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GEORGIAN MECHANICAL UNION saqarTvelos meqanikosTa kavSiri VIII ANNUAL MEETING Dedicated to the 110th Birthday Anniversary of Ilia Vekua OF THE GEORGIAN MECHANICAL UNION saqarTvelos meqanikosTa kavSiris VIII yovelwliuri saerTaSoriso ეძღვნება ილია ვეკუას konferencia დაბადებიდან 110 წლისთავს BOOK OF ABSTRACTS moxsenebaTa Tezisebi © GTbilisi University Press თბილისის უნივერსიტეტის გამომცემლობა 27.09 – 29.09.2017, TBILISI ISSN 2233‐355X 27.09 – 29.09.2017, Tbilisi ORGANIZERS: ორგანიზატორები: • I. Javakhishvili Tbilisi State University • ი. ჯავახიშვილის სახელობის თბილისის სახელმწიფო უნივერსიტეტის − I. Vekua Institute of Applied Mathematics − ი. ვეკუას სახელობის გამოყენებითი მათემატიკის ინსტიტუტი − Faculty of Exact and Natural Sciences − ზუსტ და საბუნებისმეტყველო მეცნიერებათა ფაკულტეტი • Georgian Mechanical Union • საქართველოს მექანიკოსთა კავშირი − Georgian National Committee of Theoretical and Applied Mechanics − საქართველოს ეროვნული კომიტეტი თეორიულ და გამოყენებით მექანიკაში • Tbilisi International Centre of Mathematics and Informatics • თბილისის საერთაშორისო ცენტრი მათემატიკასა და ინფორმატიკაში SCIENTIFIC COMMITTEE: საერთაშორისო სამეცნიერო კომიტეტი: CHAIR: Jaiani, George (Georgia) თავმჯდომარე: ჯაიანი გიორგი წევრები: მეგრელიშვილი ზურაბი MEMBERS: Kvitsiani, Tariel (Georgia) აფციაური ამირანი მეუნარგია თენგიზი Apt iauri, Amirani (Georgia) Makhviladze, Nodar (Georgia) s გაბრიჩიძე გურამი მიულერი ვოლფგანგი (გერმანია) Chinchaladze, Natalia (Georgia) Megrelishvili, Zurabi (Georgia) გულუა ბაკური პატარაია დავითი Davitashvili, Teimuraz (Georgia) Meunargia, Tengizi (Georgia) დავითაშვილი თეიმურაზი ფარცხალაძე გაიოზი Gabrichidze, Guram (Georgia) Müller, Wolfgang H. (Germany) ვაშაყმაძე თამაზი ყიფიანი გელა Gilbert, Robert (USA) Pataraia, David (Georgia) კაპანაძე გიორგი შავლაყაძე ნუგზარი Gulua, Bakuri (Georgia) Phartskhaladze, Gaioz (Georgia) კვიციანი ტარიელი შრივასტავა სურეში (კანადა) Kapanadze, George (Georgia) Shavlakadze, Nugzar (Georgia) კიკვიძე ომარი ნატალია ჩინჩალაძე Kikvidze, Omar (Georgia) Shrivastava, Suresh (Canada) Kipiani, Gela (Georgia) Vashakmadze, Tamaz (Georgia) მახვილაძე ნოდარი ჯილბერტი რობერტი (აშშ) საორგანიზაციო კომიტეტი: ORGANIZING COMMITTEE: თავმჯდომარე: ჯაიანი გიორგი CHAIR: Jaiani, George თავმჯდომარის მოადგილეები: ავაზაშვილი ნიკოლოზი, ყიფიანი გელა, VICE-CHAIRS: Nikoloz Avazashvili, Natalia Chinchaladze, Gela Kipiani ჩინჩალაძე ნატალია წევრები: MEMBERS: ბერიაშვილი მარიამი პაპუკაშვილი არჩილი Mariam Beriashvili Archil Papukashvili გაბელაია მირანდა რუხაია მიხეილი Miranda Gabelaia Mikheil Rukhaia გულუა ბაკური შარიქაძე მერი Bakuri Gulua Mary Sharikadze თევდორაძე მანანა ცუცქირიძე ვარდენი Roman Janjgava Manana Tevdoradze ნოზაძე გიორგი ჯანჯღავა რომანი George Nozadze Varden Tsutskiridze TOPICS OF THE MEETING: კონფერენციის თემატიკა: • Solid Mechanics • მყარ დეფორმად სხეულთა მექანიკა • Fluid Mechanics • ჰიდროაერომექანიკა • Solid-Fluid Interaction Problems • დრეკად მყარ და თხევად გარემოთა ურთიერთქმედების ამოცანები • Applied Mechanics • გამოყენებითი მექანიკა • Technical Mechanics • ტექნიკური მექანიკა • Related Problems of Analysis • ანალიზის მონათესავე საკითხები CONFERENCE WEB-PAGE: კონფერენციის ვებ-გვერდი: http://www.viam.science.tsu.ge/others/gnctam/annual8.htm http://www.viam.science.tsu.ge/others/gnctam/annual8.htm 1 2 LIFE, ACTIVITIES, AND SCIENTIFIC HERITAGE OF ILIA VEKUA George Jaiani Ilia Vekua Institute of Applied Mathematics & Department of Mathematics of Ivane Javakhishvili Tbilisi State University [email protected] I.Vekua’s parents: Vekua’s Parents house in The present talk is devoted to a con- Liza (Memu) Apshilava and Shesheleti cise survey of scientific, pedagogical, Nestor Vekua and educational activities of the out- Ilia Vekua was born on April 23rd, 1907 in a pic- standing Georgian mathematician and turesque region of Georgia, in Abkhazia, namely in mechanist Ilia Vekua. Biographical data the district Samurzakano in the village Shesheleti. are also given. After finishing a secondary school in the capital of If we look through the Mathematical another ravishing region of Georgia Samegrelo, Ili Vekua Subject Classification, we find Ilia Ve- Zugdidi, in 1925 Ilia Vekua was enrolled at the Fa- kua’s name among the names of a few culty of Physics and Mathematics of the Tbilisi State outstanding mathematicians and mechanists of the world: “30G20 – Ilia Vekua in University (TSU). Generalizations of Bers and Vekua type (pseudoanalytic, p-analytic, 1925 etc, functions). It’s remarkable that only Ilia Vekua has such an honour amidst Georgian scientists. In this connection we quote here Tbilisi State University an outstanding American mathematician L. Bers: “While “Vekua’s generalized analytic functions” are basically the same as my “pseudoanalytic functions”, our motivation and our aims