**AUTHORS:**Kalimoldayev Maksat, Kalizhanova Аliya, Kozbakova Ainur, Kartbayev Timur, Aitkulov Zhalau, Abdildayeva Assel, Akhmetzhanov Maxat, Kopbosyn Leila

**Download as PDF**

**ABSTRACT:**
This work is devoted to the asymptotic solutions of integral boundary value problem for the Interlinear second order differential equation of Fredholm type. Studying an integral boundary value task, obtaining solution assessment of the set singular perturbed integral boundary value problem and difference estimate between the solutions of singular perturbed and unperturbed tasks; determination of singular perturbed integral boundary value problem solution behavior mode and its derivatives in discontinuity (jump) of the considered section and determination of the solution initial jumps values at discontinuity and of an integral member of the equation, as well, creation of asymptotic solution expansion assessing a residual member with any range of accuracy according to a small parameter by means of Cauchy task with an initial jump, at that selection of initial conditions due to singular perturbed boundary value problem solution behavior mode and its derivatives in the jump point. In the paper there applied methods of differential and integral equations theories, boundary function method, method of successive approximations and method of mathematical induction

**KEYWORDS:**
Singular, differential equations, asymptotic solutions, integral boundary problem, small parameter.

**REFERENCES:**

[1] A.N. Tikhonov, On the dependence of solutions of differential equations on a small parameter // Mathematical Collection. S.193- 204.− №2, −1948. 22/64.

[2] A.N. Tikhonov, Systems of differential equations containing parameters // Mathematical sbornik.1950. S.147-156.−№1, −27/69.

[3] A.N. Tikhonov, Systems of differential equations containing small parameters // Mathematical sbornik.1952. №3.−31/73.

[4] M.I. Vishik and LA Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter. // UMN. 1957.12. №5.− S.3-122.

[5] M.I. Vishik and LA Lyusternik, The asymptotic behavior of solutions of boundary value problems for quasi-linear differential equations with small parameter. S.778-781.− №5, −// DAN SSSR, 1958, T.121.

[6] A.B. Vasil'eva The asymptotic behavior of solutions of some boundary value problems for quasi-linear equations with a small parameter at the highest derivative // Dokl. 1958 T.123. №4.− S.583-586.

[7] Vasilyeva A.B. Construction of uniform approximations to solutions of differential equations with a small parameter at the highest derivative. // Mathematical Collection. 1960. T.50. №1.− S.43-58.

[8] A.B. Vasil'eva uniform approach to solving a system of differential equations with a small parameter at the derivative and an application to boundary problems. // Dokl. 1959 T.124. №3.− S.509-512.

[9] A.B. Vasil'eva asymptotic behavior of solutions of some boundary value problems for equations with a small parameter at the highest derivative // Dokl. 1960. T.35. №6.− S.1303- 1306.

[10] Vasilyeva A.B. asymptotic behavior of solutions of some problems for nonlinear ordinary differential equations with a small parameter at the highest derivative // UMN. 1963. T.18. №3.− S.15-86.

[11] A.B. Vasil'eva, V.F. Butuzov asymptotic expansions of solutions of singularly perturbed equations. p.272.−M .: Nauka, 1973.

[12] A.B. Vasil'eva, V.F. Butuzov singularly perturbed equations in critical cases. M .:. p.106.−Moscow University, 1978.

[13] M.I .Imanaliev, Asymptotic methods in the theory of singularly perturbed integrodifferential systems. Frunze: Ilim. 1972.

[14] M.I. Imanaliev, Oscillations and stability of singularly perturbed integro-differential systems. Frunze: Ilim. p.352.−1974.

[15] Trenogin V.A. Development and application of the asymptotic method of Lyusternik-Vishik. // UMN, 1970. T.25. №4 (154).− S.123-156.

[16] Tupchiev V.A. The asymptotic solution of boundary value problem for first order differential equations with a small parameter in the derivatives .// Akad. 1962 T.143. №6.− S.1296-1299.

