Uniform asymptotic expansion of the solution for the initial value problem with a piecewise constant argument
Mirzakulova A.E. Konisbayeva K.T.
2024E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2024#116Issue 4138 - 148 pp.
The article is devoted to the study of a singularly perturbed initial problem for a linear differential equation with a piecewise constant argument second-order for a small parameter. This paper is considered the asymptotic expansion of the solution to the Cauchy problem for singularly perturbed differential equations with piecewise-constant argument. The initial value problem for first order linear differential equations with piecewise-constant argument was obtained that determined the regular members. The Cauchy problems for linear nonhomogeneous differential equations with a constant coefficient were obtained, which determined the boundary layer terms. An asymptotic estimate for the remainder term of the solution of the Cauchy problem was obtained. Using the remainder term, we construct a uniform asymptotic solution with accuracy O(ϵN+1) on the θi ≤ t ≤ θi+1; i = 0; p segment of the singularly perturbed Cauchy problem with a piecewise constant argument.
asymptotics , boundary layer part , piecewise constant argument , singular perturbation , small parameter
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Department of mathematics, Al-Farabi Kazakh National University, 71 Al-Farabi, Almaty, 050038, Kazakhstan
Department of mathematics
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