About Us

 Extensive experience in applied mathematics, solid, and fluid mechanics. Combination of strong theoretical background and mathematical simulation skills. Fields of activity include mathematical modeling of physical phenomena in solid mechanics, dynamic fracture mechanics, and two phase flow of incompressible fluid. Intensive applications of boundary element methods, finite element methods, and finite difference methods. Regularization and numerical calculation of singular and hypersingular integrals. Solid background in statistics. Good computer programming skills (Fortran, C/C++, PowerBuilder, and Matlab), UNIX experience (on SUN and SGI platform), and proficient in Windows applications (Word, Excel, Excess) and Software tools (HTML, Minitab, SAS, R, LaTeX, CASs).

Scalar wave propagation problems with the Dirichlet or Neumann boundary conditions have been extensively studied. The mixed problem for the wave equation with an oblique derivative is ill-posed. The conditions of normal solvability were found. The exact solutions of initial value problems for wave equation have been constructed. Main publications: Dobrushkin, V.A. (1994). On solvability of mixed problem in a domain with an angle for wave equation. Sib. J. Diff. Equations, 3, No. 2, 3 - 10 (1994).

The shock phenomena in solid mechanics is simulated as an initial-boundary value problem for PDE with discontinuous initial data. A rigorous solution of such problem was obtained and the uniqueness theorem was proved. Main publications: Dobrushkin, V.A., & Gaiduk, S.I. (1979). Solution of a problem in the theory of thermoelasticity related to mechanical and thermal shocks. English transl. in Differential equations, 15, 1165 - 1174; Dobrushkin, V.A. (1977). The uniqueness of the generalized solution of a certain problem in shock theory. English transl. in Differential Equations, 13 , 1061 - 1063.

A new method for solving linear boundary value problems for abstract operator -differential equations was developed, a variant of the boundary element method. The main advantage of the new method in comparison with this technique is the reduction of singularities. Main publication: Dobrushkin, V.A. (1983). A general boundary value problem for an abstract differential-operator equation, Dokl. Akad. Nauk BSSR, 27, 776 - 779 (in Russian; English summary). The dynamic problems of elasticity for domains with nondifferentiable boundaries pose special difficulties since the standard methods cannot be used reliably for the analysis of wave effects in these geometries. In these domains the effects of wave reflection and diffraction are strongly influenced by the irregular boundary. The solutions of several mixed problems in the dynamic theory of elasticity for the wedge-shaped domains with the coupled boundary conditions (when elastic potentials are not separated) were established. The necessary algorithms for the approximate calculation of these solution have been developed. Main publications: Dobrushkin, V.A. (1988). The boundary value problems in the dynamic theory of elasticity for the wedge-shaped domains. Minsk: Nauka I Technika (in Russian, monograph); Dobrushkin, V.A. (1984). The second boundary value problem of the plane theory of elasticity for a wedge. Dokl. Akad. Nauk SSSR, 1984, 279, No. 11 77 -79 (in Russian; English transl. in Soviet Phys. Dokl. 1984, 29, No. 11, 905 - 907)., 1061 - 1063.

Numerical Methods
Approximation of continual integrals:

The appropriate numerical procedure to approximate the value of the continual integral with respect to several measures was developed. Main publication: Dobrushkin, V.A., & Likhoded, N.A. (1982). Approximate calculation of a continual integral with respect to a Cauchy measure with a weight in Hilbert space, 1982, No. 1, 104 - 106 (in Russian; English summary).

A new method of regularization and approximate calculation of hypersingular integrals and integro-differential expressions with singularities was proposed. The estimates of quadrature errors were obtained. Main publications: Dobrushkin, V.A. (1992). Regularization in Marchaud form and approximate calculation of a class of integro- differential expressions. Izvestiya vuzov. Matematika, 1992, No. 9, .38 - 41 (in Russian; English transl. in Russian Mathematics. Izvest. VUZ, Matematika, 1992, No. 9, 34 - 37); Dobrushkin, V.A. (1993). Approximate calculation of a class of integro-differential expressions that describe the propagation of elastic waves. Prikladn. Mat. i Mech., No. 2, 164 -167. (in Russian; English transl. J. Appl. Math. Mechs. 1993, 57, 393 - 397).

