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Asymptotic solution of first-order equation with small parameter under the derivative with perturbed operator

Annotation

The paper is devoted to the Cauchy problem for a differential equation with a small parameter when using a Fredholm operator in a Banach space with a certain method. The investigated effect of this parameter. The solution is in the form of an asymptotic expansion. When solving the problems of using the cascade decomposition method for equations, which allows us to split the equation into equations in subspaces.

Keywords

differential equation; asymptotic solution; small parameter; perturbation in the right-hand side; Fredholm operator; boundary layer phenomenon

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DOI

10.20310/1810-0198-2018-23-124-784-796

UDC

517.928

Pages

784-796

References

1. Vishik M.I., Lyusternik L.A. Regulyarnoye vyrozhdeniye i pogranichnyy sloy dlya lineynykh differentsial’nykh uravneniy s malym parametrom [Regular degeneracy and boundary layer for linear differential equations with a small parameter]. Uspekhi matematicheskikh nauk – Russian Mathematical Surveys, 1957, vol. 12, no. 5 (77), pp. 3-122. (In Russian). 2. Zubova S.P., Uskov V.I. Prilozheniya matrichno-differentsial’nogo operatora k resheniyu zadach dlya uravneniy v chastnykh proizvodnykh [Applications of the matrix-differential operator to the solution of problems for partial differential equations]. Izbrannyye trudy Mezhdunarodnogo simpoziuma po fundamental’nym i prikladnym problemam nauki «Itogi nauki» [Selected Works of the International Symposium on Fundamental and Applied Problems of Science “The Results of Science”]. Moscow, RAS Publ., 2017, no. 31, 253 p. (In Russian). 3. Krein S.G., Ngo Zuy Kan Asimptoticheskiy metod v zadache o kolebaniyakh sil’no vyazkoy zhidkosti [Asymptotic method in the problem of oscillations of a highly viscous fluid]. Prikladnaya matematika i mekhanika – Journal of Applied Mathematics and Mechanics, 1969, vol. 33, no. 3, pp. 456-464. (In Russian). 4. Trenogin V.A. Razvitiye i prilozheniya asimptoticheskogo metoda Lyusternika-Vishika [Development and applications of the Lyusternik-Vishik asymptotic method]. Uspekhi matematicheskikh nauk – Russian Mathematical Surveys, 1970, vol. 25, no. 4 (154), pp. 123-156. (In Russian). 5. Lomov S.A., Lomov I.S. Osnovy matematicheskoy teorii pogranichnogo sloya [Fundamentals of the Mathematical Theory of the Boundary Layer]. Moscow, Moscow State University Publ., 2011, 456 p. (In Russian). 6. Zubova S.P., Uskov V.I. Asimptoticheskoye resheniye singulyarno vozmushchennoy zadachi Koshi dlya uravneniya pervogo poryadka v banakhovom prostranstve [The asymptotic solution of a singularly perturbed cauchy problem for the first-order equation in a Banach space]. Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika – Proceedings of Voronezh State University. Series: Physics. Mathematics, 2016, no. 3, pp. 143-155. (In Russian). 7. Nikolsky S.M. Lineynyye uravneniya v lineynykh normirovannykh prostranstvakh [Linear equations in linear normed spaces]. Izvestiya AN SSSR. Seriya matematicheskaya – Mathematics of the USSR – Izvestiya, 1943, vol. 7, no. 3, pp. 147-166. (In Russian). 8. Zubova S.P., Chernyshov K.I. O lineynom differentsial’nom uravnenii s fredgol’movskim operatorom pri proizvodnoy [On a linear differential equation with a Fredholm operator under the derivative]. Differentsial’nyye uravneniya i ikh primeneniye – Differential Equations and Its Applications, 1976, no. 14, pp. 21-39. (In Russian). 9. Zubova S.P., Uskov V.I. Asimptoticheskoye resheniye zadachi Koshi dlya uravneniya pervogo poryadka s malym parametrom v banakhovom prostranstve. Regulyarnyy sluchay [Asymptotic Solution of the Cauchy Problem for a First-Order Equation with a Small Parameter in a Banach Space. The Regular Case]. Matematicheskie zametki – Mathematical Notes, 2018, vol. 103, no. 3, pp. 392-403. (In Russian). 10. Zubova S.P. O roli vozmushcheniy v zadache Koshi dlya uravneniya s fredgol’movym operatorom pri proizvodnoy [On the role of perturbations in the Cauchy problem for an equation with a Fredholm operator under the derivative]. Doklady Akademii nauk – Proceedings of the Russian Academy of Sciences, 2014, vol. 454, no. 4, pp. 383-386. (In Russian). 11. Krein S.G. Lineynyye differentsial’nyye uravneniya v banakhovom prostranstve [Linear Differential Equations in a Banach Space]. Moscow, Nauka Publ., 1967, 464 p. (In Russian). 12. Vasil’eva A.B., Butuzov V.F. Asimptoticheskiye razlozheniya resheniy singulyarno vozmushchennykh uravneniy [Asymptotic Expansions of Solutions of Singularly Perturbed Equations]. Moscow, Nauka Publ., 1973, 272 p. (In Russian).

Received

23.04.2018

Section of issue

Scientific articles

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