## Journal of Applied Mathematics

### Asymptotic Estimates of the Solution of a Restoration Problem with an Initial Jump

Duisebek Nurgabyl

#### Abstract

The asymptotic behavior of the solution of the singularly perturbed boundary value problem ${L}_{\epsilon }y=h(t)\lambda ,{L}_{i}y+\sigma i\lambda ={a}_{i},i=\stackrel{̅}{1,n+1}$ is examined. The derivations prove that a unique pair $(\stackrel{̃}{y}(t,\stackrel{̃}{\lambda }(\epsilon ),\epsilon ),\stackrel{̃}{\lambda }(\epsilon ))$ exists, in which components $y(t,\stackrel{̃}{\lambda }(\epsilon ),\epsilon )$ and $\stackrel{̃}{\lambda }(\epsilon )$ satisfy the equation ${L}_{\epsilon }y=h(t)\lambda$ and boundary value conditions ${L}_{i}y+\sigma i\lambda ={a}_{i},i=\stackrel{̅}{1,n+1}$. The issues of limit transfer of the perturbed problem solution to the unperturbed problem solution as a small parameter approaches zero and the existence of the initial jump phenomenon are studied. This research is conducted in two stages. In the first stage, the Cauchy function and boundary functions are introduced. Then, on the basis of the introduced Cauchy function and boundary functions, the solution of the restoration problem ${L}_{\epsilon }y=h(t)\lambda ,{L}_{i}y+\sigma i\lambda ={a}_{i},i=\stackrel{̅}{1,n+1}$ is obtained from the position of the singularly perturbed problem with the initial jump. Through this process, the formula of the initial jump and the asymptotic estimates of the solution of the considered boundary value problem are identified.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 956402, 11 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305705

Digital Object Identifier
doi:10.1155/2014/956402

Mathematical Reviews number (MathSciNet)
MR3206901

#### Citation

Nurgabyl, Duisebek. Asymptotic Estimates of the Solution of a Restoration Problem with an Initial Jump. J. Appl. Math. 2014 (2014), Article ID 956402, 11 pages. doi:10.1155/2014/956402. https://projecteuclid.org/euclid.jam/1425305705

#### References

• A. N. Tihonov, “On independence of the solutions of differential equations from a small parameter,” Matematicheskii Sbornik, vol. 22, no. 2, pp. 193–204, 1948.
• A. N. Tihonov, “Systems of differential equations containing small parameters within derivatives,” Matematicheskii Sbornik, vol. 31, no. 3, pp. 575–586, 1952.
• M. I. Višik and L. A. Lyusternik, “Regular degeneration and boundary layer for linear differential equations with a small parameter,” Uspekhi Matematicheskikh Nauk, vol. 12, no. 5, pp. 3–122, 1957.
• A. B. Vasilyeva, “Asymptotics of the solutions of some boundary value problems for quasilinear equations within a small parameter and a senior derivative,” Doklady Akademii Nauk SSSR, vol. 123, no. 4, pp. 583–586, 1958.
• V. F. Butuzov, “Angular boundary layer in the mixed singularly perturbed problems for hyperbolic equations of the second order,” Doklady Akademii Nauk SSSR, vol. 235, no. 5, pp. 997–1000, 1977.
• M. I. Višik and L. A. Lyusternik, “On initial jump for non-linear differential equations containing a small parameter,” Doklady Akademii Nauk SSSR, vol. 132, no. 6, pp. 1242–1245, 1960.
• K. A. Kasymov, “On asymptotics of the solutions of Cauchy problem with boundary conditions for non-linear ordinary differential equations containing a small parameter,” Uspekhi Matematicheskikh Nauk, vol. 17, no. 5, pp. 187–188, 1962.
• Z. N. Zhakupov, “Asymptotic behavior of the solutions of boundary value problem for some class of non-linear equation systems containing a small parameter,” Izvestiya Akademii Nauk Kazakhskoĭ SSR, no. 5, pp. 42–49, 1971.
• D. N. Nurgabylov, “Asymptotic expansion of the solution of a boundary value problem with internal initial jump for nonlinear systems of differential equations,” Izvestiya Akademii Nauk Kazakhskoĭ SSR, no. 3, pp. 62–65, 1984.
• Yu. I. Neĭmark and V. N. Smirnova, “Singularly perturbed problems and the Painlevé paradox,” Differentsial'niye Uravneniya, vol. 36, no. 11, pp. 1493–1500, 2000.
• V. F. Butuzov and A. B. Vasiliyeva, “On asymptotics of the solution of a contrast structure type,” Matematicheskie Zametki, vol. 42, no. 6, pp. 881–841, 1987.
• A. V. Kibenko and A. M. Perov, “On a two-point boundary-value problem with a parameter,” Akademiya Nauk Ukrainskoĭ SSR, no. 10, pp. 1259–1265, 1961.
• D. S. Dzhumabayev, “On the unique solvability of the linear point-to-point problems with a parameter,” Izvestiya Akademii Nauk SSSR, no. 1, pp. 31–37, 1999.
• L. Schlesinger, “Über asymptotische darstellungen der lösungen linearer differentialsysteme als funktionen eines parameters,” Mathematische Annalen, vol. 63, no. 3, pp. 277–300, 1907.
• G. D. Birkhoff, “On the asymptotic character of the solutions of certain linear differential equations containing a parameter,” Transactions of the American Mathematical Society, vol. 9, no. 2, pp. 219–231, 1908.
• S. A. Lomov, Introduction to the General Theory of Singular Perturbations, Nauka, Moscow, Russia, 1981.
• K. A. Kasymov and D. N. Nurgabyl, “Asymptotic estimates for the solution of a singularly perturbed boundary value problem with an initial jump for linear differential equations,” Differential Equations, vol. 40, no. 5, pp. 641–651, 2004. \endinput