History of Department

The theory of differential equations and function theory Department was founded in Physico-Mechanical Institute of the Academy of Sciences of the Ukrainian SSR in 1969. The head of the Department from 1969 was Doctor of Physics and Mathematics Sciences, Professor V.Ya. Skorobohat'ko. Since 1996 the head of the Department is Doctor of Physics and Mathematics Sciences V.O. Pelykh.

The main areas of the Department researches are: - development of methods of the qualitative theory of ordinary differential equations and differential equations with partial derivatives and methods of differential geometry in relation to the problems of relativistic physics and astrophysics; - development of analytical theory of branched continued fractions and methods of approximation of several variables functions by branched continued fractions.

Researches in the theory of differential equations since 1989 were also continued at the Department of mathematical physics, formed on the base of the laboratory of the theory of differential equations and function theory.

Directions of the Department investigations were formed by Honored Scientist of Ukraine, Doctor of Physics and Mathematics Sciences, Professor Vitaly Yakovych Skorobohat'ko. With the formation of the theory of differential equations Department the activity combined by V.Ya. Skorobohat'ko mathematicians from the Lviv University and the Polytechnic Institute has received recognition and support from the Academy of Sciences of Ukraine. After leaing Lviv by Academician Ukr SSR Ya.B. Lopatynskyy these mathematicians rallied for their researches in the Creative Mathematicians Club (1964). The Club activity was maintained by the reports of well known in the world mathematicians, particularly V.Petryshyn, L.D. Kudryavtsev, Y.A. Dubinskyy, V.I. Bernik, V.O. Yefremovych, O.O. Dezin, H. Waadeland, I.I. Danylyuk, I.V. Skrypnyk and others.

Except of V.Ya.Skorobohat'ko, B.Yo. Ptashnyk, I.V. Korobchuk, I.F. Klyuynyk, Kh.Yo. Kuchmins'ka, P.I. Bodnarchuk, I.P. Pustomelnykov were the first researchers of the Department.

During the Department existence about 50 scientists were its researchers. They defended 35 candidate and 10 doctor (B.Yo. Ptashnyk, M.S. Syavavko, P.I. Kalenyuk, D.I. Bodnar, B.I. Hnatyk, M.O. Nedashkovskiy, R.M. Plyatsko, V.O. Pelykh, O.L. Petruk, Kh.Yo. Kuchmins'ka) theses. Many of them are now leading experts at the Institutions of the Academy of Sciences of Ukraine and higher educational establishments in Lviv and other cities of Ukraine.

Unusual scientific activity of Prof. Ya. Skorobohat'ko contained a wide range of mathematical ideas and diverse interests for their practical applications. Among his pupils were students educated as mathematicians, physicists, engineering graduate students, solved the problem posed by him.

In 1970-1990 new directions of the qualitative theory of differential equations are developing in the Department:

1) The correctness investigation of some non-classical boundary value problems, the solvability of which is connected with the problem of small denominators (B.Yo. Ptashnyk); 2) Development of a generalized method of separation of variables and study the spectral properties of boundary value problems, including multiparametric arising from the application of this method (P.I. Kalenyuk); These directions have been established and actively supported by Professor V.Ya. Skorobohat'ko.

Since 1975 the Department began intensive development of multipoint geometry and its applications to relativistic physics and started solving the problems of special and general relativity through the development and application of the theory of differential equations and differential geometry. In this connection a variant of the multipoint geometry with a metric that is invariant of fractional-linear mappings was developed and also group invariants were constructed together with Yu. S. Vladimirov, and it was proved that this geometry corresponds to the binary structure in the binary geometrophysics developed by Yu.S. Vladimirov (V.Ya. Skorobohat'ko, V.O. Pelykh).

In the collaboration with the Academician of Belarus' F.I.Fedorov the method of representation of the general solution of the universal matrix Fedorov's equations in the power series form was proposed (V. Ya.Skorobohat'ko, O.O. Myakinnik).

The correctness of the Cauchy problem for the equations of Einstein-Hilbert depending on connected to them additional conditions was investigated; the covariant formulation of the Cauchy problem for these equations was also justified and investigated. The justification of the possibility for adding gauge (coordinate) conditions to the Hilbert-Einstein equations was proposed, it replaced, as proved incorrect application of one of the statements of Riemann's memoir. The differential geometry of spinor fields associated with non-integrated distributions was developed; it allowed proving the generalized theorem on the positive definiteness of the gravitational energy in the general theory of relativity. The conditions of absence of nodal points of double-covariant solutions of elliptic type equation systems are established, and on this basis, the correspondence between the existence in asymptotically flat Riemannian space Sen-Witten spinor field and the special orthonormal frames with defined properties. Because of this the actual problem of establishing the relationship between methods of Witten and Nestor for the theorem proof on positive definiteness of the gravitational energy was solved and there is also proposed the correct proof of the theorem of positive definiteness of the total energy of the gravitational field at maximal hypersurface in the method of the local orthonormal frame and extended the theorem tensor proof for a certain class no maximal hypersurfaces (V.O. Pelykh).

