{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T09:42:17Z","timestamp":1775209337787,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2018,5,6]],"date-time":"2018-05-06T00:00:00Z","timestamp":1525564800000},"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>The time-fractional diffusion equation with mass absorption is studied in a half-line domain under the Dirichlet boundary condition varying harmonically in time. The Caputo derivative is employed. The solution is obtained using the Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate. The results of numerical calculations are illustrated graphically.<\/jats:p>","DOI":"10.3390\/e20050346","type":"journal-article","created":{"date-parts":[[2018,5,7]],"date-time":"2018-05-07T03:12:21Z","timestamp":1525662741000},"page":"346","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Time-Fractional Diffusion with Mass Absorption in a Half-Line Domain due to Boundary Value of Concentration Varying Harmonically in Time"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7492-5394","authenticated-orcid":false,"given":"Yuriy","family":"Povstenko","sequence":"first","affiliation":[{"name":"Institute of Mathematics and Computer Sciences, Faculty of Mathematical and Natural Sciences, Jan D\u0142ugosz University in Cz\u0229stochowa, al. Armii Krajowej 13\/15, 42-200 Cz\u0229stochowa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tamara","family":"Kyrylych","sequence":"additional","affiliation":[{"name":"Institute of Law, Administration and Management, Faculty of Philology and History, Jan D\u0142ugosz University in Cz\u0229stochowa, Zbierskiego 2\/4, 42-200 Cz\u0229stochowa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,5,6]]},"reference":[{"key":"ref_1","first-page":"513","article-title":"Neue Methode, das W\u00e4rmeleitungsverm\u00f6ogen der K\u00f6orper zu bestimmen","volume":"144","year":"1861","journal-title":"Ann. Phys. Chem."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1063\/1.1310118","article-title":"Diffusion waves and their uses","volume":"53","author":"Mandelis","year":"2000","journal-title":"Phys. Today"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Mandelis, A. (2001). Diffusion-Wave Fields: Mathematical Methods and Green Functions, Springer.","DOI":"10.1007\/978-1-4757-3548-2"},{"key":"ref_4","unstructured":"Vrentas, J.S., and Vrentas, C.M. (2013). Diffusion and Mass Transfer, CRC Press."},{"key":"ref_5","first-page":"145","article-title":"State of stress in an elastic space due to a source of heat varying harmonically as function of time","volume":"5","author":"Nowacki","year":"1957","journal-title":"Bull. Acad. Polon. Sci. S\u00e9r. Sci. Techn."},{"key":"ref_6","unstructured":"Nowacki, W. (1986). Thermoelasticity, Pergamon Press. [2nd ed.]."},{"key":"ref_7","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Baehr, H.D., and Stephan, K. (2006). Heat and Mass Transfer, Springer. [2nd ed.].","DOI":"10.1007\/3-540-29527-5"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zudin, Y.B. (2007). Theory of Periodic Conjugate Heat Transfer, Springer.","DOI":"10.1007\/978-3-540-70725-7"},{"key":"ref_10","unstructured":"Bergman, T.L., Lavine, A.S., Incropera, F.P., and DeWitt, D.P. (2011). Fundamentals of Heat and Mass Transfer, John Wiley & Sons. [7th ed.]."},{"key":"ref_11","unstructured":"Carslaw, H.S., and Jaeger, J.C. (1959). Conduction of Heat in Solids, Oxford University Press. [2nd ed.]."},{"key":"ref_12","unstructured":"Crank, J. (1975). The Mathematics of Diffusion, Oxford University Press. [2nd ed.]."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Polyanin, A.D. (2002). Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall\/CRC.","DOI":"10.1201\/9781420035322"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Shitzer, A., and Eberhart, R.C. (1985). Heat Transfer in Medicine and Biology, Plenum Press.","DOI":"10.1007\/978-1-4684-8285-0"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1137\/S0036139994279177","article-title":"Cooperative mechanisms of self-regulation in hierarchical living systems","volume":"60","author":"Lubashevsky","year":"2000","journal-title":"SIAM J. Appl. Math."},{"key":"ref_16","first-page":"31","article-title":"Modelling of heat transfer in biomechanics\u2014A review. Part I. Soft tissues","volume":"4","author":"Telega","year":"2002","journal-title":"Acta Bioengng. Biomech."},{"key":"ref_17","unstructured":"Jack, J., Noble, D., and Tsien, R.W. (1983). Electric Current Flow in Excitable Cells, Oxford University Press. [2nd ed.]."},{"key":"ref_18","unstructured":"Gabbiani, F., and Cox, S.J. (2010). Mathematics for Neuroscientists, Academic Press. [2nd ed.]."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Wazwaz, A.-M. (2009). Partial Differential Equations and Solitary Waves Theory, Springer.","DOI":"10.1007\/978-3-642-00251-9"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1119\/1.3559500","article-title":"Classical applications of the Klein\u2013Gordon equation","volume":"79","author":"Gravel","year":"2011","journal-title":"Am. J. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Alber, H.-D. (1998). Materials with Memory: Initial-Boundary Value Problems for Constitutive Equations with Internal Variables, Springer.","DOI":"10.1007\/BFb0096275"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Amendola, G., Fabrizio, M., and Golden, J.M. (2012). Thermodynamics of Materials with Memory: Theory and Applications, Springer.","DOI":"10.1007\/978-1-4614-1692-0"},{"key":"ref_23","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_24","unstructured":"Magin, R.L. (2006). Fractional Calculus in Bioengineering, Begell House Publishers, Inc."