{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T10:35:22Z","timestamp":1760783722705,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2013,9,18]],"date-time":"2013-09-18T00:00:00Z","timestamp":1379462400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We investigate solutions of a generalized diffusion equation that contains nonlinear terms in the presence of external forces and reaction terms. The solutions found here can have a compact or long tail behavior and can be expressed in terms of the q-exponential functions present in the Tsallis framework. In the case of the long-tailed behavior, in the asymptotic limit, these solutions can also be connected with the L\u00b4evy distributions. In addition, from the results presented here, a rich class of diffusive processes, including normal and anomalous ones, can be obtained.<\/jats:p>","DOI":"10.3390\/e15093931","type":"journal-article","created":{"date-parts":[[2013,9,18]],"date-time":"2013-09-18T11:52:17Z","timestamp":1379505137000},"page":"3931-3940","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Solutions of Some Nonlinear Diffusion Equations and Generalized Entropy Framework"],"prefix":"10.3390","volume":"15","author":[{"given":"Ervin","family":"Lenzi","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maike","family":"Dos Santos","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Flavio","family":"Michels","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Renio","family":"Mendes","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luiz","family":"Evangelista","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual de Maring\u00e1, Av. Colombo 5790, Maring\u00e1-PR 87020-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,9,18]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Pekalski, A., and Sznajd-Weron, K. (1999). Anomalous Diffusion: From Basics to Applications, Springer. Lecture Notes in Physics.","key":"ref_1","DOI":"10.1007\/BFb0106828"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/S0378-4371(98)00614-1","article-title":"Anomalous transport in disordered systems under the influence of external fields","volume":"266","author":"Metzler","year":"1999","journal-title":"Physica A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5015","DOI":"10.1103\/PhysRevLett.80.5015","article-title":"Non-Markovian configurational diffusion and reaction coordinates for protein folding","volume":"80","author":"Plotkin","year":"1998","journal-title":"Phys. Rev. Lett."},{"doi-asserted-by":"crossref","unstructured":"Muskat, M. (1937). The Flow of Homogeneous Fluid Through Porous Media, McGraw-Hill.","key":"ref_4","DOI":"10.1063\/1.1710292"},{"doi-asserted-by":"crossref","unstructured":"Polubarinova-Kochina, P.Y. (1962). Theory of Ground Water Movement, Princeton University Press.","key":"ref_5","DOI":"10.1515\/9781400876044"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1017\/S0022112077002328","article-title":"Viscous sheets advancing over dry bed","volume":"81","author":"Buckmaster","year":"1977","journal-title":"J. Fluid Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.physa.2005.09.007","article-title":"Nonextensive statistics in viscous fingering","volume":"362","author":"Grosfils","year":"2006","journal-title":"Physica A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"260601","DOI":"10.1103\/PhysRevLett.105.260601","article-title":"Thermostatistics of overdamped motion of interacting particles","volume":"105","author":"Andrade","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1088\/0305-4470\/30\/4\/006","article-title":"Anomalous diffusion in linear shear flows","volume":"30","author":"Compte","year":"1997","journal-title":"J. Phys. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1007\/s100510170114","article-title":"Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and entropy","volume":"22","author":"Giordano","year":"2001","journal-title":"Eur. Phys. J. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"R2197","DOI":"10.1103\/PhysRevE.54.R2197","article-title":"Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and entropy","volume":"54","author":"Tsallis","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1417","DOI":"10.1103\/PhysRevE.61.1417","article-title":"Anomalous diffusion with absorption: Exact time-dependent solutions","volume":"61","author":"Drazer","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"021107","DOI":"10.1103\/PhysRevE.67.021107","article-title":"Derivation of nonlinear Fokker-Planck equations by means of approximations to the master equation","volume":"67","author":"Curado","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1140\/epjb\/e2009-00172-9","article-title":"Dynamics of normal and anomalous diffusion in nonlinear Fokker-Planck equations","volume":"70","author":"Curado","year":"2009","journal-title":"Eur. Phys. J. B"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"036108","DOI":"10.1103\/PhysRevE.68.036108","article-title":"Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology","volume":"68","author":"Chavanis","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1063\/1.1421062","article-title":"On a general link between anomalous diffusion and nonextensivity","volume":"43","author":"Frank","year":"2002","journal-title":"J. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/0378-4371(95)00211-1","article-title":"Non-Extensive statistical mechanics and generalized Fokker-Planck equation","volume":"222","author":"Plastino","year":"1995","journal-title":"Physica A"},{"key":"ref_18","first-page":"16","article-title":"Nonlinear diffusion equation, Tsallis formalism and exact solutions","volume":"342","author":"Silva","year":"2004","journal-title":"Physica A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/S0378-4371(00)00006-6","article-title":"A nonextensive maximum entropy approach to a family of nonlinear reaction-diffusion equations","volume":"280","author":"Plastino","year":"2000","journal-title":"Physica A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1063\/1.2828730","article-title":"On the non-linear Fokker-Planck equation associated with \u03ba-entropy","volume":"965","author":"Wada","year":"2007","journal-title":"AIP Conf. Proc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1140\/epjb\/e2009-00159-6","article-title":"Asymptotic solutions of a nonlinear diffusive equation in the framework of \u03ba-generalized statistical mechanics","volume":"70","author":"Wada","year":"2009","journal-title":"Eur. Phys. J. B"},{"unstructured":"Frank, T.D. (2005). Nonlinear Fokker-Planck Equations, Springer.","key":"ref_22"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"912","DOI":"10.1016\/j.jcp.2012.08.027","article-title":"A spacetime smooth artificial viscosity method for nonlinear conservation laws","volume":"235","author":"Reisner","year":"2013","journal-title":"J. Comput. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1016\/S0378-4371(98)00437-3","article-title":"The role of constraints within generalized nonextensive statistics","volume":"261","author":"Tsallis","year":"1998","journal-title":"Physica A"},{"doi-asserted-by":"crossref","unstructured":"Coffey, W.T., Kalmykov, Y.P., and Waldron, J.T. (2004). The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, World Scientific Publishing Company.","key":"ref_25","DOI":"10.1142\/9789812795090"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"426","DOI":"10.2478\/s13540-012-0031-2","article-title":"Velocity and displacement correlation functions for fractional generalized Langevin equations","volume":"15","author":"Sandev","year":"2012","journal-title":"Fract. Calc. Appl. Anal."},{"unstructured":"Weiss, G.W. (1994). Aspects and Applications of the Random Walk, North Holland.","key":"ref_27"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","article-title":"The random walk\u2019s guide to anomalous diffusion: A fractional dynamics approach","volume":"339","author":"Metzler","year":"2000","journal-title":"Phys. Rep."},{"doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","key":"ref_29","DOI":"10.1142\/9789812817747"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1080\/10407790.2013.778719","article-title":"Higher-order numerical scheme for the fractional heat equation with dirichlet and neumann boundary conditions","volume":"63","author":"Priya","year":"2013","journal-title":"Numer. Heat Transf. Part B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2527","DOI":"10.1016\/j.physa.2011.12.035","article-title":"Generalized space-time fractional diffusion equation with composite fractional time derivative","volume":"391","author":"Tomovski","year":"2012","journal-title":"Physica A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.physa.2006.09.001","article-title":"Fractional nonlinear diffusion equation, solutions and anomalous diffusion","volume":"375","author":"Silva","year":"2007","journal-title":"Physica A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1137\/110834342","article-title":"Continuous dependence estimates for nonlinear fractional convection-diffusion equations","volume":"44","author":"Alibaud","year":"2012","journal-title":"SIAM J. Math. Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1007\/s10543-011-0327-3","article-title":"The discontinuous Galerkin method for fractional degenerate convection-diffusion equations","volume":"51","author":"Cifani","year":"2011","journal-title":"BIT Numer. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1088\/0305-4470\/25\/4\/029","article-title":"A nonlinear model of heat conduction","volume":"25","author":"Pascal","year":"1992","journal-title":"J. Phys. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"056303","DOI":"10.1103\/PhysRevE.70.056303","article-title":"Similarity solutions of nonlinear diffusion problems related to mathematical hydraulics and the Fokker-Planck equation","volume":"70","author":"Daly","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"6960","DOI":"10.1103\/PhysRevE.60.6960","article-title":"Theory of creeping gravity currents of a non-Newtonian liquid","volume":"60","author":"Gratton","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1016\/j.physa.2006.10.010","article-title":"A family of evolution equations with nonlinear diffusion, Verhulst growth, and global regulation: Exact time-dependent solutions","volume":"375","author":"Troncoso","year":"2007","journal-title":"Physica A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3589","DOI":"10.1103\/PhysRevLett.75.3589","article-title":"Statistical mechanical foundation of the ubiquity of Lvy distributions in nature","volume":"75","author":"Tsallis","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"5442","DOI":"10.1103\/PhysRevLett.77.5442","article-title":"Errata: Statistical mechanical foundation of the ubiquity of Lvy distributions in nature [Phys. Rev. Lett. 75, 3589 (1995)]","volume":"77","author":"Tsallis","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2398","DOI":"10.1103\/PhysRevE.60.2398","article-title":"Nonextensive foundation of Levy distributions","volume":"60","author":"Prato","year":"2000","journal-title":"Phys. Rev. E"},{"doi-asserted-by":"crossref","unstructured":"Murray, J. (1989). Mathematical Biology, Springer.","key":"ref_42","DOI":"10.1007\/978-3-662-08539-4"},{"key":"ref_43","first-page":"1","article-title":"Recherches mathematiques sur la loi daccroissement de la population","volume":"18","author":"Verhulst","year":"1845","journal-title":"Mem. Acad. R. Belg."},{"doi-asserted-by":"crossref","unstructured":"Mazumdar, J. (1999). An Introduction to Mathematical Physiology and Biology, Cambridge University Press.","key":"ref_44","DOI":"10.1017\/CBO9781139173278"},{"unstructured":"Hutchinson, G.E. (1978). An Introduction to Population Ecology, Yale University Press.","key":"ref_45"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1016\/j.chaos.2004.03.035","article-title":"An oscillatory population model","volume":"22","author":"Leach","year":"2004","journal-title":"Chaos Soliton. Fract."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/9\/3931\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:49:24Z","timestamp":1760219364000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/9\/3931"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,9,18]]},"references-count":46,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2013,9]]}},"alternative-id":["e15093931"],"URL":"https:\/\/doi.org\/10.3390\/e15093931","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2013,9,18]]}}}