A method for solving the inverse variational problem was proposed involving the properties of the exterior differential system in conjunction with the Sophus Lie symmetry approach. The method was then applied to the Ostrohrads'kyj mechanics. The relativistic uniformly accelerated motion was considered together with the radiation friction and helical motion. Special attention was paid to the variational analysis of the relativistic top motion. On this way a formula for some variational generalization of a test particle mass dependency upon the value of the semi-classical spin was discovered. A procedure of a generalized Legendre mapping was suggested for the models above and a generalized Hamiltonian formalism together with the corresponding Poisson structure was obtained. By means of the differential geometric methods the higher-order connection coefficients were build for the only possible variational prolongation of the three dimensional (pseudo)Euclidean space metric to a second-order Kawaguchi space metric and the corresponding one-parametric family of geodesic autoparallel curves was obtained. (R.Ya. Matsyuk).

New properties of the gravitational interaction for a highly relativistic test mass with inner rotation (spin) in the Schwarzschild and Kerr backgrounds are investigated. By analysis of the Mathison-Papapetrou (MP) in their traditional and in terms of local comovinglues, the theory of the gravitational ultrarelativistic spin-orbit interaction is developed. A method for separation of the P equations solutions under the Mathisson-Pirani supplementary condition, which descri be the motion of the proper center of mass in Schwarzschild’s field, is proposed. The integrals of energy and angular momentum in Kerr’s background are used for obtaining a new representation of the exact MP equations without the third coordinate derivatives. The analytical solutions of these equations are investigated, as the common for the Mathisson-Pirani and Tulczyjew-Dixon conditions and significantly different. It was shown that for highly relativistic motions the adequate condition is just the Mathisson-Pirani one. The essentially non-geodesic trajectories of a spinning particle which manifest the effects of the significant gravitational repulsion or the additional attraction, depending on the spin orientation, are analyzed. The possible role of effects of the highly relativistic spin-gravity coupling in the real astrophysical processes are stressed (R.M. Plyatsko, O.B. Stefanyshyn, M.T. Fenyk).

In the middle of 60-ies of XX century V.Ya. Skorobohat'ko began a new direction in the function approximations theory, namely the theory of branched continued fractions (BCFs), developed by him and his students, mostly in an analytic way. With the establishment of the difference formula between the approximants of the BCF the intensive study of the convergence problem was begun. At the end of 70-ies of XX century Kh.Yo. Kuchmins'ka proposed the construction of corresponding two-dimensional continued fractions (TDCFs). For multidimensional generalizations of continued fractions (BCFs, TDCFs) analogues of many important classical convergence criteria, such as theorems of Seidel, Worpitzky, Sleszynski - Pringsheim, Van Vleck, parabolic theorem were obtained; also was investigated their computational stability, and some approximate properties. Branched continued fractions were also used to construct approximants for ratio of several variables hypergeometric functions, in partiular the Appell and Lauricella functions (D.I. Bodnar, Kh.Yo. Kuchmins'ka, O.M. Sus', N. P. Hoyenko).

Effectiveness of applications of continued fractions and their multivariate generalizations in computational mathematics associated with the property of their computational stability. Using the presentation of equation solutions in the form of fractional rational expressions obtained from the function expansions into continued fractions or their multidimensional generalization, analogues of methods of Gauss and Jordan for solving systems of linear algebraic equations, nonlinear analogues of Runge-Kutta method for solving ordinary differential equations, nonlinear difference methods for numerical solution of the Cauchy problem for the parabolic type equations, algorithms for calculating multiple integrals, presentations of real and p-adic numbers were constructed (M.O. Nedashkovskiy, Ya.M. Hlyns'kyy, Ya.M. Pelekh , O.I. Ohirko, Yu.V. Melnychuk). By means of integral continued fractions (M.S. Syavavko) solutions of Ambartsumian-Chandrasekhar, Len-Emden and other integral and integro-differential equations are presented.

The main tool in the study of singular matrix and differential matrix equations are the generalized inverse matrices. Using established new determinantal representations of the generalized inverse matrices of Moore-Penrose, Drazin and group inverse are obtained explicit determinantal representations of least squares solutions of some complex matrix equations and Drazin inverse solutions of some matrix and differential matrix equations. Solving similar problems for matrices over a non-split quaternion algebra were introduced and developed a theory of new matrix functionals - row, column and double determinants of square q1uaternionic matrices. Within the framework of a theory of the row and column determinants there were obtained determinantal representations of the inverse and Moore-Penrose inverse matrices, and (under certain conditions) Drazin inverse. There were obtained determinantal representations (analogues Cramer's rule) as solutions such thst least squares solutions with minimal norm of the left and right systems of linear equations and some matrix equations.(I.I.Kyrchei)

Doctor dissertations completed in the Department:

Ptashnyk B.Yo. "Non-classical boundary value problems for differential equations with partial derivatives," 1989; specialty - differential equations.

