Laboratory of the theory of integro-differential equations Print
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Head of Laboratory

Doctor of Physical and Mathematical Sciences

Professor Iskandarov Samandar

The laboratory began its activities with the Department of Physics and Mathematics at the Presidium of the Academy of Sciences of the Kyrgyz SSR, organized in 1955. In 1960, the sector of the theory of integro-differential equations was created, which was later transformed into a laboratory.

The following research methods are being developed:

- general and qualitative theory of integro-differential equations (briefly: IDE), integral equations (IE), differential equations (DE) in ordinary and partial derivatives;

- study of the influence of integral perturbations on the qualitative properties of the solutions of DE and functions - free members;

- establishment of specific signs of stability and asymptotic stability of solutions of linear homogeneous Volterra type IDE, including high orders.

The laboratory has developed the following new research methods:

  • method of transition from homogeneous to non- homogeneous IDE (Y.V. Bykov);
  • the method of integral inequalities with nondegenerate kernels (Ya.V. Bykov, Yu.A. Ved', Z. Pakhyrov);
  • the method of special successive approximations for the IDE and the DE with deviating arguments (Yu.A. Ved');
  • the method of integral transformations for the study of the influence of integral perturbations of the Volterra type in the theory of the stability of scalar and vector DE (M.I. Imanaliev, Y.A. Ved');
  • the method of additional argument for DE in partial derivatives with an integral coefficient (M.I. Imanaliev, Yu.A. Ved');
  • finite methods in the theory of multidimensional branching (A.I. Botashev);
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  • the method of introducing parameters for studying the problem of periodic solutions of Volterra type IDE (A.I. Botashev);
  • weighting functions method for Volterra type second order IDE (Yu.A. Ved', Z. Pakhyrov);
  • the method of weighting and cutting functions for the IDE and IE of Volterra type (S. Iskandarov);
  • method of partial cutting (S. Iskandarov, D. N. Shabdanov);
  • the method of squaring equations for the second order IDE and the implicit first order IDE of Volterra type (S. Iskandarov);
  • a method based on transformations according to scheme A) → B) → C), which allows to study the asymptotic properties of first-order Volterra type IDE solutions in the critical case (with a zero coefficient of the function sought) and to establish specific signs of the presence of various speakers (estimates, limitations, stability , striving for zero, including the exponential and power law, power absolute integrability on the half-line, and other qualitative properties) of the solutions of first-and second-order IDE of Volterra type (S. Iskandarov);
  • non-standard method of reducing to a system consisting from one DE of the first order and one IDE of the second order with the introduction of some positive weight function for a third-order IDE of Volterra type (S. Iskandarov, A.T. Khalilov);
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  • non-standard methods of reduce to the system: for a second-order linear Volterra type IDE (with the introduction of some auxiliary parameter and a positive weight function, the system consists of one first-order DE and one first-order IDE); in three versions for the linear third-order DE and the Volterra type IDE, including the method that generalizes the method S.Iskandarov, A.T. Khalilov; in various variants for the linear fourth, fifth, sixth, seventh, eighth-order Voltara IDE (S.Iskandarov);
  • the method of integral inequalities with delays (Yu.A.Ved', L.N. Kitaeva, S. Iskandarov, M.A. Temirov);
  • the method of integral inequalities of the first kind, which allows to study the influence of integral perturbations of the Volterra type on the boundedness and vanishing of solutions of scalar and vector DE (S. Iskandarov);
  • method for estimating the below solutions of scalar and vector IDE types Volterra (Yu.A. Ved', S. Iskandarov, G.T. Khalilova).
  • auxiliary kernel method of studying asymptotic properties solutions of IDE of Volterra type (S. Iskandarov).
  • A new construction of Lyapunov functionals was proposed, which allows one to study the asymptotic properties of solutions on semi-axes of the new classes of IDE and Volterra type IE (S. Iskandarov).
    According to the results of the research, more than 900 scientific works were published, including 10 monographs (Y.V. Bykov - 7, A.I. Botashev - 2, S. Iskandarov - 1);

At the scientific forums of various levels (congresses, symposia, International, All-Union, Regional, local conferences, non-resident seminars), the laboratory's staff performed more than 380 scientific reports and reports.
Since 1961, the laboratory has been publishing the thematic collection "Studies on Integro-Differential Equations", 47 issues of this collection have been published, the collection is an official periodical, known in our country and abroad. Prepared 28 candidates of science. He has scientific ties with KSTU. I.Razzakov, KNU them. J. Balasagyn, BSU them. K.Karasaev, KRSU them. Boris N. Yeltsin, Batu State University, Moscow State University. Mv University, David Ben-Gurion University in the Negev (Israel) and Ariel University (Israel), Perm State University (Perm), Kazakh National University. Al-Farabi (Almaty), ENU. L.N. Gumilyov (Astana).

Until 1967, the laboratory (sector) of the theory of integro-differential equations was headed by Dr. Phys-mat. Sc., Professor, Corresponding Member. AN Kirg. SSR Ya.V. Bykov, from 1967 to 2006, the head of the laboratory was Cand. Phys-mat. Sc., Senior Researcher. Yu.A.Ved', from 2006 to the present, the head of the laboratory is Dr. Phys-mat. Sc., Professor S. Iskandarov.