Averaging method and it's applications.
Generalization of Ljapunov's second method.
Сomputational methods in micromagnetic problems.
Variational method in optimal control problems.
Main publications:
Khapaev M.M., Usrednenie v teorii ustoichivosti: Issledovanie rezonansnykh mnogochastotnykh sistem, Glavnaya redaktsiya fiziko-matematicheskoi literatury, M., 1986
Khapaev M.M., “Stability investigation in 3-body problem based on hydrodynamic model of planets”, Doklady Akademii Nauk, 231:5 (1976), 1092–1095
Khapaev M.M., Osipov S.G., Ternovskii V.V., “Modelirovanie trekhmernykh periodicheskikh struktur v ferromagnitnykh plenkakh”, Doklady Akademii Nauk, 305:4 (1989), 831–834
Ternovskii V.V., Khapaev M.M., “Direct numerical method for solving optimal control problems”, Doklady Mathematics, 420:40 (2008), 463–466
A. S. Ionin, S. V. Egorov, M. S. Sidelnikov, L. N. Karelina, N. S. Shuravin, M. M. Khapaev, V. V. Bol'ginov, “Numerical simulation of the superconducting sigma neuron design”, Fizika Tverdogo Tela, 66:7 (2024), 1019–1025
2023
2.
I. G. Mileshin, V. M. Goloviznin, M. M. Khapaev, “Numerical simulation of atmospheric electricity problem with unknown ionosphere potential”, Num. Meth. Prog., 24:3 (2023), 305–315
V. V. Ternovskii, M. M. Khapaev, T. M. Khapaeva, “Application of the variational method for solving inverse problems of optimal control”, Dokl. Akad. Nauk, 483:4 (2018), 370–373; Dokl. Math., 98:3 (2018), 603–606
2016
4.
M. M. Khapaev, T. M. Khapaeva, “Computation of eigenfunctions and eigenvalues for the Sturm–Liouville problem with Dirichlet boundary conditions at the left endpoint and Neumann conditions at the right endpoint”, Zh. Vychisl. Mat. Mat. Fiz., 56:10 (2016), 1750–1753; Comput. Math. Math. Phys., 56:10 (2016), 1732–1736
2015
5.
V. V. Ternovski, M. M. Khapaev, “Revised solution of ill-posed algebraic systems for noise data”, Comp. nanotechnol., 2015, no. 2, 14–19
6.
M. M. Khapaev, T. M. Khapaeva, “On the Computation of Eigenfunctions and Eigenvalues in the Sturm–Liouville Problem”, Mat. Zametki, 97:4 (2015), 604–608; Math. Notes, 97:4 (2015), 616–620
V. V. Ternovskii, M. M. Khapaev, “Reconstruction of periodic functions from noisy input data”, Dokl. Akad. Nauk, 424:4 (2009), 452–454; Dokl. Math., 79:1 (2009), 81–82
V. V. Ternovskii, A. M. Khapaev, M. M. Khapaev, “Resonance effects in the field of a stationary electromagnetic wave and a constant magnetic field”, TMF, 142:1 (2005), 64–71; Theoret. and Math. Phys., 142:1 (2005), 54–60
2004
9.
M. M. Khapaev, “Construction of the Solution of an Ill-Posed Singularly Perturbed Problem for the Heat Equation with a Nonlinearity”, Differ. Uravn., 40:6 (2004), 848–849; Differ. Equ., 40:6 (2004), 908–910
2003
10.
M. M. Khapaev, “Constructing Solutions of an Ill-Posed Nonlinear Singularly Perturbed Problem for an Equation of Elliptic Type”, Mat. Zametki, 73:2 (2003), 318–320; Math. Notes, 73:2 (2003), 293–295
2002
11.
M. M. Khapaev, “On one ill-posed singularly perturbed problem”, Fundam. Prikl. Mat., 8:4 (2002), 1251–1254
1999
12.
M. M. Khapaev, A. A. Tsygankov, “An algorithm for solving the conditional extremum problem”, Differ. Uravn., 35:3 (1999), 310–312; Differ. Equ., 35:3 (1999), 310–312
M. M. Khapaev, “Differential equations that contain singular manifolds in control and minimization problems”, Differ. Uravn., 31:11 (1995), 1886–1892; Differ. Equ., 31:11 (1995), 1854–1860
14.
