Prof. Irina Burova

Saint Petersburg State University, Russia



Title of the paper: Features of the application of local splines for solving mathematical physics problems


Abstract: This report discusses the features of constructing minimal local splines. The features of constructing approximations of Lagrangian and Hermitian types include, first of all, the fact that approximations are constructed as a sum of products of basis functions and values of a certain function in grid nodes. Approximations have the following properties: they are constructed separately on each grid interval, interpolate the function at grid nodes, have a given order of approximation, and can be polynomial or non-polynomial expressions. These basis splines are then used as coordinate functions when solving boundary value problems and integral equations using variational methods. In addition, these basis splines can be used to solve integral equations by replacing the unknown function with a linear combination of basis functions with unknown coefficients (which are approximations to the solution values at the grid nodes). In this way, we obtain a framework of an approximate solution, and then we can construct a continuous or piecewise-smooth approximate solution on the entire interval using one or another type of basis splines. The construction of local continuous splines up to the 11th order of accuracy is considered. The results of using splines to construct approximations and solving integral equations and boundary value problems are discussed. The formulas of basis splines and the results of numerical experiments are given.

Bio: Prof. Irina Burova was born in 1955 in Leningrad (now St. Petersburg) Russia. Irina Burova graduated from St. Petersburg State University [SPbU], Mathematics and Mechanics Faculty (in 1978). She initially conducted research, in a government department, before moving to SPbU, where she earned her candidate degree of science (PHD) in 1986, before receiving a doctorate degree (Habilitation) in 2001. Since 2001 she has been working as a professor in the Department of Computational Mathematics at SPbU. Her research interests are in the areas of computing mathematics with an emphasis on the approximations of functions, splines and their applications. She has delivered lectures on the subjects of computing mathematics, and parallel calculations. She has had more than 240 peer reviewed papers published. Out of 250 publications, 70 papers are indexed in the Scopus database. The Hirsch index in the Scopus database is 8. The topics of the papers are related to the use of the spline approximations for solving problems of mathematical physics, including numerical solving the boundary value problems, numerical solving the integral equations, numerical solving the integro-differential equations, the approximation and the interpolation of functions. The region of interests Dynamical Systems; Non-polynomial splines; Polynomial splines; Systems Theory; Fredholm integral equations of the first and the second kind, Volterra integral equations of the first and the second kind, ill- posed problems, Boundary value problems




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