Boundary-Value Problems with Initial Jumps for Singularly Perturbed Integrodifferential Equations

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

doi:10.1007/s10958-017-3294-7

Cited by 9 publications

(9 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By applying (18)-(20) in (16), we obtain asymptotic estimations of the solution (17). Theorem 2 is proved.…”

Section: Resultsmentioning

confidence: 95%

“…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%

“…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%

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.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…By applying (18)-(20) in (16), we obtain asymptotic estimations of the solution (17). Theorem 2 is proved.…”

Section: Resultsmentioning

confidence: 95%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Control problems, also referred to as boundary value problems with parameters or as parameter identification problems, for ordinary differential and integro-differential equations have been extensively studied by many authors Akhmetov et al (2002); Alimhan et al (2015); Dauylbayev and Atakhan (2015); Dauylbaev and Mirzakulova (2017); Kiguradze (1987); Luchka and Nesterenko (2008); Nesterenko (2014); Ronto and Samoilenko (2000). Various methods have been applied to study these problems, such as methods of qualitative theory of differential equations, the calculus of variations and optimization theory, the method of upper and lower solutions, etc.…”

Section: Introductionmentioning

confidence: 99%

A computational method for solving a problem with parameter for linear systems of integro-differential equations

et al. 2020

View full text Add to dashboard Cite

This article presents a computational method for solving a problem with parameter for a system of Fredholm integro-differential equations. Some additional parameters are introduced and the problem under consideration is reduced to solving a system of linear algebraic equations. The coefficients and right-hand side of the system are calculated by solving the Cauchy problems for ordinary differential equations. We establish a criterion for the unique solvability of the problem under consideration. A numerical algorithm is offered for solving the problem with parameter. The results are illustrated by numerical examples. Keywords Problem with parameter • System of integro-differential equations • Solvability criteria • Algorithm • Computational method Mathematics Subject Classification 34B08 • 34H05 • 45J05 • 34K28 Communicated by Antonio José Silva Neto.

show abstract

“…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.

View full text Add to dashboard Cite

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.

show abstract

Boundary-Value Problems with Initial Jumps for Singularly Perturbed Integrodifferential Equations

Cited by 9 publications

References 3 publications

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

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

A computational method for solving a problem with parameter for linear systems of integro-differential equations

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

Contact Info

Product

Resources

About