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, cilt.9, sa.1, ss.50-59, 2018 (ESCI) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 9 Sayı: 1
  • Basım Tarihi: 2018
  • Dergi Adı: INTERNATIONAL JOURNAL OF MATHEMATICS AND PHYSICS
  • Derginin Tarandığı İndeksler: Emerging Sources Citation Index (ESCI)
  • Sayfa Sayıları: ss.50-59
  • Anahtar Kelimeler: singular perturbation, small parameter, the initial functions, asymptotics, passage to the limit
  • Orta Doğu Teknik Üniversitesi Adresli: Evet

Özet

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.