Asymptotic convergence of the solution of the initial value problem for singularly perturbed higher-order integro-differential equation


Dauylbayev M. K. , AKHMET M., Mirzakulova A. E. , Uaissov A. B.

INTERNATIONAL JOURNAL OF MATHEMATICS AND PHYSICS, vol.9, no.1, pp.50-59, 2018 (ESCI) identifier

  • Publication Type: Article / Article
  • Volume: 9 Issue: 1
  • Publication Date: 2018
  • Journal Name: INTERNATIONAL JOURNAL OF MATHEMATICS AND PHYSICS
  • Journal Indexes: Emerging Sources Citation Index (ESCI)
  • Page Numbers: pp.50-59
  • Keywords: singular perturbation, small parameter, the initial functions, asymptotics, passage to the limit
  • Middle East Technical University Affiliated: Yes

Abstract

The article is devoted to research the Cauchy problem for singularly perturbed higher-order linear integro-differential equation with a small parameters at the highest derivatives, provided that the roots of additional characteristic equation have negative signs. An explicit analytical formula of the solution of singularly perturbed Cauchy problem is obtained. The theorem about asymptotic estimate of a solution of the initial value problem is proved. The nonstandard degenerate initial value problem is constructed. We find the solution of the nonstandard degenerate initial value problem. An estimate difference of the solution of a singularly perturbed and nonstandard degenerate initial value problems is obtained. The asymptotic convergence of solution of a singularly perturbed initial value problem to the solution of the nonstandard degenerate initial value problem is established.