[17] Tupchiev V.A. The existence, uniqueness and asymptotic behavior of the solution of the boundary problem for a system of differential equations with a small parameter at the highest derivative. // Dokl. 1962 T.142. №6.− S.1261- 1264.

[18] Tupchiev V.A. Angular solving boundary value problems with a small parameter in the system of equations of the first order. // Vestnik MGU. Mathematics and Mechanics. 1963. №3. S.17- 24.

[19] V.F. Butuzov The asymptotic behavior of solutions of singularly perturbed elliptic equations in a rectangular area. // Differential Equations. 1975. V.2. №6.− S.1030-1041.

[20] V.F. Butuzov angular boundary layer in mixed singularly perturbed problems for hyperbolic equations. // Mathematical Collection. 1977 T.104. №3.− S.460-485.

[21] V.F. Butuzov angular boundary layer in mixed singularly perturbed problems for hyperbolic equations of second order. // Dokl. 1977 T.235. №5.− S.997-1000.

[22] Butuzov V.F. Construction of boundary functions in some singularly perturbed problems of elliptic type. // Differential Equations. 1977. T.13. №10.− S.1829-1835.

[23] V.P. Maslov Complex WKB method in nonlinear equations. M .: Science, 1977.

[24] S.A. Lomov power boundary layer in problems with a singular perturbation .// Proceedings of the USSR Academy of Sciences. A series of mathematical. 1968. T.30. №4.− S.525-572.

[25] S.A. Lomov, Perturbation theory of singularly perturbed boundary value problems. S.65.−Alma-Ata, 1975.

[26] S.A. Lomov method of perturbations for singular problems .// Proceedings of the USSR Academy of Sciences, mathematical series. 1972. T.36. S.635-651.

[27] S.A. Lomov, Introduction to the general theory of singular perturbations. M .: Nauka, 1981. S.400.

[28] L.S. Pontryagin, Asymptotic behavior of solutions of systems of differential-equations with a small parameter in the higher derivatives. // Proceedings of the Academy of Sciences of the USSR. A series of mathematical. 1957. T.21. №5.− S.605-626.

[29] L.S. Pontryagin, systems of ordinary differential equations-tions with small parameters in the highest derivatives. Proceedings of the 3rd All-Union Math. Congress. T.111. Publ. USSR Academy of Sciences.

[30] E.F. Mishchenko Asymptotic calculation of periodic solutions of differential equations containing small parameters in the derivatives. // Proceedings of the USSR Academy of Sciences, mathematical series. 1957. T.21. №5.− S.627-654.

[31] E.F. Mishchenko, Asymptotic methods in the theory of relaxation oscillations. // UMN, 1959. №6. S.229-236.

[32] E.F. Mishchenko, L.S. Pontryagin Derivation of certain asymptotic estimates for solutions of differential equations with a small parameter on the derivatives. // Proceedings of the USSR Academy of Sciences, mathematical series. 1969. T.23. №5.− S.643-660.

[33] E.F. Mishchenko, N. Kh Rozov, Differential equations with a small parameter and relaxation oscillations. S.248.−M .: Nauka, 1975.

[34] Rozov N. Kh. asymptotic computation close to discontinuous periodic solutions of systems of second order differential equations. // Dokl. 1962 T.145. №1.− S.38-40.

[35] Rozov N. Kh. On the asymptotic theory of relaxation oscillations in systems with one degree of freedom. Calculation of the period of the limit cycle. // Bulletin of Moscow State University, mathematics and mechanics. 1964. №3. S.56-65.

[36] A.M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems. -M .: Nauka, 1989.

[37] Rozhkov V.I. Asymptotics of solutions of equations of neutral type with a small delay. // Proceedings of the Seminar on the theory of differential-equations with deviating argument. 1963. V.2. S.208-222.−Publishing House of Peoples' Friendship University.

[38] Rozhkov V.I. equation of neutral type with a small delay variable. // Differential Equations. 1966. №3.−T.2 S.407-416.

[39] Khapaev M.M. Linear equations with a small parameter at the highest derivative in a neighborhood of a regular singular point. // Dokl. 1959 T.129. №2.− S.268-271.

