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SOURCE:   Almanac of Modern Science and Education. Tambov: Gramota, 2017. № 4-5. P. 76-80. ISSN 1993-5552.
SCIENTIFIC AREA: Physical-Mathematical Sciences
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ASYMPTOTICS OF THE SOLUTION OF THE BOUNDARY-VALUE PROBLEM WITH THE INTERNAL INITIAL JUMP

Nurgabyl Duisebek Nurgabyluly, Kanatkyzy Zere
Zhetysu State University named after I. Zhansugurov


Abstract. In the paper the authors consider the boundary-value problem for the second-order linear differential equation with the small parameter at the derivative. An algorithm for constructing the asymptotic expansion of the solution of the boundary-value problem is described. Uniform asymptotic approximation of the solution of the singularly perturbed boundary-value problem is constructed accurate within random order as the small parameter tends to zero. On the basis of asymptotics of the solution of the initial problem, existence and uniqueness of the solution of the boundary-value problem are proved. A degenerate problem is formulated. In case of sufficiently small the authors find difference estimate between solutions of perturbed and unperturbed problems. The phenomenon of the internal initial jump is investigated.
Key words and phrases: дифференциальные уравнения, краевые задачи, малый параметр, начальный скачок, асимптотическое разложение, differential equations, boundary-value problems, small parameter, initial jump, asymptotic expansion.
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References:
  1. Biyadilov N. B. Singulyarno vozmushchennoe kvazilineinoe differentsial'noe uravnenie vtorogo poryadka s vnutrennim nachal'nym skachkom // Izvestiya Akademii nauk Kazakhskoi SSR. Seriya fiziko-matematicheskikh nauk. 1990. № 5. S. 42-49.
  2. Vasil'eva A. B. Asimptotika reshenii nekotorykh kraevykh zadach dlya kvazilineinykh uravnenii s malym parametrom pri starshei proizvodnoi // Doklady Akademii nauk SSSR. 1958. T. 123. № 4. S. 583-586.
  3. Vishik M. I., Lyusternik L. A. O nachal'nom skachke dlya nelineinykh differentsial'nykh uravnenii, soderzhashchikh malyi parametr // Doklady Akademii nauk SSSR. 1960. T. 132. № 6. S. 1242-1245.
  4. Vishik M. I., Lyusternik L. A. Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsial'nykh uravnenii s malym parametrom // Uspekhi matematicheskikh nauk. 1957. T. 12. № 5. S. 3-122.
  5. Galakhov M. A. Resheniya differentsial'nykh uravnenii s malym parametrom, imeyushchie v peredele razryv // Zhurnal vychislitel'noi matematiki i matematicheskoi fiziki. 1969. T. 9. № 1. S. 96-107.
  6. Imanaliev M. I. Asimptoticheskie metody v teorii singulyarno vozmushchennykh integro-differentsial'nykh sistem // Issledovaniya po integro-differentsial'nym uravneniyam: v 2-kh t. Frunze: Ilim, 1962. T. 2. S. 21-39.
  7. Kasymov K. A. Ob asimptotike resheniya zadachi Koshi s bol'shimi nachal'nymi usloviyami dlya nelineinykh obyknovennykh differentsial'nykh uravnenii, soderzhashchikh malyi parametr // Uspekhi matematicheskikh nauk. 1962. T. 17. № 5. S. 187-188.
  8. Kasymov K. A., Nurgabylov D. N. Asimptoticheskoe razlozhenie resheniya kraevoi zadachi s vnutrennim nachal'nym skachkom // Materialy II Vsesoyuznoi konferentsii po asimptoticheskim metodam: v 2-kh t. Alma-Ata: Nauka, 1979. T. 1. S. 97-99.
  9. Yarkin A. N. Kraevye zadachi dlya singulyarno vozmushchennykh sistem obyknovennykh differentsial'nykh uravnenii, imeyushchie resheniya, blizkie k razryvnym: avtoref. diss. … k. f.-m. n.: 01.01.02. M.: MGU, 1977. 7 s.

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