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Asymptotics of solutions of a linear singularity perturbed system with degeneracy

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Abstract

Asymptotic expansions with respect to a small parameter e are constructed for a fundamental system of solutions of a linear singularity perturbed system of equations of the form

$$\varepsilon ^h A(t,\varepsilon )\frac{{dx}}{{dt}} = B(t,\varepsilon )x$$

on a finite interval of variation of the independent variable. It is assumed here that the (n×n) -matricesA (t, ɛ) and B(t,ɛ) can be expanded in series in powers of\(A(t,\varepsilon ) = \sum\limits_{k = 0}^\infty {A_k } (t)\varepsilon ^k , B(t,\varepsilon ) = \sum\limits_{k = 0}^\infty {B_k } (t)\varepsilon ^k ,\) where the matrix A0(t)is degenerate on the given interval and the pencil of matrices B∩(t)−ΩA0 (t) has several “finite” and “infinite” elementary divisors of the same multiplicity.

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Literature cited

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Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1559–1566, November, 1990.

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Yakovets, V.P. Asymptotics of solutions of a linear singularity perturbed system with degeneracy. Ukr Math J 42, 1403–1409 (1990). https://doi.org/10.1007/BF01066199

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Keywords

  • Asymptotic Expansion
  • Small Parameter
  • Finite Interval
  • Fundamental System
  • Elementary Divisor