","author":"Jafari","year":"2014","journal-title":"Int. J. Math. Comput. Res."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., Golmankhaneh, A.K., Baleanu, D., and Yang, X.J. (2014). Local fractional Sumudu transform with application to IVPs on Cantor set. Abstr. Appl. Anal., 2014.","DOI":"10.1155\/2014\/620529"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Zhao, C.G., Yang, A.M., Jafari, H., and Haghbin, A. (2014). The Yang-Laplace transform for solving the IVPs with local fractional derivative. Abstr. Appl. Anal., 2014.","DOI":"10.1155\/2014\/386459"},{"key":"ref_20","unstructured":"Yang, X.J. (2012). Advanced Local Fractional Calculus and Its Applications, World Science Publisher."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Yang, Y.J., and Hua, L.Q. (2014). Variational iteration transform method for fractional differential equations with local fractional derivative. Abstr. Appl. Anal., 2014.","DOI":"10.1155\/2014\/760957"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"19","DOI":"10.2298\/TSCI160112019H","article-title":"Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey\u2019s kernel to the Caputo-Fabrizio time-fractional derivative","volume":"20","author":"Hristov","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3390\/e18020001","article-title":"New derivatives on the fractal subset of real-line","volume":"18","author":"Golmankhaneh","year":"2016","journal-title":"Entropy"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4028","DOI":"10.3390\/e17064028","article-title":"Generalized boundary conditions for the time-fractional advection diffusion equation","volume":"17","author":"Povstenko","year":"2015","journal-title":"Entropy"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2350","DOI":"10.3390\/e16042350","article-title":"Fractional order generalized information","volume":"16","author":"Machado","year":"2014","journal-title":"Entropy"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"109","DOI":"10.5890\/JAND.2012.03.001","article-title":"Entropy analysis of fractional derivatives and their approximation","volume":"1","author":"Machado","year":"2012","journal-title":"J. Appl. 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