Plenary Lecture
Reduction Method for the Solution of Weakly Singular Integro-Differential Equations
Professor Iurie Caraus
Faculty of Mathematics and Informatics
Moldova State University
Chisinau, MOLDOVA
E-mail: ciurie.caraus@gmail.com
Abstract: Approximation of functions of a complex variable by various finite-dimensional aggregates is an important problem not only in constructive function theory and approximation but also in the justification of direct approximate methods for functional equations. This problem has been well studied for the case of functions defined on standard contours (a straight line segment, the unit circle, and so on). In the case of an arbitrary closed smooth contour Γ in the complex plane, the problem is less studied. It should be noted that conformal mapping from the arbitrary smooth closed contours to the unit circle does not solve the problem. Moreover, it makes more difficulties:
• The coefficients, kernel and right part of the transformed equation lose their smoothness;
• The power of smoothness appears in convergence speed of collocation method. So that the evaluations of convergence speed will depend from particular contour;
• The numerical schemes of researched methods become more difficult. The singularity appears in new kernel and we are not able to use the numerical schemes of mechanical quadrature method because of a singularity for new kernel.
We suggest the numerical schemes of the reduction method over the system of Faber-Laurent polynomials for the approximate solution of weakly singular integro- differential equations defined on smooth closed contours in the complex plane. We use the cut-off technique kernel to reduce the weakly singular integro- differential equation to the continuous one. Our approach is based on the Krykunov theory and Zolotarevski results.
We obtain the theoretical justification in Generalized Holder spaces.
Brief biography of the speaker: From 1996 Dr. Iurie Caraus started working at the Faculty of Mathematics and Informatics, Moldova State University.
From 2004-2011, Dr. Iurie Caraus was Associate Professor at the Faculty of Mathematics and Informatics, Moldova State University, Chisinau.
In 1998 he obtained PhD in Numerical Mathematics.
Visiting Universities
• October 2010- July 2011, Fulbright Scholar, Department of Mathematics, NC State University, Raleigh, USA;
• August 2010-September 2010,University of Boudreaux1, France;
• August 2009-August 2010, PostDoc, Center of Mathematics and Application, Lisbon, Portugal;
• May 2007-August 2008, Visiting Researcher, Department of Computer Science, Leuven, Belgium;
• April 2006-June 2006 Junior Visiting Researcher, Department of Mathematics and its Applications, Central European University, Budapest, Hungary;
• February 2005- August 2005 Visiting Researcher, Department of Mathematics and Informatics, University of Trieste, Trieste, Italy;
• 15.09.04-14.12.04 Visiting Researcher, Technische Universitat;
• Faculty of Mathematics, Chemnitz, Germany, DAAD scholarship;
• January 2003-April 2003 Visiting Researcher, NC State University, Department of Mathematics, Raleigh, USA.
Fields of Scientific Interests Collocation Methods, Cauchy Singular Integral Equations, Finite Elements Methods, Optimization, Information Security, Mathematical Economics
Publications: more than 40 articles in journals and proceedings, 3 didactical materials