Plenary Lecture

Investigation and Approximate Resolution of One Nonlinear Parabolic Integro-Differential Model

Professor Temur Jangveladze
Ilia Vekua Institute of Applied Mathematics
Ivane Javakhishvili Tbilisi State University
Department of Mathematics
Georgian Technical University
E-mail: tjangv@yahoo.com

Abstract: The relevance of partial integro-differential equations can be motivated by their applications in mathematical modeling of many natural processes. Most of these mathematical equations and systems are nonlinear and this circumstance significantly complicates study of such models. Investigation and approximate resolution of initial-boundary value problems for these models are the actual sphere of contemporary mathematical physics and numerical analysis. One such type parabolic integro-differential model is obtained at mathematical modeling of process of electro-magnetic field penetration in the substance. In the quasistationary approximation, after reduction of corresponding nonlinear well known system of Maxwell equations, one new kind parabolic integro-differential model is obtained in the work - Gordeziani D.G., Dzhangveladze T.A., Korshiya T.K. Existence and Uniqueness of a Solution of Certain Nonlinear Parabolic Problems. Differential’nye Uravnenyia, 1983, V.19, N7, p.1197-1207. Maxwell model and its abovementioned integro-differential variant is complex for investigation and practical study of certain diffusion problems. Therefore, its comparatively simplified versions are often used and studied. Investigation of this type integro-differential models began in the abovementioned work. Thereafter many scientific works were dedicated to these type integro-differential models. There are still many open questions in this direction. We study existence, uniqueness and asymptotic behavior as t->∞ v of solution of initialboundary value problems for scalar case of abovementioned model as well as numerical solution of these problems. Special attention is paid to construction of discrete analogs corresponding to one-dimensional model, as well as to construction, analysis and computer realization of discrete analogs of considered initial-boundary value problems. Theoretical results to numerical ones are compared. Wider class of nonlinearity is studied than one has been investigated before.

Brief Biography of the Speaker: Temur Jangveladze (Dzhangveladze) graduated from Ivane Javakhishvili Tbilisi State University (TSU) Faculty of Applied Mathematics and Cybernetics in 1977. Professor Jangveladze has earned his PhD (candidate degree) in “Computational Mathematics” in 1984. In 1998 he defended Doctor of Science (Habilitation) Degree in specialty “Theoretical Bases of Mathematical Modeling, Numerical Methods, Program Complexes”. He was Junior Scientific Researcher (1977-1983); Scientific Researcher (1983-1988); Senior Scientific Researcher (1988-1998); Leading Scientific Researcher (1998-present) of Ilia Vekua Institute of Applied Mathematics (VIAM) of TSU; Professor (1984-2009) and Invited Professor (2009-present) at TSU; Professor (2006-2010) at Ilia State University; Professor (2010-2014) at Caucasus University and Professor (2013-present) at Georgian Technical University. Since 1977 till now Professor Jangveladze gives lectures at various universities. Field of his scientific interests is Nonlinear Differential and Integro-Differential Equations, Numerical Analysis, Nonlocal Boundary and Initial Value Problems, Mathematical Modeling, etc. The full list of his publications comprises more than 180 scientific papers and books. Professor Jangveladze is editor and member of editorial board of several international scientific journals. He was the member of scientific and program committees and the participant of many conferences. He is chair of the Enlarged Sessions of the Seminar of VIAM, Section of Partial Differential Equations and chair of the regular monthly seminar which is dedicated to Academician A. Bitsadze. He is holder of various national and international grant awards. In 2012-2013 he was awarded grant of Fulbright Visiting Scholar Program which gave him opportunity to visit Department of Applied Mathematics at Naval Postgraduate School in Monterey, CA, USA as Visiting Professor and to prepare main part of the monograph - Jangveladze T., Kiguradze Z., Neta B. Numerical Solutions of Three Classes of Nonlinear Parabolic Integro-Differential Equations. Elsevier. Academic Press, 2016. Certain part of this monograph is dedicated to the model which will be considered in our plenary lecture.

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