Syavavko M.S. "Theory and application of integral continued fractions with respect to measure", 1990; specialty-numerical mathematics;

Kalenyuk P.I. "Generalized method of separation of variables and its application", 1992, specialty - differential equations.

Bodnar D.I. "The questions on the analytic theory of branched continued fractions", 1992; specialty - mathematical analysis.

Nedashkovskiy M.O. "The methods and algorithms of the for computer algebra for systems of linear algebraic equations with polynomial coefficients," 1995; specialty - mathematical modeling and computing methods.

Hnatyk B.I. "Non-stationary high-temperature processes and shock waves in space plasmas", 1997; specialty - astrophysics, radio astronomy.

Plyatsko R.M. "The theory of gravitational ultrarelativistic spin-orbit interaction," 2003; specialty - theoretical physics.

Pelykh V.O. "Locally-covariant methods and problem of the energy positiveness in general relativity theory," 2006; specialty - theoretical physics.

Petruk O. "Acceleration of cosmic rays in shell supernova remnants", 2011; specialty - astrophysics, radio astronomy.

Kuchmins'ka Kh.Yo. "Development of the analytic theory of two-dimensional continued fractions," 2012; specialty - mathematical analysis.

Monographs published by the Department researches:

Bobyk O.I., Bodnarchuk P.I., Ptashnyk B.Yo., Skorobohat'ko V.Ya. The elements of the qualitative theory of differential equations with partial derivatives. - K.: "Naukova Dumka", 1972. - 176 p. (in Ukrainian)

Bodnarchuk P.I., Skorobohat'ko V.Ya. Branched continued fractions and their application. - K.: "Naukova Dumka", 1974. - 272 p. (in Ukrainian)

Kalenyuk P.I., Skorobohat'ko V.Ya. Qualitative methods of the theory of differential equations. - K.: "Naukova Dumka", 1977. - 123 p. (in Ukrainian)

Skorobohat'ko V.Ya. Investigations in the quality theory of differential equations with partial derivatives. - K.: "Naukova Dumka", 1980. - 244 p. (in Russian)

Skorobohat'ko V.Ya. Theory of branched continued fractions and its application in computational mathematics. - Moscow: Nauka, 1983. - 312 p. (in Russian)

Bodnar D.I. Branched continued fractions. - K.: "Naukova Dumka", 1986. - 176 p. (in Russian)

Novikov L.O., Skorobohat'ko V.Ya. Methods of scientific investigations research in mathematics. - K.: UMK VO IN USSR, 1988. - 132 p. (in Ukrainian)

Plyatsko R.M. Manifestations of gravitational ultrarelativistic spin-orbit interaction. - K.: "Naukova Dumka", 1988. - 148 p. (in Ukrainian)

Skorobohat'ko V.Ya. I look at the world as a mathematician. - Lviv: "Aphisha", 1994. - 75 p. (in Ukrainian)

Syavavko M.S. Integral continued fractions. - K.: "Naukova Dumka", 1994. - 205 p. (in Ukrainian)

Novikov L.O., Skorobohat'ko V.Ya. Mathematics methods: development, use, public echo. - Lviv: Word and Commerce, 1995. - 218 p. (in Ukrainian)

Gavrilov, M.I. , Skorobohatko V.Ya., Syavavko M.S. Resistance mathematical model of solar system and high convergence method of small parameter. - Lviv, 1996. - 180 p. (in Ukrainian)

Kuchmins'ka Kh.Yo. Two-dimensional continued fractions. - Lviv: Pidstryhach Institute of applied problems of mechanics and mathematics of NAS of Ukraine, 2010. - 218 p. (in Ukrainian)

In 1999, B. Hnatyk (consisting author) awarded by Barabashov's Prize of National Academy of Sciences of Ukraine.

The reports of department coworkers were pronounced at international conferences in Antwerp, Baku, Bratislava, Varna, Warsaw, Washington, Vienna, Gottingen, Dnipropetrovs'k, Yokohama, Kazan', Kyoto, Kyiv, Krakiw, Moscow, Prague, Rome, Trondheim, Florence.

Through the efforts of department's coworkers held scientific conferences: "Continued Fractions and their Applications", September, 1975, Lviv; International School - Seminar "Continued Fractions, their generalizations and applications", 18-25 September 1994, Verkhne Synyevydne – Lviv ; 19-24 August 2002, Uzhgorod.

The periodic conference "New approaches to solving differential equations," launched by V.Ya.Skorobohat'ko, which since 2004 was named after him, is regularly held in Drohobych together with the department of mathematical physics. In 2011 there was the 9-th International Mathematical Conference behalf of V. Ya. Skorobohat'ko.