Abstract:
The mankind, having improved in various fields of science and technology and having liberalized, using environment resources and more deeply interfering in the outer world, destroys the existing balance of the earth ecosystem. Research of the ways of its prevention and rehabilitation is one of the most important tasks of contemporary world. Via computer simulation mathematical modelling and application of numerical analysis make possible to forecast these or those parameters of water quality, to control and manage pollution processes. That kind of observation and prediction are cost-effective and preserve expenses that would be needed for arrangement and conduction of experiments; sometimes such approach appears to be the only way of studying relevant phenomena. Thus, mathematical modelling of diffusion processes in the environment and investigation of pollution problems is one of the most actual and interesting challenge of applied and computational mathematics. Therefore, mathematical modelling and models themselves are being constantly improved, refined and in some cases even simplified. Actually, a big variety of non-linear mathematical models describing pollution processes exist, but in the current work we only focus on linear mathematical models describing pollution transfer and diffusion in water bodies. The literature concerning the research of problems and mathematical modelling issues on the basis of classical equations of mathematical physics with classical initial-boundary conditions is quite rich. In some works concerning mathematical modelling of admixture diffusion processes in various environments, the authors have encountered with the specific type of equations that until recently were not used to describe the above mentioned processes. Such equations are known under the name of “pluri-parabolic” equations. Theoretical issues and algorithms of numerical solution of these types of equations with classical initial-boundary conditions are poorly studied, though investigation of the mentioned problems has substantial theoretical and practical value. Here should be emphasized that in some cases during the process of mathematical modelling of pollution problems we deal with initial-boundary value problems with non-classical boundary conditions as well. Quite often the questions of investigation of mathematical problems describing pollution dissemination processes get down to classical equations of mathematical physics with non-classical (e.g. non-local) initial-boundary conditions. Finally, we would like to present mathematical models with non-classical equations and non-classical boundary conditions (conditions of Cannon, Bitsadze-Samarskii, their generalization and others). In the present work some mathematical models of the mentioned type are considered, problems of their numerical analysis and respective difference methods are developed and studied.

Brief Biography of the Speaker:
David Gordeziani in 1961 graduated from Tbilisi State University, Department of Mathematics and Mechanics. In 1961-1964 post-graduate studentship; then he was junior research worker of A. Razmadze Institute of Mathematics, Georgian Academy of Sciences (1964-1968); Senior scientific worker at I. Vekua Institute of Applied Mathematics (1968-1969); Probationer of the Laboratory of Numerical Analysis in the Paris University; head of the department of Numerical Methods of I. Vekua Institute of Applied Mathematic; Supervised the scientific work of the department in shell theory, meteorology, ecology, magnetic hydro-dynamics, computation and optimization of gas pipelines, took part in special scientific works(1969-1979); Deputy Director of I. Vekua Institute of Applied Mathematics (1979-1985) where he supervised the Institute scientific-research works; in 1985-2006 director of I. Vekua Institute of Applied Mathematics; head of Department of Computational mathematics of Tbilisi State University; supervised the Institute scientific-research works concerning investigation and realization of different problems of mathematical physics and mechanics of continuum media, problems of the theory of elastic mixtures, nonlocal-in-time problems for some equations of mathematical physics, mathematical models for computation of thermo-elastic state of some energetic plants; full professor of Tbilisi State University (2006-2009); from September 2009 Emeritus at the Iv. Javakhishvili Tbilisi State University. Since 1963 till now read the lectures in programming and computational mathematics, mathematical modeling, functional analysis and computational mathematics, numerical methods of partial differential equations etc. at the Tbilisi State University. In 1966 defended Ph.D. thesis in specialty "Computational mathematics". In 1981 defended a thesis for a Doctor of Science (habilitation) degree in specialty "Computational mathematics" at the Moscow State University. Supervised preparation and defenses of 17 Candidate thesis and 7 thesis for a Doctor of Science degree (habilitation). He participated in organization and holding of many international, republic congresses, symposiums, conferences, schools on mathematics, computational mathematics, mechanics, theory of shells, hydro-dynamics, magnetic hydrodynamics, informatics (International Congresses of Mathematics, Athens Interdisciplinary Olimpia, IUTAM Symposium, etc.). Was invited to carry lecture courses and scientific researches, participation in congresses, conferences and symposiums in scientific centers of many countries. Hi is the author of more than 170 scientific publications, 4 inventions (USSR), 2 patents (USA, Sweden) and 3 Monographs; the member of the Engineering Academy of Georgia; the member of the International Academy of Computer Sciences and Systems; the honorary president of Georgian Academy of Natural Sciences, etc.; the member of editorial board of mathematical journals, supervisor and team member in various international grants and owner of different scientific and governmental awards, prizes, medals and diplomas.