O. V. Anashkin, M. M. Khapaev, “On the partial stability of nonlinear systems of ordinary differential equations with a small parameter”, Differ. Uravn., 31:3 (1995), 371–381; Differ. Equ., 31:3 (1995), 341–351
M. M. Khapaev, “Singular differential equations in control and minimization problems”, Trudy Mat. Inst. Steklov., 211 (1995), 411–418; Proc. Steklov Inst. Math., 211 (1995), 371–377
1993
16.
M. M. Khapaev, “On taking into account measurement errors by the mathematical
processing of them according to the Gauss least squares method”, Dokl. Akad. Nauk, 330:4 (1993), 444; Dokl. Math., 47:3 (1993), 538–539
17.
O. V. Anashkin, M. M. Khapaev, “On the stability of nonlinear systems with a small parameter”, Differ. Uravn., 29:8 (1993), 1301–1307; Differ. Equ., 29:8 (1993), 1127–1132
1992
18.
M. M. Khapaev, “Singular differential equations in control problems with
constraints”, Dokl. Akad. Nauk, 324:6 (1992), 1166–1168; Dokl. Math., 45:3 (1992), 706–709
19.
M. M. Khapaev, “Singular differential equations in problems on the investigation
of functions for a conditional extremum”, Dokl. Akad. Nauk, 323:2 (1992), 250–252; Dokl. Math., 45:2 (1992), 322–324
O. V. Anashkin, M. M. Khapaev, “The method of Lyapunov functions for systems with perturbations”, Differ. Uravn., 28:12 (1992), 2027–2029; Differ. Equ., 28:12 (1992), 1675–1677
1991
21.
M. M. Khapaev, “Conditions for the controllability of singularly perturbed systems
that contain singular controls”, Dokl. Akad. Nauk SSSR, 320:2 (1991), 300–302; Dokl. Math., 44:2 (1992), 459–461
22.
O. P. Filatov, M. M. Khapaev, “On the problem of the washing out of admixtures from a cavity”, Dokl. Akad. Nauk SSSR, 317:6 (1991), 1373–1375; Dokl. Math., 36:4 (1991), 303–304
23.
S. G. Osipov, M. M. Khapaev, “Modelling of micromagnetic structures”, Mat. Model., 3:11 (1991), 12–38
M. M. Khapaev, “On the determination of averaged physical parameters in periodic
media”, Dokl. Akad. Nauk SSSR, 315:4 (1990), 857–859; Dokl. Math., 35:12 (1990), 1031–1032
25.
M. M. Khapaev, “On the evolution of planetary orbits”, Dokl. Akad. Nauk SSSR, 312:3 (1990), 599–603; Dokl. Math., 35:5 (1990), 431–433
26.
S. G. Osipov, M. M. Khapaev, “On the nonlinear dynamics of a domain boundary”, Dokl. Akad. Nauk SSSR, 311:6 (1990), 1351–1355; Dokl. Math., 35:4 (1990), 344–346
27.
O. P. Filatov, M. M. Khapaev, “Averaging of differential inclusions with ‘fast’ and ‘slow’ variables”, Mat. Zametki, 47:6 (1990), 102–109; Math. Notes, 47:6 (1990), 596–601
O. P. Filatov, M. M. Khapaev, “Mutual $\varepsilon$-approximation of solutions of a system of differential inclusions and of the averaged”, Mat. Zametki, 47:5 (1990), 127–134; Math. Notes, 47:5 (1990), 509–515
S. A. Trushin, M. M. Khapaev, G. V. Sholin, “Employing the methods from the theory of singularly perturbed systems to an analysis of the conditions under which a magnetic field is frozen into an electron fluid”, Prikl. Mekh. Tekh. Fiz., 31:4 (1990), 3–10; J. Appl. Mech. Tech. Phys., 31:4 (1990), 505–511
1989
30.
S. G. Osipov, V. V. Ternovskii, M. M. Khapaev, “Modelling of $3$-dimensional periodic structures in ferromagnetic films”, Dokl. Akad. Nauk SSSR, 305:4 (1989), 831–834
31.
