ὅδε οἶκος, ὦ ἑταῖρε, μνημεῖον ἐστιν ζῴων τῶν σοφῶν ἀνδρῶν, καὶ τῶν ἔργων αὐτῶν

Approximation of integral and differential operators and the corresponding applications are the subject of research. Since it belongs to the following areas: approximation theory, numerical analysis and functional analysis, we expect new results in these areas of mathematics, software implementation, as well as significant applications in telecommunications, computer sciences, physics and economics. Research will be focused to approximation of various classes of integral and differential operators, construction and analysis of interpolation and quadrature processes and solving integral equations and ordinary and partial differential equations. Besides linear operators, the problems with nonlinear operators will be treated in order to solve nonlinear problems. Special attention is paid to the methods for solving boundary and initial-boundary problems for partial differential equations. Constructive problems and stability and convergence of difference schemes will be investigated. A recent progress in the weighted polynomial approximation will be used to obtain efficient and stable methods for solving certain classes of integral equations and contour problems with differential equations. Approximation and development of stable algorithms for unbounded operators will be based on a regularization process. Integral representations of special functions will enable constructions of fast and efficient algorithms for calculating special functions and integral transformations.

**Keywords: ** approximation of operators; interpolation; orthogonality;
quadratures; differential equations

Approximations of functions, functionals and operators are the main part of the Approximation Theory and a basis for the development of numerical and symbolic methods. Research in these areas, beside a significant theoretical progress in mathematics (approximation theory, numerical analysis, functional analysis), provide the main tools for creating new mathematical models and computer simulations of high accuracy. It can influence a variety of new techniques for treating very complex problems in applied and computational sciences. Construction of efficient and stable algorithms and their implementation lead to a high quality software for solving such complex problems. Our research will be focused in that direction: the approximation of various classes of integral and differential operators defined on the important functional spaces, as well as the design and analysis of interpolation operators, including numerous applications in the field of interpolation and quadrature processes, integral equations and ordinary and partial differential equations. Besides linear (bounded and unbounded) operators, some problems with nonlinear operators will be treated in order to solve the corresponding nonlinear problems. Thus, an important part of our research will take interpolation operators and processes, including interpolation as a general principle (e.g. interpolation spaces). For polynomial systems we follow the monograph [G. Mastroianni, G.V. Milovanović, Interpolation Processes - Basic Theory and Applications, Springer Verlag, 2008]. Weighted polynomial approximations, as well as special non-polynomial systems (Muntz's and other generalized systems), can be used in the new interpolation and quadrature processes and thereby provide methods for solving problems with singularities, quasi-singularities, and so on. A recent progress in weighted polynomial approximation will be used to obtain efficient and stable methods for solving certain classes of integral equations and some boundary problems for differential equations. Based on our previous work [G.V. Milovanović, A.S. Cvetković, Nonstandard Gaussian quadrature formulas based on operator values, Adv. Comput. Math. 32 (2010), 431-486], an idea of numerical construction and analysis of non-standard quadrature formulas of Gaussian type in one and more dimensions, for some special families of linear operators acting in subspaces of polynomials (algebraic, trigonometric, Muntz's), splines, etc. will be developed. Special cases with average operator (Steklov operator), difference and differential operators will be treated separately, where the theory of orthogonality plays an important role. Using the so-called multiple-orthogonal polynomials, related to Hermite-Pade's approximants [A.I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 423-447], it is possible to obtain generalized Birkhoff-Young's quadrature for analytic functions in the complex plane, to give characterization of such quadratures in terms of multiple-orthogonal polynomials and to prove the existence and uniqueness of these quadratures. Also, some constructive problems of optimal quadrature formulae in the sense of Sard in certain Hilbert spaces will considered. Starting from the integral representation of special functions and series and constructing the corresponding quadrature processes we will obtain some efficient algorithms for a fast and stable computation of some classes of special functions, integral transforms, summation of slowly convergent series, and the generalized inverses of linear transformations. Taking into consideration its significant applications in many scientific fields, the integration of fast oscillatory functions and numerical methods for solving differential equations will be a permanent subject of the research. A special attention will be paid to methods for solving boundary and initial-boundary problems for partial differential equations. The problems of construction and stability of difference schemes, as well as their convergence will be considered. Three issues will be treated separately: (1) Boundary and initial-boundary problems with weak solutions (construction of difference schemes and testing their properties, convergence rate estimates compatibles with the smoothness of input data and different methods for obtaining such estimates, difference schemes on non-uniform meshes, coefficient stability of difference schemes), (2) Problems with interfaces and the problems with singular coefficients, transmission problems, especially problems in disjoint domains (compatibility conditions, conjugation conditions, properties of solutions, the corresponding functional spaces, a priori estimates, difference schemes and their characteristics), (3) Non-linear boundary and initial-boundary problems, especially "blow-up" problems (a priori estimates, stability, estimates of the moment when the solution becomes unlimited ("blow-up"), difference schemes and their properties - a priori estimates, stability, convergence, numerical "blow up"). Fredholm integral equations of second kind with compact integral operators (with a locally smooth and weakly singular kernel), and the corresponding equations with noncompact integral operators (Abel equation, Cauchy singular equation, Winer-Hopf equation, etc.) will be investigated. The new projection and Nystr's methods will be developed, including some efficient iterative methods. By development of appropriate multi-dimensional interpolation, integral equations in various domains (polygonal area, triangular area, simplex, etc.), interesting in various applications, will be considered. Special attention will be given to boundary problems for ordinary differential equations that can be treated via integral equations using a previous reformulation, as well as to integral equations (boundary integral equations) representing a reformulation of boundary problems for partial differential equations. Also, the subject of our research will be approximation and development of stable algorithms for unbounded operators, such as derivatives. Using the general principles of stabilization (Tikhonov-Morozov, Lavrentiev) it will be developed stable numerical methods for calculating the derivatives in the case of data with numerical "noise", using a suitable reformulation of ill-conditioned problems to Volterra and Fredholm integral equations. Finally, for algorithms, developed on this project, a software implementation will be implemented. Also, some results will be applied to particular problems in the field of telecommunications, computerized tomography, and financial and actuarial mathematics. In order to popularize the science and to provide a wide availability of the research project, we will organize a site that will be constantly updated.

