Asymptotic behavior of the solution of the integral boundary value problem for singularly perturbed integro-differential equations
DOI:
https://doi.org/10.26577/JMMCS.2021.v112.i4.02Keywords:
singular perturbation, small parameter, the initial jump, asymptoticsAbstract
The work is devoted to clarifying asymptotic with respect to a small parameter behavior of the solution of the integral boundary value problem for singularly perturbed linear integro-differential equation. In the work are obtained analytical formula and asymptotic estimates of the solution for the integral boundary value problem. It is established that the solution of the considered boundary value problem at the ends of a given segment has the phenomena of boundary jumps of the same orders. A modified degenerate boundary value problem is constructed, to the solution of which approaches the solution of assumed singularly perturbed integral boundary value problem. The value of the jump of integral terms is found.
References
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[2] Birkhoff GD. "On the asymptotic character of the solutions of certain linear differential equations containing a parameter" , Transactions of the American Mathematical Society, (1908); 9(2): 219–231.
[3] Noaillon P. "Developpements asymptotiques dans les equations differentielles lineaires a parametre variable" , Mem. Soc. Sci. Liege, (1912); 3(11): 197.
[4] Wasow W. "Singular perturbations of boundary value problems for nonlinear differential equations of the second order" , Comm. On Pure and Appl. Math., (1956); 9 : 93–113.
[5] Nayfeh A. "Perturbation Methods" , New York, USA: John Wiley, (1973).
[6] Tihonov AN. "O zavisimosti reshenij differencial’nyh uravnenij ot malogo parametra" , Matematicheskij sbornik (1948); 22(2): 193–204 (in Russian).
[7] Tihonov AN. "O sistemah differencial’nyh uravnenij soderzhashhih parametry" , Matematicheskij sbornik, (1950); 27(69): 147–156 (in Russian).
[8] Vishik MI, Lyusternik LA. "Regular degeneration and boundary layer for linear differential equations with small parameter multiplying the highest derivatives" , Usp.Mat. Nauk, (1957); 12: 3–122 (in Russian), Amer. Math. Soc. Transl. 1962; 20(2): 239–364.
[9] Vishik MI, Lyusternik LA. "On the initial jump for non-linear differential equations containing a small parameter" ,Doklady Akademii Nauk SSSR, (1960); 132(6): 1242–1245 (in Russian).
[10] Bogoliubov N, Mitropolskii YA. "Asymptotic methods in the theory of nonlinear oscillations" ,Delhi: Hindustan Publ. Corp., (1961).
[11] Vasil’eva A, Butuzov V. "Singularly perturbed differential equations of parabolic type in Asymptotic Analysis II" , Lecture Notes in Math. Berlin: Springer-Verlag, (1983).
[12] Vasil’eva A, Butuzov V, Kalachev L. "The boundary function method for singular perturbation problems" ,Philadeplhia: SIAM Studiesin Applied Mathematics, (1995).
[13] O’Malley R. "Introduction to singular perturbations" ,New York, USA: Academic Press, (1974).
[14] O’Malley R. "Singular perturbations methods for ordinary differential equations" , Berlin: Springer-Verlag, (1991).
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Published
2021-12-31
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Mathematics