O. V. Anashkin, M. M. Khapaev, “The comparison method and investigation of the stability of systems of ordinary differential equations containing perturbations. II”, Differ. Uravn., 25:2 (1989), 187–192; Differ. Equ., 25:2 (1989), 133–137
1988
32.
S. A. Trushin, M. M. Khapaev, G. V. Sholin, “Investigation of singularly perturbed equations that describe
nonlinear magneto-acoustic waves”, Dokl. Akad. Nauk SSSR, 303:1 (1988), 74–77; Dokl. Math., 33:11 (1988), 801–803
33.
K. V. Mal'kov, M. M. Khapaev, “On the suppression of soliton-like solutions of shallow water
equations by the outstripping resistance”, Dokl. Akad. Nauk SSSR, 300:5 (1988), 1052–1059; Dokl. Math., 37:3 (1988), 777–783
34.
S. G. Osipov, V. V. Ternovskii, M. M. Khapaev, “Properties of the solution of a nonlinear variational problem”, Differ. Uravn., 24:7 (1988), 1274–1276
35.
M. M. Khapaev, O. P. Filatov, “Stability of differential inclusions with multivalued perturbations”, Mat. Zametki, 43:3 (1988), 346–355; Math. Notes, 43:3 (1988), 199–204
K. V. Mal'kov, M. M. Khapaev, “On the evolution and stability of soliton-like solutions of
perturbed equations of Boussinesq type”, Dokl. Akad. Nauk SSSR, 292:1 (1987), 68–73
37.
K. V. Mal'kov, M. M. Khapaev, “Some methods in perturbation theory connected with averaging”, Differ. Uravn., 23:10 (1987), 1693–1704
38.
M. M. Khapaev, V. V. Balandin, “Construction of perturbations of Lyapunov functions and an investigation of stability under small random actions”, Differ. Uravn., 23:4 (1987), 675–680
1986
39.
M. M. Khapaev, K. V. Mal'kov, “Asymptotic behavior of solutions of differential equations with
perturbations”, Dokl. Akad. Nauk SSSR, 290:4 (1986), 800–805
M. M. Khapaev, G. V. Sholin, “Analysis of plasma hydrodynamic equations by the methods of
singularly perturbed systems”, Dokl. Akad. Nauk SSSR, 286:3 (1986), 602–605
O. V. Anashkin, M. M. Khapaev, “The comparison method and investigation of the stability of systems of ordinary differential equations containing perturbations”, Differ. Uravn., 22:9 (1986), 1604–1606
43.
M. M. Khapaev, V. N. Shinkin, “Extremal properties of the Lyapunov function on unstable resonance regimes in the multifrequency case”, Differ. Uravn., 22:8 (1986), 1463–1466
44.
M. M. Khapaev, K. V. Mal'kov, “A class of methods for studying the asymptotic behavior of the solutions of differential equations with closed operators”, Differ. Uravn., 22:2 (1986), 255–267
1985
45.
I. V. Kuznetsova, M. M. Khapaev, “Investigation of the stability of systems containing
perturbations”, Dokl. Akad. Nauk SSSR, 283:4 (1985), 810–813
46.
M. M. Khapaev, “A generalization of Lyapunov's second method and an investigation of the stability of applied problems”, Differ. Uravn., 21:10 (1985), 1730–1733
1984
47.
M. M. Khapaev, “An algorithm for the integration of singularly perturbed systems”, Dokl. Akad. Nauk SSSR, 278:6 (1984), 1313–1315
1983
48.
M. M. Khapaev, “A. N. Tikhonov's theorem for singularly perturbed systems”, Dokl. Akad. Nauk SSSR, 271:5 (1983), 1074–1077
49.
M. M. Khapaev, O. P. Filatov, “The averaging principle for systems with “fast” and “slow” variables”, Differ. Uravn., 19:9 (1983), 1640–1643
1982
50.
M. M. Khapaev, “On linear systems of ordinary differential equations”, Dokl. Akad. Nauk SSSR, 266:2 (1982), 299–301
51.
M. M. Khapaev, I. V. Kuznetsova, “Multifrequency systems that have lag”, Differ. Uravn., 18:2 (1982), 354–356
1981
52.
