Plenary Lecture

Green’s Function Method for Connected Regular Domains and Applications of the Conservative Averaging Method for Transient Processes

Professor Andris Buikis
Institute of Mathematics and Computer Science
University of Latvia

Abstract: Green’s functions have long been used to create integral representations for boundary value problems. Although the method is usually associated with canonical domains, we propose to use this method for non-canonical adjacent domains. We generalize the Green’s function method by conjugation conditions along the surface connecting two or more neighbour domains. This generalization is being used for elliptic, parabolic and hyperbolic types of partial differential equations. The idea of conservative averaging method is at least 100 years old; the origins can be found in papers of Knezer A., Samarskii A. and Tikhonov A. This idea was employed in the modeling of heat transfer processes and improvement of oil recovering process. Practically in all papers non-classical boundary conditions will be obtained in the intuitive form by obtaining some specific peculiarities of concrete task, without well-founded mathematical basis. The mathematical basis was given in thesis of doctor of mathematics Buikis A. In the thesis the method was named “conservative averaging method.” In this publication of 1980ies and other papers was used the function approximation with polynomial, including polynomial and rational splines. Later exponential approximation for electric wires and car fuses was developed. Now, together with colleagues Professor Kalis H. and others we use new hyperbolic approximation with numeral parameters. In the lecture are given numerical calculations for ordinary and partial differential equations as samples of several mathematical models.

Brief Biography of the Speaker: Andris Buikis received the M.S. in numerical mathematics from University of Latvia (Faculty of Physics and Mathematics) in 1963 and Dr.math. (Candidate of Science in former USSR), University of Latvia, in 1970. He was Junior Researcher, Senior Researcher, Computing Centre, University of Latvia, 1962 – 1972. Assistant Professor and Head of Chair of Applied Mathematics, 1972 – 1976 and Head of Chair of Differential Equations and Numerical Methods, Faculty of Physics and Mathematics, University of Latvia, 1976 – 1984. Dr. habil. math. (Doctor of Science in former USSR), University of Kasan, Russia, 1988.Professor, University of Latvia, 1991. Director, Institute of Mathematics, Latvian Academy of Sciences and Latvian University, 1991 - 1996; 2003 – 2006 and Director, Science and Dialogue Centre of Latvia, 1993 -2007. Head of Laboratory of Mathematical Technologies, Institute of Mathematics and Computer Science, University of Latvia 2006-2010. From 2010 he was thrice elected as Member of Saeima (Latvian Parliament), now till year 2018. Full Member, Latvian Academy of Sciences, 1997.Member, Senate of the Latvian Academy of Sciences, 1994 –2012.The Latvian Academy of Sciences Piers Bohl Prize for a cycle of papers “Method of Conservative Averaging, Theory and Applications”, 2005. Member of Editorial Advisory Board, Journal Mathematical Modelling and Analysis (The Baltic Journal on Mathematical Applications, Numerical Analysis and Differential Equations), Lithuania 1999- Associate Editor of WSEAS Transactions in Heat and Mass Transfer 2007-, Associate Editor ofJournal of Vortex Science and Technology 2012-,Fluid Mechanics 2015-and Academic Editor of Current Research International 2014-.

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