The solution of initial-boundary value problem which simulates half-plane with a semi-infinite crack have been developed. Main publication: Clifton, R.J., & Dobrushkin, V.A. Elastic wave propagation in a half-space containing a buried crack. Submitted to Wave Motion.
Simulation of the common motion of spherical bubbles in incompressible viscous unbounded fluid was developed. Main publications: Dobrushkin V.A., & Maxey M.R. . An improved boundary element formulation for the mothion of spherical bubbles. Submitted to J. Comp. Phys.. Dobrushkin, V.A., Maxey, M.R., & Likhoded, N.A. Fixed-size processor array for the implementation of the boundary element method in potential flow of spherical bubbles. Submitted to IEEE Transactions on Computers.
Segregation of Mg Implanted into InAs: Influence of the Internal Elastic Stresses. Main publication: Dobrushkin, V.A., Velichko, O.I., Fedotov, A.K., Tsurko V.A. Diffusion and Defect Data, Part B: Solid State Phenomena, 2002, Vols.82-84, pp.569-574

1999 -- 2000 "Simulation of Coupled Diffusion of Impurity Atoms and Point Defects in the Vicinities of Interfaces Insulator-Semiconductor and Heterojunctions Semiconductor-Semiconductor" funded by the National Research Council's Collaboration in Basic Science and Engineering (COBASE).

Objective: To formulate basic assumptions for the generalized models of impurity - defect system evolution in the vicinity of interfaces dielectric-semiconductor and heterojunctions semiconductor - semiconductor; and to develop an algorithm to solve a system of generalized impurity atoms and point defects diffusion equations.

1995 -- 1998 "Interaction of finite-sized particles and turbulence in dispersed two-phase flow" funded by the National Science Foundation (CTS-94-24169)

Objective: To determine the potential flow of particles of arbitrary form and of ellipsoidal form. Description: An improved boundary element method is applied to simulate two-phase flow of particles of arbitrary form and of ellipsoidal form.

1994 -- 1996 Dynamics of bubbly flows and the modification of turbulence by micro-bubbles" funded by the Office of Naval Research (ONR-N00014-91-J1340)

Objective: To simulate multi-bubble motion with touching in viscous incompressible flow.

Description: To solve Navier-Stokes equations an improved boundary element method was applied. It allows to reduce the problem under consideration to the Fredholm integral equation of the second kind without singularities.

1989 -- 1992 Computation of the stress-strain fields of rock under drill bits'', funded by the Petroleum University, Grozny, USSR (The leader of five-man team of developers).

Objective: To determine the stress-strain field of rock under drill bits in order to accelerate the speed of drill in deep bore hole.

Description: The process of drilling was simulated by the solution of the system of partial differential equations of elastodynamics and Newtonian flow to describe the problem of crack propagation under penetrating viscous liquid.

1987 -- 1989 Analysis and calculations of thermal processes in optical non-linear semiconductor structures'' funded by the Ministry of Defense, Minsk, USSR (The leader of four-man team of developers).

Objective: To analyze the temperature and translucency effects of a laser beam penetrating a multi-layer semiconductor material.

Description: The temperature distribution in a multi-layer semiconductor material was numerically analyzed by using finite difference schemes. This knowledge was applied to figure out the hysteresis of the phenomena. Experimental results agreed well with theoretical calculations.

1985 -- 1987 Calculation of the stress-strain fields of metal-ceramic solids in thermal flux'' funded by the Soviet Air Force for Space Shuttle ``Buran''. Leningrad, USSR (The leader of four-man team of developers).

Objective: To determine the high temperature expansion properties of metal-ceramic solids.

Description: The focus of the project was to allow the ``Buran'' Shuttle to deflect reentry atmosphere friction heat. The Lame method and the Finite Element Method were used to determine the stress-strain fields in a solid consisted of metal and ceramic. The numerical results matched the experimental ones.

1978 -- 1980 Numerical analysis of branching pipeline oscillations of for nuclear power stations'' funded by the Polzunov Power Research Institute, Leningrad, USSR (The leader of four-man team of developers).

Objective: To determine the locations of supporters for branching pipelines in order to prevent their damage due to earthquake

Description: The dynamic behavior of oscillated pipelines were modeled as a system of ordinary differential equations and numerical simulated with feedback constrains algorithm. This algorithm used the calculated eigenvalues to analyze and determine the necessary positions of pipeline supporters.

