Plenary
Lecture
Conservative Averaging Method for Linear and NonLinear
Heat Problems
Professor Andris Buikis
University of Latvia
Member of Latvian Parliament
RIGA, Raina bulv. 29310, LV1459
LATVIA
Email: buikis@lu.lv
Abstract: This conservative averaging method was
developed 30 years ago in my doctor dissertation. Now I
and my collaborators had generalized this method for
Descartes’, cylindrical and spherical system of
coordinates.
By modeling practically interesting processes, very
often we need to consider the situation, when the medium
has a layered structure. Speaking mathematically, such
situation can be described by PDE (or its system) with
piecewise constant/continuous coefficients. In this
lecture I’ll show how conjugations conditions, nonideal
contact conditions etc. and their generalizations can be
obtained by our original method of conservative
averaging (CAM). The usage of CAM for separate
relatively thin subdomain or/and for subdomain with
large medium characteristic, leads to reduction of
domain, in which the solution must be found. To apply
CAM procedure for several layers, it is necessary to
construct special type of polynomial and rational
spline, which interpolates the integral averaged values
of the solution. New type of spline was developed for
noncontinuous solutions. In this case the original
problem with discontinuous coefficients from transforms
to problem with continuous coefficients in . The usage
of CAM for illposed inverse problems in some cases
transforms them to wellposed inverse problems.
We demonstrate this method for linear, nonlinear heat
transfer problems: convective conductive stream in
layered system, intensive steel quenching, heat exchange
in system with extended surface and electrical fuse.
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, 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 20062010.
In 2010 he was elected as Member of Saeima (Latvian
Parliament). Full Member, Latvian Academy of Sciences,
1997. Member, Senate of the Latvian Academy of Sciences,
1994 –. 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 and Associate Editor of
WSEAS Transactions in Heat and Mass Transfer 2007.