[40] Khapaev M.M. Asymptotic expansions of the solutions of ordinary differential equations with small coefficients on the highest derivatives in the neighborhood of a regular singular point. // Dokl. 1960 T.135. №6.− S.1338-1341.

[41] Khapaev M.M. Asymptotic methods and stability in the theory of nonlinear oscillations. M .: Higher School., 1988, p.184.

[42] Vladimir Vazov asymptotic expansions for ordinary differential equations. - M .: Mir, 1968.

[43] Cole J. Perturbation methods in applied mathematics. -M .: Mir, 1972.

[44] Nayfe A.H. Introduction to perturbation. - M .: Mir, 1984.

[45] Nayfe A. Howes F. Metody disturbances. S.456.−-M .: Mir, 1976.

[46] K. Chang, F. Howes nonlinear singularly perturbed boundary value problems. Theory and Applications. -M .: Mir, 1988.

[47] N.N. Bogolyubov, A. Mitropol'skii asymptotic methods in the theory of nonlinear oscillations. M .: Nauka, 1979. S.503.

[48] Filatov A.N. averaging method in systems integral-differential equations. // Dokl. 1965. - №3. - S.490-492.

[49] Filatov A.N. An averaging method for differential and integral-differential equations. Tashkent. 'FAN'. 1971.

[50] M.I. Vishik and L.A. Lyusternik, On the initial jump of non-linear differential equations containing a small parameter. // Dokl. 1960 T.132. -№6. -S.1242-1245.

[51] K.A. Kasymov The asymptotic behavior of solutions of the Cauchy problem with large initial conditions for nonlinear ordinary differential equations containing a small parameter. // UMN, 1962 vol.12. -№5. -S.187- 188.

[52] K.A. Kasymov The asymptotic behavior of solutions of the Cauchy problem with the initial jump for a system of differential equations with a small parameter in the derivative .// Proc. 'The equations of mathematical physics and functional analysis.' S.16-24.−Alma-Ata: Science, 1966.

[53] K.A. Kasymov The problem with an initial jump of nonlinear systems of differential equations with a small parameter at the highest derivative. // Dokl. 1968 T.179. -№2. - S.275- 278.

[54] K.A. Kasymov The asymptotic solution of the problem with the initial jump for nonlinear systems integral-differential equations containing a small parameter. // Math. KazSSR series of physical and mathematical. 1968. - №5. -S.69-72.

[55] K.A. Kasymov The asymptotic solution of the problem with the initial jump for nonlinear systems of differential equations with a small parameter at the highest derivative. // DAN SSSR, 1969. T.189. -№5. -S.941-944.

[56] K.A. Kasymov The asymptotic solution of the problem with the initial jump for the second order hyperbolic equations containing a small parameter. // Dokl. 1970 T.195. - №1. - S.28- 30.

[57] K.A. Kasymov On a problem for nonlinear systems of integro-differential equations with a small parameter. // Proceedings of the Kazakh SSR, series of physical and mathematical. 1984. - №3. - S.31-35.

[58] K.A. Kasymov, Mameshev DA An estimate of solutions of singularly perturbed boundary value problems for a system of integraldifferential equations of Fredholm type. // Reports of MH RK NAS, №2, -1997, S.20-27.

[59] K.A. Kasymov, Mameshev DA The asymptotic solution of boundary value problem for a system of linear singularly perturbed integrodifferential equations. S.95-103.−// Bulletin of the Kazakh State University, a series of mathematics, mechanics and informatics, №7, - 1997,

[60] K.A. Kasymov, Burkitbayev NI, Kigali AK Estimates of solutions of integral boundary value problems for linear singularly perturbed integro-differential equations of Fredholm type. S.26-29.−// Bulletin of MH RK NAS, №6, - 1998.

[61] Kigai A.K. The asymptotic solution of a singularly perturbed boundary value problem for integro-differential equations of the third order. // Bulletin of the Kazakh State University, a series of mathematics, mechanics and informatics, №14, -1999, -S.94-101.