Dobrushkin V.A. (2022). Applied Differential Equations. The Primary Course, Chapman & Hall/CRC, 2022, second edition, ISBN-13: 978-1439851043 

Zwillinger, D. and Dobrushkin V.A. (2021). Handbook of Differential Equations, Fourth Edition, Chapman & Hall/CRC, 2021, fourth edition, ISBN-13: 978-0-367-25257-1 

Dobrushkin V.A. (2019). Mathematics of Social Choice and Finance, Kendall Hunt Publishing, 2019, second edition, ISBN-13: 978-1-5249-8959-0 

Dobrushkin V.A. (2017). Applied Differential Equations with Boundary Value Problems, Chapman & Hall/CRC, 2017, ISBN-13: â€Ž 1498733654 

Dobrushkin V.A. (2014). Applied Differential Equations. The Primary Course, Chapman & Hall/CRC, 2014, ISBN-10:1439851042, ISBN-13: 978-1439851043 

Dobrushkin V.A. (2014). Mathematics of Social Choice and Finance, Kendall Hunt Publishing, 2014, ISBN-10: 1465250999, ISBN-13: 978-1465250995 

Dobrushkin V.A. (2009). Methods in Algorithmic Analysis (CRC Computer and Information Science Series), Chapman & Hall/CRC, 2009, ISBN-10: 1420068296, ISBN-13: 978-1420068290  

Dobrushkin V.A. (1988). The boundary value problems in the dynamic theory of elasticity for the wedge-shaped domains. Minsk: Nauka i Technika (in Russian).