M. M. Khapaev, O. P. Filatov, “On averaging and stability in systems with singularities”, Dokl. Akad. Nauk SSSR, 261:1 (1981), 67–70
53.
M. M. Khapaev, O. P. Filatov, “Stability of resonance problems”, Differ. Uravn., 17:5 (1981), 949–953
1980
54.
M. M. Khapaev, A. I. Falin, “On the investigation of stability in systems of integrodifferential equations by the averaging method”, Dokl. Akad. Nauk SSSR, 250:2 (1980), 295–299
55.
M. M. Khapaev, “Problems of stability in systems of ordinary differential equations”, Uspekhi Mat. Nauk, 35:1(211) (1980), 127–170; Russian Math. Surveys, 35:1 (1980), 145–197
M. M. Khapaev, O. V. Anashkin, “Stability in systems of ordinary differential equations with almost periodic coefficients”, Differ. Uravn., 15:7 (1979), 1216–1224
1978
57.
M. M. Khapaev, O. V. Anashkin, “On the investigation of stability in systems of ordinary differential equations with almost-periodic coefficients”, Dokl. Akad. Nauk SSSR, 240:5 (1978), 1028–1031
1976
58.
M. M. Khapaev, “A study of stability in the three body problem on the basis of a hydrodynamical model of the planets”, Dokl. Akad. Nauk SSSR, 231:5 (1976), 1092–1095
1974
59.
M. M. Khapaev, “On averaging in multifrequency systems”, Dokl. Akad. Nauk SSSR, 217:5 (1974), 1021–1024
M. M. Khapaev, “A stability analysis of resonance problems”, Differ. Uravn., 10:3 (1974), 447–457
1973
61.
M. M. Khapaev, “A generalization of Ljapunov's second method”, Differ. Uravn., 9:11 (1973), 2020–2028
1970
62.
M. M. Khapaev, “Stability in the problem of three bodies”, Dokl. Akad. Nauk SSSR, 195:2 (1970), 300–302
63.
M. M. Khapaev, “Generalization of Ljapunov’s second method and the investigation of some resonance problems”, Dokl. Akad. Nauk SSSR, 193:1 (1970), 46–49
M. M. Khapaev, “Investigation of stability in the theory of nonlinear oscillations”, Mat. Zametki, 3:3 (1968), 307–318; Math. Notes, 3:3 (1968), 194–200
M. M. Khapaev, “A theorem of Ljapunov type”, Dokl. Akad. Nauk SSSR, 176:6 (1967), 1262–1265
1966
68.
M. M. Khapaev, “On the averaging method in certain problems connected with averaging”, Differ. Uravn., 2:5 (1966), 600–608
69.
M. M. Khapaev, “Жесткая фокусировка гофрированным магнитным полем”, Zh. Vychisl. Mat. Mat. Fiz., 6:supplement to № 4 (1966), 75–79
1965
70.
M. M. Khapaev, “Non linear theory of the motion of fast charged particles in helical toroidal magnetic fields”, Dokl. Akad. Nauk SSSR, 163:2 (1965), 343–346
1962
71.
M. M. Khapaev, “Asymptotic behavior in the neighborhood of an irregular singular point of the solutions of ordinary differential equations with small coefficients on the highest derivatives”, Mat. Sb. (N.S.), 57(99):2 (1962), 187–200
1961
72.
M. M. Khapaev, “Asymptotic expansions of hypergeometric and confluent hypergeometric functions”, Izv. Vyssh. Uchebn. Zaved. Mat., 1961, no. 5, 98–101
73.
M. M. Khapaev, “Linear differential equations with small coefficients of several higher derivatives in the neighborhood of regular singular points of the equation”, Uspekhi Mat. Nauk, 16:4(100) (1961), 187–194
1960
74.
M. M. Khapaev, “Asymptotic expansions of the solutions of ordinary linear differential equations in the neighborhood
of an irregular singularity when the coefficients of the higher derivatives in the equations are small”, Dokl. Akad. Nauk SSSR, 135:6 (1960), 1338–1341
1986
75.
I. V. Kuznetsova, M. M. Khapaev, “Correction: “Investigation of the stability of systems containing perturbations””, Dokl. Akad. Nauk SSSR, 288:6 (1986), 1288
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