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press.","DOI":"10.1142\/9781848163300"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Tarasov, V.E. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer.","DOI":"10.1007\/978-3-642-14003-7"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1016\/j.cnsns.2010.05.027","article-title":"Recent history of fractional calculus","volume":"16","author":"Kiryakova","year":"2011","journal-title":"Comm. Nonlinear Sci. Numer. Simul."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Uchaikin, V.V. (2013). Fractional Derivatives for Physicists and Engineers, Springer.","DOI":"10.1007\/978-3-642-33911-0"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Povstenko, Y. (2015). Fractional Thermoelasticity, Springer.","DOI":"10.1007\/978-3-319-15335-3"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.chaos.2016.10.005","article-title":"Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems","volume":"93","author":"Owolabi","year":"2016","journal-title":"Chaos, Solitons Fractals"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1438","DOI":"10.4208\/aamm.OA-2016-0115","article-title":"Analysis of mathematics and numerical pattern formation in superdiffusive fractional multicomponent system","volume":"9","author":"Owolabi","year":"2017","journal-title":"Adv. Appl. Math. Mech."},{"key":"ref_32","first-page":"1","article-title":"On the solutions of fractional order of evolution equations","volume":"132","year":"2017","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1515\/fca-2018-0015","article-title":"Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point","volume":"21","author":"Datsko","year":"2018","journal-title":"Fract. Calc. App. Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/0893-9659(96)00089-4","article-title":"The fundamental solutions for the fractional diffusion-wave equation","volume":"9","author":"Mainardi","year":"1996","journal-title":"Appl. Math. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1461","DOI":"10.1016\/0960-0779(95)00125-5","article-title":"Fractional relaxation-oscillation and fractional diffusion-wave phenomena","volume":"7","author":"Mainardi","year":"1996","journal-title":"Chaos, Solitons Fractals"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Povstenko, Y. (2015). Linear Fractional Diffusion-Wave Equation for Scientists and Engineers, Birkh\u00e4user.","DOI":"10.1007\/978-3-319-17954-4"},{"key":"ref_37","first-page":"601","article-title":"The fractional Schr\u00f6dinger\u2013Klein\u2013Gordon equation and intermediate relativism","volume":"3","author":"Blackledge","year":"2013","journal-title":"Math. Aeterna"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1471","DOI":"10.4028\/www.scientific.net\/AMR.1049-1050.1471","article-title":"Analytical solution for the time-fractional Pennes bioheat transfer equation on skin tissue","volume":"1049\u20131050","author":"Cui","year":"2014","journal-title":"Adv. Mater. Res."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1007\/s00231-014-1300-x","article-title":"Fractional modeling of Pennes\u2019 bioheat transfer equation","volume":"50","author":"Ezzat","year":"2014","journal-title":"Heat Mass Transf."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1080","DOI":"10.1515\/fca-2015-0062","article-title":"Fractional Pennes\u2019 bioheat equation: Theoretical and numerical studies","volume":"18","author":"Ford","year":"2015","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"5061","DOI":"10.22436\/jnsa.009.07.09","article-title":"Numerical solution of fractional bioheat equation by quadratic spline collocation method","volume":"9","author":"Qin","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1186\/s40064-016-1743-2","article-title":"Solution of fractional bioheat equation in terms of Fox\u2019s H-function","volume":"5","author":"Damor","year":"2016","journal-title":"SpringerPlus"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/j.chaos.2017.04.043","article-title":"Time fractional cable equation and applications in neurophysiology","volume":"102","author":"Vitali","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1515\/fca-2018-0008","article-title":"Time-fractional diffusion with mass absorption under harmonic impact","volume":"21","author":"Povstenko","year":"2018","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1442","DOI":"10.1080\/01495739.2016.1209991","article-title":"Fractional heat conduction in a space with a source varying harmonically in time and associated thermal stresses","volume":"39","author":"Povstenko","year":"2016","journal-title":"J. Thermal Stresses"},{"key":"ref_46","unstructured":"Abramowitz, M., and Stegun, I.A. (1972). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover."},{"key":"ref_47","unstructured":"Doetsch, G. (1967). Anleitung zum praktischen Gebrauch der Laplace-Transformation und der Z-Transformation, Springer. (In German)."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/5\/346\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:03:29Z","timestamp":1760195009000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/5\/346"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,6]]},"references-count":47,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2018,5]]}},"alternative-id":["e20050346"],"URL":"https:\/\/doi.org\/10.3390\/e20050346","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,5,6]]}}}