Significant theoretical results in approximation theory and numerical analysis, new algorithms and software implementations are expected, as well as applications in the area of mathematics, physics, telecommunications, economy, etc. More precise, results are expected in the following areas: (1) interpolation and quadrature processes for problems with singularities in one dimension and several dimensions; (2) nonstandard Gaussian quadrature formulae based on operator values, integration of the fast oscillating functions, optimal quadrature processes - characterization and construction, integration of analytic functions in the complex plane - characterization in terms of multiple orthogonality, proof of existence and uniqueness; (3) approximation of compact and non-compact integral operators and applications in integral equations, multivariable integral equations on polygonal regions, over triangles, etc.; (4) boundary problems for ordinary and partial differential equations treated via integral equations using a previous reformulation; (5) boundary and initial-boundary problems for partial differential equations with weak solutions - construction of difference schemes on uniform and non-uniform meshes and qualitative analysis of properties, convergence analysis, stability analysis; (6) problems with interfaces and problems with singular coefficients, transmission problems; (7) nonlinear boundary and initial-boundary problems for partial differential equations; (8) nonlinear integral equations; (9) approximation and developments of stable algorithms for unbounded operators . regularizations and obtaining stable methods for differentiation; (10) integral representations of the special functions and series - construction of the fast and efficient algorithms for the computation of some classes of special functions, integral transformations, summation of slowly convergent series, as well as a computation of the generalized inverses of linear transformations. Results of the research are going to be published in the leading international journals, and there is expectation that there is going to appear another monograph as the continuation of the existing one [G. Mastroianni, G.V. Milovanović, Interpolation Processes - Basic Theory and Applications, Springer Verlag, 2008], which should include some of the problems which are subject of the research in the present proposal. Also, as results of research couple of software packages are going to be constructed which are related to approximation of the solutions of integral and differential equations. Finally, for five students participating we are expecting their successful joining to the scientific work and scientific community.

