Asymptotic Behavior of Solutions of Singular Integro-differential Equations

Dauylbaev, M. K.; Mirzakulova, A.E.

doi:10.5890/dnc.2016.06.004

Cited by 5 publications

(7 citation statements)

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“…Similarly the Cauchy problem (1), (2) for ordinary differential equation was considered in [19]. In the particular case, similarly boundary value problem with initial jumps for this case 2 , 2   l m [16,17]. We consider the following homogeneous singularly perturbed differential equation associated with (1):…”

Section: Resultsmentioning

confidence: 99%

“…By applying the asymptotic estimations of the initial functions (18), (19) in (15), we obtain the following asymptotic estimations for the functions…”

Section: Resultsmentioning

confidence: 99%

“…It is established that this property essentially depends on the order of the derivatives entering under the integral sign. M. K. Dauylbayev [16,17] studied boundary value problems with two boundary layers possessing the phenomena of initial jumps. The novelty of these studies is that when the small parameter tends to zero, the fast solution variable grows unlimitedly, not only at one, the so-called initial point, but also at the other end of the considered segment.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, a class of singularly perturbed integro-differential equations with initial jump phenomena at both ends of the given segment is singled out. He also developed a method for studying and constructing the asymptotics of the solution of the Cauchy problem with initial jump for singularly perturbed linear differential equations with impulse action [18].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Dauylbayev

Akhmet

Mirzakulova

et al. 2018

Int. j. math. phys.

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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.

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Section: Resultsmentioning

confidence: 99%

“…By applying the asymptotic estimations of the initial functions (18), (19) in (15), we obtain the following asymptotic estimations for the functions…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Dauylbayev

Akhmet

Mirzakulova

et al. 2018

Int. j. math. phys.

View full text Add to dashboard Cite

show abstract

“…Note that other mathematical school of singularly perturbed equations in Kazakhstan and abroad investigate only boundary value problems, which does not have an initial jump. In our previous works in [1][2][3][4][5][6][7][8][9][10], we considered the initial and boundary value problems that are equivalent to the Cauchy problem with the initial jump for differential and integro-differential equations in the stable case.…”

Section: Introductionmentioning

confidence: 99%