1. (with R.A. Beauregard) Powers of a Class of Generating Functions. Mathematics Magazine, 89/5, 359 -- 363 (2016).
2. (with R.A. Beauregard) Multisection of series. The Mathematical Gazette, 100/549, 460 -- 470 (2016).
3. (with R.A. Beauregard) A discrete view of Faa di Bruno. The Rocky Mountain Journal of Mathematics , 46, No. 1, 73 -- 83 (2016).
4. (with R.A. Beauregard) Differential equations vs power series. The Mathematical Gazette, 99/546, 499 -- 503 (2015).
5. (with R.A. Beauregard) Finite sums of the Alcuin Numbers. Mathematics Magazine, 86/4, 280 -- 287 (2013).
6. (with V.A. Tsurko and G.M. Zayats) On convergence of difference schemes for two-dimensional second-order hyperbolic equations with discontinuous coefficients. Matematychni Studii, 31/1, 91 -- 101 (2009).
7. (with M.R. Maxey) An improved boundary element formulation for the motion of spherical bubbles. Differential Equations, 44/9, 1305 -- 1312 (2008).
8. (with F.F. Komarov, O.I. Velichko, and A.M. Mironov) Mechanisms of arsenic clustering in silicon. Physical Review B, 74, July, 035205-1 -- 035205-10 (2006).
9. (with O. I. Velichko and L. Pakula) A Model of Clustering of Phosphorus Atoms in Silicon. Materials Science and Engineering: B, 123/2, 176 -- 180 (2005).
10. (with I. I. Abramov, V.A. Tsurko, and V.A. Zhuk) Numerical ''Renaissance'' Procedure of Device and Process Parameters in Integrated Circuits. Nonlinear Phenomena in Complex Systems, 78/3, 296 -- 301 (2005)
11. (with A.K. Fedotov and O. I. Velichko) A set of equations for stress-mediated evolution of the nonequilibrium dopant-defect system in semiconductor crystals, Journal of Alloys and Compounds, 382, Issue 1-2, pp. 283-287 (2004).
12. (with O. I. Velichko, A. N. Muchynski, V. A. Tsurko, and V. A. Zhuk) Simulation of Coupled Diffusion of Impurity Atoms and Point Defects under Nonequilibrium Conditions in Local Domain. Journal of Computational Physics, 178, 1 - 14 (2002).
13. (with O. I. Velichko, A. K. Fedotov, V. A. Tsurko) Segregation of Mg Implanted into InAs: Influence of the Internal Elastic Stresses, Diffusion and Defect Data, Part B: Solid State Phenomena, 2002, Vols.82-84. pp.569 - 574.
14. (1994). On solvability of mixed problem in a domain with an angle for wave equation. Sib. J. Diff. Equations, 1994, 3, No. 2, 3 - 10.
15. (with Nitrebych, Z.N.) (1993). Initial and boundary value problems for hyperbolic partial differential operators with constant coefficient. Sib. J. Diff. Equations, 1993, 2, No. 2, 39 - 46.
16. (1993). Approximate calculation of a class of integro-differential expressions that describe the propagation of elastic waves. Prikladn. Mat. i Mech., No. 2, 164 -(in Russian; English transl. J. Appl. Math. Mechs. 1993, 57, 393 - 397).
17. (with Kolesnikov, A.N., Podkopaev, P.A., & Shestokov, V.N.) (1993). Algorithm for calculation of strain-stress state of elastic semi-plane. Vesty Akad Nauk Belarus, ser. fiz.-mat. nauk, 1993, No. 2, 219.
18. (1992). Regularization in Marchaud form and approximate calculation of a class of integro- differential expressions. Izvestiya vuzov. Matematika, 1992, 36, No. 9, 38 - 41 (in Russian; English transl. in Russian Mathematics. Izvest. VUZ, Matematika, 1992, No. 9, 34 - 37).
19. (with Djum, T.R.) (1992). On the solution of the Cauchy problem for the Lame operator and its regularization. Differ. Uravn., 1992, 28, No. 10, 1830 - 1833.(in Russian; English summary).
20. (1990). Regularization and approximate calculation of the Marchaud's derivative of a two-dimensional integral with sliding singularities. In: Some applications of functional analysis to the problems of mathematical physics. Novosibirsk: Mathematical institute of Siberian branch of the Academy of Science of the USSR, 1990, 81 - 91 (in Russian).
21. (1989). Approximate calculation of the Marchaud's derivative of a two-dimensional integral with singularities. Dokl. Akad. Nauk BSSR, 1989, 33, 101 - 103 (in Russian; English summary).
22. (with Podkopaev, P.A.) (1988). On the solution of the first boundary value problem of elastodynamics for the plane with cracks. Minsk, 1988. 38 pp. (Preprint /Mathematical Institute, Byelorussian Academy of Science, No. 9 (319)) (in Russian).
23. (with Podkopaev,P.A.) (1986). Mathematical investigation of the first boundary value problem of elasticity for the plane with two semi-infinite cracks. Minsk, 1986. 28 pp. (Preprint /Mathematical Institute, Byelorussian Academy of Science; No. 20 (256)) (in Russian).
24. (with Podkopaev, P.A.) (1986). Mathematical investigation of the first boundary value problem of elasticity for the plane with semi-infinite crack. Minsk, 1986. 32 pp. (Preprint /Mathematical Institute, Byelorussian Academy of Science; No. 19 (255)) (in Russian).
25. (1986). Solution of the second boundary value problem of elasticity theory for a semiband. Differ. Uravn., 1986, 22, No. 6, 987 -993 (in Russian; English transl. in Differential Equations, 1986, 22, 697 - 702).
