Tikhonov theorem for differential equations with singular impulses

Akhmet, Marat; Çağ, Sabahattin

doi:10.5890/dnc.2018.09.007

Cited by 1 publication

(8 citation statements)

References 19 publications

(51 reference statements)

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“…Since there exist an infinite number of discontinuity moments in a finite time, the possibility of the blow up of solutions occurs here. Akhmet and Çag [1] first time in literature considered differential equations when impulses are also singular beside the differential equation. They presented the following problem…”

Section: Introductionmentioning

confidence: 99%

“…which prevents impulsive function to blow up as the parameter ε tends to zero. The main novelty of the paper [1] is the extension of Tikhonov's theorem so that the system (3) has the small parameter of the impulse function and the instant of discontinuity is different for each dependent variables. The singularity of the impulse part of the system can be handled using methods of perturbation theory.…”

Section: Introductionmentioning

confidence: 99%

“…The singularity of the impulse part of the system can be handled using methods of perturbation theory. In our present research, we apply the ideas of the paper [1].…”

Section: Introductionmentioning

confidence: 99%

“…Жүйенiң импульстiк бөлiгiндегi сингулярлықты ауытқу теориясының әдiстерiн қолдану арқылы қарастыруға болады. Бұл мақала [1] жұмыстың жалғасы болып табылады. Осы зерттеудеде [1] жұмыста сипатталған әдiс қолданылады.…”

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“…Бұл мақала [1] жұмыстың жалғасы болып табылады. Осы зерттеудеде [1] жұмыста сипатталған әдiс қолданылады. Жұмыстың мақсаты-жуықтауды жоғары дәлдiкпен құру және толық асимптотикалық жiктелудi алу.…”

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A case of impulsive singularity

Akhmet

Aviltay²,

Dauylbayev³

et al. 2023

JMMCS

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The paper considers an impulsive system with singularities. Different types of problems with singular perturbations have been discussed in many books. In Bainov and Kovachev's book [4]several articles cited therein consider impulse systems with small parameter involving only differential equations. The parameter is not in the impulsive equation of the systems. In our present the small parameter is inserted into the impulse equation. This is the principal novelty of our study. Furthermore, for the impulsive function, we found a condition that prevents the impulsive function to blow up as the parameter tends to zero. So we have significantly extended the singularity concept for discontinuous dynamics.The singularity of the impulsive part of the system can be treated in the manner of perturbation theory methods. This article is a continuation of [1] work. In our present research, we apply the method of the paper [1]. Our goal is to construct an approximation with higher accuracy and to obtain the complete asymptotic expansion. We construct a uniform asymptotic approximation of the solution that is valid in the entire close interval by using the method of boundary functions [22]. An illustrative example using numerical simulations is given to support the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The singularity of the impulse part of the system can be handled using methods of perturbation theory. In our present research, we apply the ideas of the paper [1].…”

Section: Introductionmentioning

confidence: 99%

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