Homogenization of the time-dependent heat equation on planar one-dimensional periodic structures

References

• Kuchment P. Graph models for waves in thin structures. Waves Random Media. 2002;12(4):R1–R24.
   Web of Science ®Google Scholar
• Rubinstein J, Schatzman M. Variational problems on multiply connected thin strips. I. Basic estimates and convergence of the Laplace spectrum. Arch Ration Mech Anal. 2001;160(4):271–308.
   Web of Science ®Google Scholar
• Marušić S, Marušić-Paloka E. Two-scale convergence for thin domains and its applications to some lower-dimensional models in fluid mechanics. Asymptot Anal. 2000;23(1):23–57.
   Web of Science ®Google Scholar
• Joly P, Semin A. Construction and analysis of improved Kirchoff conditions for acoustic wave propagation in a junction of thin slots. ESAIM Proc. 2008;25:44–67.
   Google Scholar
• Bouchitté G, Butazzo G, Seppecher P. Energies with respect to a measure and applications to low dimensional structures. Calc Var Partial Differ Equ. 1997;5:37–54.
   Web of Science ®Google Scholar
• Bouchitté G, Fragalà I. Homogenization of thin structures by two-scale method with respect to measures. SIAM J Math Anal. 2001;32(6):1198–1226.
   Web of Science ®Google Scholar
• Bouchitté G, Fragalà I. Homogenization of elastic thin structures: a measure-fattening approach. Special issue on optimization (Montpellier, 2000). J Convex Anal. 2002;9(2):339–362.
   Google Scholar
• Bouchitté G, Fragalà I, Rajesh M. Homogenization of second order energies on periodic thin structures. Calc Var Partial Differ Equ. 2004;20(2):175–211.
   Web of Science ®Google Scholar
• Zhikov VV. On an extension of the method of two-scale convergence and its applications. Math Sb. 2000;191:973–1014.
   Google Scholar
• Neuss-Radu M. Some extensions of two-scale convergence. C R Acad Sci Paris Sér I Math. 1996;322(9):899–904.
   Google Scholar
• Radu M. Homogenization techniques [Diploma thesis], Cluj-Napoca/Romania: University Heidelberg/Germany; 1992.
   Google Scholar
• Allaire G, Damlamian A, Hornung U. Two-scale convergence on periodic surfaces and applications. Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media; Singapore: World Scientific Publication; 1996. p. 15–25.
   Google Scholar
• Le Dret H, Raoult A. Homogenization of hexagonal lattices. Netw Heterog Media. 2013;8(2):541–572.
   Google Scholar
• Zhikov VV. Averaging of problems in the theory of elasticity on singular structures. (Russian). Izv Ross Akad Nauk Ser Mat. 2002;66(2):81–148. translation in Izv Math 2002;66(2):299–365.
   Google Scholar
• Maz'ya VG, Slutskii AS. Averaging of a differential operator on a fine periodic curvilinear net. Math Nachr. 1987;133:107–133. (in Russian).
   Web of Science ®Google Scholar
• Maz'ya VG, Slutskij AS. Averaging of differential equations on a fine mesh. Sov Math Dokl. 1987;35:371–375.
   Google Scholar
• Cioranescu D, Donato P. An introduction to homogenization. New York: The Clarendon Press, Oxford University Press; 1999. (Oxford Lecture Series in Mathematics and its Applications; 17).
   Google Scholar
• Panasenko G. Multi-scale modelling for structures and composites. Dordrecht: Springer; 2005.
   Google Scholar
• Panasenko GP. Averaging processes in framework structures. Math USSR-Sb. 1983;122(2):220–231. (in Russian). English translation in Math.USSR Sbornik, 1985;50(1):213–225. .
   Google Scholar
• Cioranescu D, Saint-Jean-Paulin J. Homogenization of reticulated structures. New-York: Springer-Verlag; 1999.
   Google Scholar
• Bondy JA, Murty USR. Graph theory with applications. New York: American Elsevier Publishing Co. Inc.; 1976.
   Google Scholar
• Cioabă SM, Ram Murty M. A first course in graph theory and combinatorics. New Delhi: Hindustan Book Agency; 2009. (Texts and Readings in Mathematics; vol. 55).
   Google Scholar
• Dautray R, Lions J-L. Mathematical analysis and numerical methods for science and technology. Berlin: Springer-Verlag; 1992. (Evolution problems. I; vol. 5).
   Google Scholar
• Evans LC. Partial differential equations. Providence: American Mathematical Society; 1998.
   Google Scholar
• Tambača J, Žugec B. A biodegradable elastic stent model. Math Mech Solids. 2019;24(8):2591–2618.
   Web of Science ®Google Scholar
• Shafarevich IR, Remizov AO. Linear algebra and geometry. Berlin: Springer Science & Business Media; 2012.
   Google Scholar
• Jungnickel D. Graphs, networks and algorithms. New York (NY): Springer; 2011. (Algorithms and Computation in Mathematics).
   Google Scholar
• Fraunfelder P., Lage C. Concepts – an object-oriented software package for partial differential equations. ESAIM: Math Model Numer Anal. 2002;36(5):937–951.
   Web of Science ®Google Scholar