26. (1985). Analytical solution of a static boundary value problem of the plane theory of elasticity. Differ. Uravn., 1985, 21, No. 4, 717 - 720 (in Russian; English summary).
27. (1985). Solution of the second boundary value problem of the theory of elasticity elasto for a wedge. Izvest. Akad. Nauk BSSR, ser. fiz.-mat. nauk, 1985. No. 4, 8 - 13 (in Russian; English summary).
28. (1984). About certain problems of ordering of noncommutative operators and continual integration. Minsk, 1984, 18 pp. (Preprint /Mathematical Institute, Byelorussian Academy of Science; No. 14 (199)) (in Russian; English summary).
29. (1984). The second boundary value problem of the plane theory of elasticity for a wedge. Dokl. Akad. Nauk SSSR, 1984, 279, No. 11 77 -79 (in Russian; English transl. in Soviet Phys. Dokl. 1984, 29, No. 11, 905 - 907).
30. (1984). Solution of the first boundary value problem of the theory of elasticity for a quadrant. Dokl. Akad. Nauk BSSR, 1984, 28, No. 2, 115 - 118 (in Russian; English summary).
31. (1983). A general boundary value problem for an abstract operator-differential equation. Dokl. Akad. Nauk BSSR, 1983, 27, No. 9, 776 - 779 (in Russian; English summary).
32. (with Podkopaev, P.A.) (1983). Representation of the solutions of boundary value problems for the heat equation in the form of a continual integral with respect to. a Wiener measure. Izvest. Akad. Nauk BSSR, ser. fiz.-mat. nauk, 1983, No. 6, 9 - 14 (in Russian; English summary)
33. (with Lihoded, N.A.) (1982). On approximate evaluation of continual integrals with respect to measures corresponding to a Markov process of telegraph type. Dokl. Akad. Nauk BSSR, 1982, 26, No. 11. 965 - 968 (in Russian; English summary).
34. (with Lihoded, N.A.) (1982). On approximate calculation of continual integral with respect to a Cauchy's measure with weight in the Hilbert space. Izvest. Akad. Nauk BSSR, ser. fiz.-mat. nauk, 1982, No. 1, 104 - 106 (in Russian; English summary).
35. (1981). Evaluation of continual integrals with respect to Cauchy's, Abel's and Gauss's measures in the product of Hilbert spaces. Minsk, 1981, 40 pp. (Preprint /Mathematical Institute, Byelorussian Academy of Science; No. 2 (103)) (in Russian).
36. (1981). The Kolmogorov-Khinchin criterion for certain measures in the space W. Izvest. Akad. Nauk BSSR, ser. fiz.-mat. nauk, 1981, No. 6, 120 - 123 (in Russian; English summary).
37. (1981). Generalized summation of continual integrals. Izvest. Akad. Nauk BSSR, ser. fiz.-mat. nauk, 1981, No. 5. 5 - 10 (in Russian; English summary
38. (1980). Investigation of the problem on simultaneous thermal and mechanical shocks onto a composite finite rod. Izvest. Akad. Nauk BSSR, ser. fiz.-mat. nauk, 1980, No. 4, 20 - 27 (in Russian; English summary).
39. (1979). Cezaro's summation of certain series and integrals. - Minsk, 1979. 16 pp. (Preprint /Mathematical Institute, Byelorussian Academy of Science; No. 1(57)) (in Russian).
40. (with Gaiduk, S.I.). (1979). Solution of a problem in the theory of thermoelasticity related to mechanical and thermal shocks. Differ. Uravn., 1979, 15, No. 9, 1632 - 1645 (in Russian; English transl. in Differential equations, 1979, 15, No. 9, 1165 - 1174).
41. (1978). Investigation of several initial boundary value problems about the impact onto the semi-infinite beams. Minsk, 1978, 29pp. (Preprint /Mathematical Institute, Byelorussian Academy of Science; No. 6 (38))(in Russian).
42. (1978). A dynamic problem of shock theory for an elastic plate. Dokl. Akad. Nauk BSSR, 1978, 22, No. 3, 197 - 199 (in Russian; English summary ).
43. (1978). The solution of a problem of a shock on elastic semi-infinite plate. Izvest. Akad. Nauk BSSR, ser. fiz.-mat. nauk, 1978, No. 1, 134 (in Russian; English summary).
44. (1977). Solution of a problem of a longitudinal shock along an inhomogeneous viscoelastically relaxing rod. Differ. Uravn., 1977, 13, No. 9, 1658 - 1666 (in Russian; English transl. in Differential Equations, 1977, 13, No. 9, 1155 - 1160).
45. (1977). The uniqueness of the generalized solution of a certain problem in shock theory. Differ. Uravn., 1977, 13, No. 8, 1521-1523. English transl. in Differential Equations, 13, 1061 - 1063.
46. (1976). A problem in the theory of longitudinal vibrations of an elastic-viscous-relaxing rod. Izvest. Akad. Nauk BSSR, ser. fiz.-mat. nauk, 1976, No. 5, 53 - 60 (in Russian; English summary).
47. (with Gaiduk S.I.). (1975). Solution of a problem of longitudinal shock along a viscoelastically relaxing rod. Izvest. Akad. Nauk BSSR, ser. fiz.-mat. nauk, 1975. No. 6, 30 - 41 (in Russian; English summary).
48. (with Antonevich, A.B.). (1973). A general boundary value problem for an elliptic system with deviating argument. Dokl. Akad. Nauk BSSR, 1973. 17, 491 - 493 (in Russian).