Asymptotic estimates of the solution of the boundary value problem for singularly perturbed integro-differential equations

Authors

  • M. Dauylbayev Al-Farabi Kazakh National University, Institute of Information and Computational Technologies, Institute of mathematics and mathematical modelling http://orcid.org/0000-0002-4179-0374
  • N. Aviltay Al-Farabi Kazakh National University
  • B. Kadirbekov Al-Farabi Kazakh National University

DOI:

https://doi.org/10.26577/JMMCS.2020.v106.i2.04

Keywords:

singular perturbation, small parameter, initial jump, asymptotic estimates

Abstract

The mathematical models of many processes in physics, astrophysics, chemistry, biology, mechanics and technology are differential and integro-differential equations containing small parameters at the highest derivatives. Such equations currently are called singularly perturbed equations. The paper considers a two-point boundary value problem for a third-order linear integro-differential equation with a small parameter at the two highest derivatives, when the roots of the «additional
characteristic equation» are negative and the boundary conditions contain terms with small perturbations. The aim of the study is to obtain asymptotic estimates of the solution and to obtain the asymptotic behavior of the solution in a neighborhood of points where additional conditions are given that are lost during degeneracy. The boundary functions of the boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using boundary functions and Cauchy function an analytical formula for the solutions of the boundary value problem is obtained. A theorem on the asymptotic estimate of the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution and the growth order of its derivatives with respect to a small parameter are established. It is shown that the solution of the boundary value problem at the left point of given segment has the phenomenon of an initial first-order jump. Distinctive features in the asymptotic properties of the solutions of this boundary-value problem are shown in comparison with similar works in the field of singularly perturbed differential and integro-differential equations with initial jumps. The obtained results make it possible to construct a uniform asymptotic expansion of the solutions of boundary value problems for singularly perturbed integro-differential equations with any degree of accuracy with respect to a small parameter.

References

[1] Vishik M.I., Lyusternik L.A. "O nachal’nom skachke dlya nelinejnyh differencial’nyh uravnenij, soderzhashchih malyj parameter [On the initial jump for nonlinear differential equations containing a small parameter]" , Doklady Akademii Nauk SSSR. vol. 132, no 6 (1960): 1242-1245.

[2] K.A. Kasymov. "Ob asimptotike resheniya zadachi Koshi s bol’shimi nachal’nymi usloviyami dlya nelinejnogo oby- knovennogo differencial’nogo uravneniya, soderzhashchego malyj parameter [On the asymptotic behavior of the solution of the Cauchy problem with large initial conditions for a nonlinear ordinary differential equation containing a small parameter]" , Uspekhi matematicheskih nauk. vol. 5, no 17 (1962): 187-188.

[3] K.A. Kassymov, D. Nurgabyl "Asymptotic Estimates of Solution of a Singularly Perturbed Boundary Value Problem with an Initial Jump for Linear Differential Equations" , Differential Equations. vol. 40, no 5 (2004): 641-651.

[4] Dauylbayev M.K. and Atakhan N. "The initial jumps of solutions and integral terms in singular BVP of linear higher order integro-differential equations" , Miskolc Math. Notes, Hungary. vol. 16, no 2 (2015): 747-761.

[5] Vasil’eva, A., Butuzov,V., and Kalachev, L. "The boundary function method for singular perturbation problems, SIAM Studiesin Applied Mathematics" , Philadeplhia: SLAM, 1995.

[6] A.H. Nayfeh "Perturbation Methods" , USA: Wiley-Interscience, 2000.

[7] O’Malley, Robert E. "Historical Developments in Singular Perturbations" , Switzerland: Springer International Publishing, 2014.

[8] Mudvanhu, B and O’Malley, R. E., Jr. "A new renormalization method for the asymptotic solution of weakly nonlinear vector systems" , SIAM J. Appl. Math. vol. 63, no 2 (2002): 373-397.

[9] Kevorkian, J. and Cole, J.D. "Multiple Scale and Singular Perturbations Method" , New York: Springer, 1996.

[10] Sanders, J. A. and Verhulst, F. and Murdock, J. "Averaging Methods in Nonlinear Dynamical Systems" , 2nd Ed. New York: Springer-Verlag, 2007.

[11] Verhulst, F. "Methods and applications of singular perturbations: Boundary layers and multiple timescale dynamics" ,
Texts in Applied Mathematics. New York: Springer, 2005.

[12] D.R. Smith "Singular-Perturbation Theory an Introduction with Applications" , Cambridge: University Press, 2009.

[13] White, R. B. "Asymptotic Analysis of Differential Equations" , London: Imperial College Press, 2005.

[14] A.M. Wazwaz "A comparative study on a singular perturbation problem with two singular boundary points" , Applied Mathematics and Computation, 99 (1999): 179-193.

[15] S. Johnson "Singular Perturbation Theory, Mathematical and Analytical Techniques with Applications to Engineering" , New York: Springer, (2005).

[16] Skinner L.A. "Singular Perturbation Theory."New York: Springer, 2011.

[17] M. Kumar, H.K. Mishra, P. Singh "A boundary value approach for singularly perturbed boundary value problems" ,
Advances in Engineering Software, vol. 40, no 4 (2009): 298-304.
[18] W.T. Van "Horssen On integrating vectors and multiple scales for singularly perturbed ordinary differential equations" ,
Nonlinear Analysis, 46, (2001): 19-43.

[19] H. Hu, Z.G. Xiong "Comparison of two Lindstedt-Poincare type perturbation methods" , Journal of Sound and Vibration, 278, (2004): 437-444.

[20] F. Lakrad, M. Belhaq "Periodic solutions of strongly nonlinear oscillators by the multiple scales method" , Journal of Sound and Vibration, vol. 258, no 4 (2002): 677-700.

[21] Dauylbayev M.K., Zhumartov M.A., Konisbaeva K.T. "Cingylyaply ayytқyғan integpaldy diffepencialdyқ teңdeylep үshin integpaldy shettik ecep sheshiminiң acimptotikalyқ zhinaқtylyғy [Asymptotic convergence of the solution of the integral boundary value problem for singularly perturbed integral differential equations]" , Vectnik KazNPU imeni Abaya, cepiya fiz-mat. no 1 (2015): 18-24.

[22] Kasymov K.A., Sharipova Zh.U. "Asimptoticheskie ocenki resheniya kraevoj zadachi dlya singulyarno vozmushchennyh linejnyh differencial’nyh uravnenij tret’ego poryadka [Asymptotic estimates of the solution of the boundary value problem for singularly perturbed linear differential equations of the third order]" , Vestnik KazGU im. S.M. Kirova. Ser. mat. no 1, (1993): 146-150.

[23] Assiya Zhumanazarova and Young Im Cho (2020) "Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem" , Mathematics, 8 (2), 213(2020); doi:10.3390/math8020213.

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Published

2020-06-26