Research significance can be understood at first in the field of approximation theory, numerical analysis and functional analysis as a whole, but can be viewed also through the application in the other areas of science and technology. Constructive approximations of operators can be viewed through the processes of the convergence of a sequence of operators in the respective functional spaces. Problem of the construction of the convergent sequence of operators is closely related with the understanding of the functional space in which processes take place. Not rarely, the existence of the sequence of operators, with a prescribed special feature, which converges to an arbitrary element of the operator space gives strong insights in the structure of the function and operator space. Construction of the sequence of operators with a prescribed property, for example of operators of the finite range, is substantial for a construction of numerical methods. Basically, numerical analysis uses results from approximation theory and functional analysis for the constructing successful algorithms and software which can be used in the scientific computations. Construction of the efficient numerical algorithms and a revision of the existent algorithms is another aspect in which the research significance can be viewed. The project research include constructions of the numerical algorithms for approximating integral and differential operators. Approximation of the values of integral operators has significance in its own. Numerical integration is inevitable in physics, chemistry, technical sciences, economy, etc. Even more, this methods of analysis find applications in biology and other sciences which are not traditionally connected with mathematics. Approximation of integral operators provides methods for the solution of integral equations, which is subject of the research as well. Research effort is specially oriented to the construction of approximations of the solutions of integral equations of Fredholm type of the first and second kind, Abel equation, Cauchy equation, Winer-Hopf equations and others. Fredholm integral equations find their applications in the signal processing, Abel integral equation appears in many applications connected to interferometry, stereography, seismology, tomography, Cauchy integral equation has applications in mechanics, for example aeronautics, and integral operator itself has direct applications in a wide variety of inverse problems. Approximation of the solutions, which can be computed using approximation of integral operators, in applications usually has an interpolation form. In many cases solutions are not smooth, hence, if one is interested in computation of computing machines application of some modified numerical methods which use averaged values of the solutions in interpolation points is needed. Furthermore, an appearance of singularities has a direct influence the usage of some new functional spaces needed to enable the statement of the convergence results. Numerical methods for the approximation of differential operators, also, enable a construction of the approximation of the solutions of the corresponding differential equations. A class of differential operators which is going to be studied are differential operators connected to Fokker-Planck differential equation. Approximation of the solutions of the Fokker-Planck differential equation enables applications in quantum mechanics, telecommunications, economy and all areas of science and technology in which stochastic processes are involved. The other group of the results is directed to the studies of Navier-Stokes equations and the well-known problems connected to the construction of the numerical algorithm which enable efficient solution of this class of differential equations. Applications of Navier-Stokes equations are numerous, for example, those completely characterize moving of the fluid in mechanics. Research also includes the development of the numerical methods connected to the regularization problems of the computation of unbounded operators. Implementation of new algorithms developed during research and development of the high quality software are also goals in this project. It will significantly increase the application of the new numerical methods in the other areas of science, since it will enable usage of the new methods for the non specialized persons in numerical analysis. Finally, a significance of the research can be viewed in the amount of the literature which is published in the areas connected the research topics. Beside huge number of papers and the books published, we mention just few of the book titles which directly reflects the application potentials: [1] F. Sauvigny, Partial Differential Equations 1: Foundations and Integral Representations, Springer-Verlag, Berlin - Heidelberg, 2007; [2] F. Sauvigny, Partial Differential Equations 2: Functional Analytic Methods, Springer-Verlag, Berlin - Heidelberg, 2007; [3] Ch. W. Groetsch, Stable Approximate Evaluation of Unbounded Operators, Lecture Notes in Mathematics, Vol. 1894, Springer-Verlag, Berlin - Heidelberg, 2007; [4] C. Bardaro, J. Musielak, G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 9, Walter de Gruyter, Berlin - New York, 2003; [5] P.D. Lax, Hyperbolic Partial Differential Equations, AMS, 2006; [6] D. Duffy, Finite Difference Methods in Financial Engineering: A Partial Differential Equations Approach, Willey, 2006.

- G. Mastroianni, G.V. Milovanović: Interpolation Processes - Basic Theory and Applications, Springer Monographs in Mathematics, Springer - Verlag, Berlin - Heidelberg, 2008, XIV+446 pp., ISBN: 978-3-540-68346-9
- G.V. Milovanović, A.S. Cvetković: Gaussian type quadrature rules for Mtz systems, SIAM Journal on Scientific Computing 27 (2005), 893.913.
- G.V. Milovanović, A.S. Cvetković, M. Stanić: Trigonometric orthogonal systems and quadrature formulae, Computers and Mathematics with Applications 56 (2008), 2915-2931.
- G.V. Milovanović, A.S. Cvetković, M. Stanić: Quadrature formulae with multiple nodes and a maximal trigonometric degree of exactness, Numerische Mathematik 112 (2009), 425.448.
- G.V. Milovanović, M.M. Spalević, M.S. Pranić: Error estimates for Gauss-Tur quadratures and their Kronrod extensions, IMA Journal of Numerical Analysis 29 (2009), 486.507.
- G. Mastroianni, G.V. Milovanović: Some numerical methods for second kind Fredholm integral equation on the real semiaxis, IMA Journal of Numerical Analysis 29 (2009), 1046.1066.
- G. Mastroianni, G.V. Milovanović: Well-conditioned matrices for numerical treatment of Fredholm integral equations of the second kind, Numerical Linear Algebra with Applications 16 (2009), 995-1011.
- B.S. Jovanović, L.G. Vulkov: Finite difference approximations for some interface problems with variable coefficients, Applied Numerical Mathematics 59 (2009),349-372.
- B.S. Jovanović, L. G. Vulkov: Numerical solution of a two-dimensional parabolic transmission problem, International Journal of Numerical Analysis and Modeling 7 (2010), 156-172.
- G.V. Milovanović, A.S. Cvetković: Nonstandard Gaussian quadrature formulae based on operatror values, Advances in Computational Mathematics 32 (2010), 431-486.