were somewhat different”. Ilia Vekua and Nikoloz Muskhelishvili A.N. Krylov 3 4 He graduated from the university with honours in 1930 and, having Sciences. the recommendation of academician Nikoloz Muskhelishvili, left Ilia Vekua was one of the enthusiastic founders of the Razmadze Tbilisi for Leningrad (former and present Saint-Petersburg) and be- Mathematical Institute, Novosibirsk State University and its first came Ph.D. student of the well-known mathematician and mechanist A.N. Krylov. The first works of Ilia Vekua were devoted to the torsion and ben- ding of elastic bars and propagation of electric waves in an infinite layer with parallel plane boundaries. These results subsequently for- med his Ph.D thesis “Propagation of Elastic Oscillations in an Infini- te Layer” which he successfully defended at the age of 29. I.Vekua Institute of First Council of the Institute Applied Mathematics of Applied Mathematics Rector, the founder and the first director of the Institute of Applied Mathematics of the Tbilisi State University until his decease in 1977. Ilia Vekua Ilia Vekua at the age of 29 at the age of 32 When he was 32 years old, Ilia Vekua received Doctor of Science degree (A Complex Representation of Solutions of Elliptic Equations Computer’s room of the Institute of Applied and its Application to Boundary Value Problems). At the age of 39 he Mathematics (BESM-6/7) became a member of the Georgian Academy of Sciences and a Now it is called I.Vekua Institute of Applied Mathematics of TSU. corresponding member of the Soviet Academy of Sciences. In 1958 At different times he was a Rector of TSU, the second (after he was elected a member (academician) of the Soviet Academy of N. Muskhelishvili) President of the Georgian Academy of Sciences, Professor of Lomonosov Moscow State University, etc. Ilia Vekua and Ilia Vekua with the first Nikoloz Muskhelishvili I. Javakhishvili Tbilisi M. Lomonosov Georgian National graduates of Novosibirsk State University Moscow State Academy of State University University Sciences 5 6 Scientific studies of Ilia Vekua as a rule start with problems of Now, I’d like to quote the following extract from the foreword of Mechanics, continue by fundamental investigations in Complex Ana- Gaetano Fichera to the English edition of I. Vekua’s book in shell lysis, Integral and Partial Differential equations and are crowned with theory: “The present monograph by I.N. Vekua, which is now offered applications to Mechanics. Results of Ilia Vekua, published in leading scientific journals, were summarized in consequent monographs, published in Russian, English, German, Chinese (later also Georgian), and other languages. The most important of his monographs were: New Methods of Solving Elliptic Equations (1948), Generalized Analytic Equations (1948), Generalized Analytic functions (1959), Shell Theory: General Methods of Constructing (first published Russian after his decease in 1982 under the slightly different title), Ricard Courand and Ilia Vekua Ilia Vekua and Gaetano Fichera to Western applied mathematicians in an English translation, is the natural continuation of the ideological program of the Tbilisi school. It is concerned with shell theory which, as is well known, constitutes an elevated and difficult generalization of plane elasticity. The rele- vant geometric continua of shell theory, although two- dimensional, are no longer planar and their geometric properties deeply influence then in 1985 by Pitman Advanced Publishing Program, and in 2007 by the analysis of the problems that arise in this important field of elas- the Tbilisi University Press in ticity.” Georgian. All these monographs Once again L. Bers. He wrote: “In the theory were awarded by different state of elliptic partial differential equations in the pla- prizes. ne Vekua followed two different paths. One, si- Richard milar to the work of Stefan Bergman, led to Courant explicit formulas for solutions of elliptic equa- and David tions with real analytic coefficients, in terms of Hilbert in arbitrary holomorphic functions of a complex their well- variable. Vekua applied his formulas to very ge- known book “Methods of Mathematical Physics” neral boundary value problems and, in this con- were skeptical about the use of general complex nection, obtained a result which is today recog- representations of solutions of PDEs. Later, Lipman Bers nizable as a precursor of the Atiyah-Singer Courant confessed that the above opinion was index theorem. totally wrong and it was proved by Vekua’s The other main direction of Vekua’s investigations was concerned methods developed on the basis of general com- David Hilbert with properties of systems of two first order elliptic equations, viewed plex representations. as generalized Cauchy- Riemann equations. This time