The initial jumps of solutions and integral terms in singular BVP of linear higher order integro-differential equations

Dauylbayev, M. K.; Atakhan, N.

doi:10.18514/mmn.2015.1437

Cited by 10 publications

(9 citation statements)

References 24 publications

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“…where Q i (t, ε), P(t, ε) have the form (30). To determine C i , i = 1, 2, 3,, we substitute formula (35) into boundary conditions (2) and obtain a system of algebraic equations:…”

Section: The Existence and Uniqueness Theorem Of A Solutionmentioning

confidence: 99%

“…By formula (30), in view of the asymptotic representations (17) and (27), we have the following asymptotic representations for ϕ i (p, ε), i = 1, 2, 3, as ε → 0 :…”

Section: The Existence and Uniqueness Theorem Of A Solutionmentioning

confidence: 99%

“…where δ has the form from condition 4. Consequently, estimating formula (45), in view of formula (30) and condition 2, we obtain the estimates (42). Thus, Theorem 4 is proved.…”

Section: Asymptotic Estimates Of the Solutionmentioning

confidence: 99%

“…The works of Nurgabyl [24,25] were devoted to the study of boundary value problems for singularly perturbed differential equations. Dauylbaev [26][27][28][29][30][31] considered singularly perturbed initial and boundary value problems with initial jumps for linear and nonlinear integro-differential equations.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem

Zhumanazarova

Cho

2020

Mathematics

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In this study, the asymptotic behavior of the solutions to a boundary value problem for a third-order linear integro-differential equation with a small parameter at the two higher derivatives has been examined, under the condition that the roots of the additional characteristic equation are negative. Via the scheme of methods and algorithms pertaining to the qualitative study of singularly perturbed problems with initial jumps, a fundamental system of solutions, the Cauchy function, and the boundary functions of a homogeneous singularly perturbed differential equation are constructed. Analytical formulae for the solutions and asymptotic estimates of the singularly perturbed problem are obtained. Furthermore, a modified degenerate boundary value problem has been constructed, and it was stated that the solution of the original singularly perturbed boundary value problem tends to this modified problem’s solution.

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“…where Q i (t, ε), P(t, ε) have the form (30). To determine C i , i = 1, 2, 3,, we substitute formula (35) into boundary conditions (2) and obtain a system of algebraic equations:…”

Section: The Existence and Uniqueness Theorem Of A Solutionmentioning

confidence: 99%

“…By formula (30), in view of the asymptotic representations (17) and (27), we have the following asymptotic representations for ϕ i (p, ε), i = 1, 2, 3, as ε → 0 :…”

Section: The Existence and Uniqueness Theorem Of A Solutionmentioning

confidence: 99%

“…where δ has the form from condition 4. Consequently, estimating formula (45), in view of formula (30) and condition 2, we obtain the estimates (42). Thus, Theorem 4 is proved.…”

Section: Asymptotic Estimates Of the Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem

Zhumanazarova

Cho

2020

Mathematics

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“…In this case, we say that there is a phenomenon of an initial jump in the solution. K. A. Kasymov and his students in [3][4][5][6][7][8][9][10][11] continued research on initial and two-point boundary value problems with initial jumps. Three-point boundary value problems for ordinary differential and integro-differential equations with a small parameter at only the highest derivative, which have the phenomenon of initial jumps, were considered in [12,13].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of the solution of a singularly perturbed three-point boundary value problem with boundary jumps

Dauylbayev

Аrtykbayeva

Konysbaeva

2019

Int. j. math. phys.

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In this paper, the three-point boundary value problem is considered for the third-order linear differential equation with a small parameter with at the two highest derivatives when the roots of additional characteristic equation have negative signs. The aim of this paper is to bring asymptotic estimation of the solution of a singularly perturbed three-point boundary value problem with boundary jumps and the asymptotic convergence of the solution of a singularly perturbed initial value problem to the solution of an unperturbed initial value problem. In this paper the fundamental system of solutions, initial functions of a singularly perturbed homogeneous di�erential equation are constructed and their asymptotic estimates are obtained. An asymptotic behavior of the solution of the three-point boundary value problem at the points of initial jumps is established. A degenerate boundary-value problem is constructed. It is proved that the solution of the original singularly perturbed boundary value problem tends to the solution of the degenerate boundary value problem.

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Asymptotic Behavior of the Solutions of Boundary-Value Problems for Singularly Perturbed Integrodifferential Equations

Dauylbayev

Uaissov

2020

Ukr Math J

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The initial jumps of solutions and integral terms in singular BVP of linear higher order integro-differential equations

Cited by 10 publications

References 24 publications

Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem

Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem

Asymptotic behavior of the solution of a singularly perturbed three-point boundary value problem with boundary jumps

Asymptotic Behavior of the Solutions of Boundary-Value Problems for Singularly Perturbed Integrodifferential Equations

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