{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:41:17Z","timestamp":1760190077775,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,16]],"date-time":"2019-06-16T00:00:00Z","timestamp":1560643200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The heat conduction equations with Caputo fractional derivative are considered in two joint half-planes under the conditions of perfect thermal contact. The fundamental solution to the Cauchy problem as well as the fundamental solution to the source problem are examined. The Fourier and Laplace transforms are employed. The Fourier transforms are inverted analytically, whereas the Laplace transform is inverted numerically using the Gaver\u2013Stehfest method. We give a graphical representation of the numerical results.<\/jats:p>","DOI":"10.3390\/sym11060800","type":"journal-article","created":{"date-parts":[[2019,6,17]],"date-time":"2019-06-17T03:24:41Z","timestamp":1560741881000},"page":"800","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Time-Fractional Heat Conduction in Two Joint Half-Planes"],"prefix":"10.3390","volume":"11","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 Science, Faculty of Mathematical and Natural Sciences, Jan Dlugosz University in Czestochowa, Armii Krajowej 13\/15, 42-200 Czestochowa, Poland"}]},{"given":"Joanna","family":"Klekot","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Faculty of Mechanical Engineering and Computer Science, Czestochowa University of Technology, Armii Krajowej 21, 42-200 Czestochowa, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/BF00281373","article-title":"A general theory of heat conduction with finite wave speeds","volume":"31","author":"Gurtin","year":"1968","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1002\/pssb.2221230241","article-title":"To the theoretical explanation of the \u201cuniversal response\u201d","volume":"123","author":"Nigmatullin","year":"1984","journal-title":"Phys. Stat. Sol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1002\/pssb.2221240142","article-title":"On the theory of relaxation for systems with \u201cremnant\u201d memory","volume":"124","author":"Nigmatullin","year":"1984","journal-title":"Phys. Stat. Sol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1080\/014957390523741","article-title":"Fractional heat conduction equation and associated thermal stresses","volume":"28","author":"Povstenko","year":"2005","journal-title":"J. Therm. Stress."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1007\/s10958-009-9636-3","article-title":"Thermoelasticity which uses fractional heat conduction equation","volume":"162","author":"Povstenko","year":"2009","journal-title":"J. Math. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"014017","DOI":"10.1088\/0031-8949\/2009\/T136\/014017","article-title":"Theory of thermoelasticity based on the space-time fractional heat conduction equation","volume":"136","author":"Povstenko","year":"2009","journal-title":"Phys. Scr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1080\/01495739.2010.511931","article-title":"Fractional Cattaneo-type equations and generalized thermoelasticity","volume":"34","author":"Povstenko","year":"2011","journal-title":"J. Therm. Stress."},{"key":"ref_8","first-page":"1778","article-title":"Fractional thermoelasticity","volume":"Volume 4","author":"Hetnarski","year":"2014","journal-title":"Encyclopedia of Thermal Stresses"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.ijheatmasstransfer.2014.06.066","article-title":"Unphysical effects of the dual-phase-lag model of heat conduction","volume":"78","author":"Rukolaine","year":"2014","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1223","DOI":"10.1007\/s00161-017-0610-x","article-title":"Thermodynamical consistency of the dual-phase-lag heat conduction equation","volume":"30","year":"2018","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_11","unstructured":"Magin, R.L. (2006). Fractional Calculus in Bioengineering, Begell House Publishers, Inc."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Sabatier, J., Agrawal, O.P., and Tenreiro Machado, J.A. (2007). Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer.","DOI":"10.1007\/978-1-4020-6042-7"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Baleanu, D., G\u00fcven\u00e7, Z.B., and Tenreiro Machado, J.A. (2010). New Trends in Nanotechnology and Fractional Calculus Applications, Springer.","DOI":"10.1007\/978-90-481-3293-5"},{"key":"ref_14","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_15","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_16","doi-asserted-by":"crossref","first-page":"1250087","DOI":"10.1142\/S0218127412500873","article-title":"Pattern formation in fractional reaction-diffusion systems with multiple homogeneous states","volume":"22","author":"Datsko","year":"2012","journal-title":"Int. J. Bifurcat. Chaos"},{"key":"ref_17","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_18","doi-asserted-by":"crossref","unstructured":"Atanackovi\u0107, T.M., Pilipovi\u0107, S., Stankovi\u0107, B., and Zorica, D. (2014). Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, John Wiley & Sons.","DOI":"10.1002\/9781118577530"},{"key":"ref_19","unstructured":"Herrmann, R. (2014). Fractional Calculus: An Introduction for Physicists, World Scientific. [2nd ed.]."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Povstenko, Y. (2015). Fractional Thermoelasticity, Springer.","DOI":"10.1007\/978-3-319-15335-3"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1016\/j.cnsns.2014.10.028","article-title":"Solitary travelling auto-waves in fractional reaction\u2013diffusion systems","volume":"23","author":"Datsko","year":"2015","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_22","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_23","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_24","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_25","doi-asserted-by":"crossref","first-page":"418","DOI":"10.2478\/s13540-011-0026-4","article-title":"Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder","volume":"14","author":"Povstenko","year":"2011","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3183","DOI":"10.1016\/j.camwa.2012.02.064","article-title":"Neumann boundary-value problems for a time-fractional diffusion-wave equation in a half-plane","volume":"64","author":"Povstenko","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1080\/01495739.2013.770693","article-title":"Fractional heat conduction in infinite one-dimensional composite medium","volume":"36","author":"Povstenko","year":"2013","journal-title":"J. Therm. Stress."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4122","DOI":"10.3390\/e15104122","article-title":"Fractional heat conduction in an infinite medium with a spherical inclusion","volume":"15","author":"Povstenko","year":"2013","journal-title":"Entropy"},{"key":"ref_29","first-page":"1284","article-title":"Fundamental solutions to time-fractional heat conduction equations in two joint half-lines","volume":"11","author":"Povstenko","year":"2013","journal-title":"Cent. Eur. J. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1287\/opre.14.3.444","article-title":"Observing stochastic processes, and approximate transform inversion","volume":"14","author":"Gaver","year":"1966","journal-title":"Oper. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1145\/361953.361969","article-title":"Algorithm 368 Numerical inversion of Laplace transform","volume":"13","author":"Stehfest","year":"1970","journal-title":"Commun. ACM"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1145\/355598.362787","article-title":"Remark on algorithm 368 Numerical inversion of Laplace transform","volume":"13","author":"Stehfest","year":"1970","journal-title":"Commun. ACM"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2984","DOI":"10.1137\/13091974X","article-title":"On the convergence of Gaver\u2013Stehfest algorithm","volume":"51","author":"Kuznetsov","year":"2013","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Rani, D., Mishra, V., and Cattani, C. (2019). Numerical inverse Laplace transform for solving a class of fractional differential equations. Symmetry, 11.","DOI":"10.3390\/sym11040530"},{"key":"ref_35","unstructured":"Prudnikov, A.P., Bry\u010dkov, Y.A., and Mari\u010dev, O.I. (1986). Integrals and Series, Vol 1: Elementary Functions, Gordon and Breach Science Publishers."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/800\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:58:51Z","timestamp":1760187531000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/800"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,16]]},"references-count":35,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["sym11060800"],"URL":"https:\/\/doi.org\/10.3390\/sym11060800","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,6,16]]}}}