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Published: 11 September 2023

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1

Vrábeľ, Róbert. "Quasilinear and quadratic singularly perturbed Neumann's problem." Mathematica Bohemica 123, no. 4 (1998): 405–10. http://dx.doi.org/10.21136/mb.1998.125970.

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2

Yarka, Ulyana, Solomiia Fedushko, and Peter Veselý. "The Dirichlet Problem for the Perturbed Elliptic Equation." Mathematics 8, no. 12 (November 25, 2020): 2108. http://dx.doi.org/10.3390/math8122108.

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In this paper, the authors consider the construction of one class of perturbed problems to the Dirichlet problem for the elliptic equation. The operators of both problems are isospectral, which makes it possible to construct solutions to the perturbed problem using the Fourier method. This article focuses on the Dirichlet problem for the elliptic equation perturbed by the selected variable. We established the spectral properties of the perturbed operator. In this work, we found the eigenvalues and eigenfunctions of the perturbed task operator. Further, we proved the completeness, minimal spanning system, and Riesz basis system of eigenfunctions of the perturbed operator. Finally, we proved the theorem on the existence and uniqueness of the solution to the boundary value problem for a perturbed elliptic equation.
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3

Nurgabyl, D. N., and S. S. Nazhim. "Recovery problem for a singularly perturbed differential equation with an initial jump." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 100, no. 4 (December 30, 2020): 125–35. http://dx.doi.org/10.31489/2020m4/125-135.

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The article investigates the asymptotic behavior of the solution to reconstructing the boundary conditions and the right-hand side for second-order differential equations with a small parameter at the highest derivative, which have an initial jump. Asymptotic estimates of the solution of the reconstruction problem are obtained for singularly perturbed second-order equations with an initial jump. The rules for the restoration of boundary conditions and the right parts of the original and degenerate problems are established. The asymptotic estimates of the solution of the perturbed problem are determined as well as the difference between the solution of the degenerate problem and the solution of the perturbed problem. A theorem on the existence, uniqueness, and representation of a solution to the reconstruction problem from the position of singularly perturbed equations is proved. The results obtained open up possibilities for the further development of the theory of singularly perturbed boundary value problems for ordinary differential equations.
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4

Vrbik, Jan. "Two-body perturbed problem revisited." Canadian Journal of Physics 73, no. 3-4 (March 1, 1995): 193–98. http://dx.doi.org/10.1139/p95-027.

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Quaternion formulation of a perturbed two-body problem is extended to include nonconservative forces, and the resulting algorithm is demonstrated using the classical example of the precession of apses.
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5

Gekeler, E. W. "On the Perturbed Eigenvalue Problem." Journal of Mathematical Analysis and Applications 191, no. 3 (May 1995): 540–46. http://dx.doi.org/10.1006/jmaa.1995.1147.

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6

Vrábeľ, Róbert. "Upper and lower solutions for singularly perturbed semilinear Neumann's problem." Mathematica Bohemica 122, no. 2 (1997): 175–80. http://dx.doi.org/10.21136/mb.1997.125912.

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7

Akmatov, A. "The Regularization Method of Solutions a Bisingularly Perturbed Problem in the Generalized Functions Space." Bulletin of Science and Practice 8, no. 2 (February 15, 2022): 10–17. http://dx.doi.org/10.33619/2414-2948/75/01.

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When singularly perturbed problems are investigated, in the case of a change in stability, all work was performed in the space of analytical functions. Naturally, questions will arise whether it is possible to obtain an estimate of solutions to a singularly perturbed problem without moving to the complex plane. In the work, the first results obtained are the solutions of the singularly motivated task, not moving into the complex plane. For this purpose, a method of regularization in the space of generalized functions has been developed and corresponding estimates have been obtained. If we choose the starting point in a stable interval, then up to the transition point, the asymptotic proximity of solutions to the perturbed and undisturbed problem is in the order of a small parameter ε. The problem will appear when the point belongs to an unstable interval. Therefore, prior to this, the works moved to the complex plane. In such problems, there is a concept of the delay time of solutions to the perturbed and undisturbed problem. Level lines will appear in complex planes. In such problems, there is a concept of the delay time of solutions to the perturbed and undisturbed problem. Level lines will appear in complex planes. At special points, these lines have critical level lines. Therefore, it is impossible to choose the starting point so as to get the maximum delay time. But the asymptotic proximity of solutions of perturbed and undisturbed problems is possible with limited time delays. If we study the solution in the space of generalized functions, then we can choose the starting point with the maximum time delay. And also, without passing to the complex plane, it is possible to establish the asymptotic proximity of solutions to the perturbed and undisturbed problem. For this purpose, a method of regularization of solutions of a singularly perturbed problem has been developed for the first time.
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8

Han, Xinli, and Lijun Pan. "The Perturbed Riemann Problem with Delta Shock for a Hyperbolic System." Advances in Mathematical Physics 2018 (September 5, 2018): 1–11. http://dx.doi.org/10.1155/2018/4925957.

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In this paper, we study the perturbed Riemann problem with delta shock for a hyperbolic system. The problem is different from the previous perturbed Riemann problems which have no delta shock. The solutions to the problem are obtained constructively. From the solutions, we see that a delta shock in the corresponding Riemann solution may turn into a shock and a contact discontinuity under a perturbation of the Riemann initial data. This shows the instability and the internal mechanism of a delta shock. Furthermore, we find that the Riemann solution of the hyperbolic system is instable under this perturbation, which is also quite different from the previous perturbed Riemann problems.
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9

PERJAN, ANDREI, and GALINA RUSU. "Two parameter singular perturbation problems for sine-Gordon type equations." Carpathian Journal of Mathematics 38, no. 1 (November 15, 2021): 201–15. http://dx.doi.org/10.37193/cjm.2022.01.16.

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In the real Sobolev space $H_0^1(\Omega)$ we consider the Cauchy-Dirichlet problem for sine-Gordon type equation with strongly elliptic operators and two small parameters. Using some {\it a priori} estimates of solutions to the perturbed problem and a relationship between solutions in the linear case, we establish convergence estimates for the difference of solutions to the perturbed and corresponding unperturbed problems. We obtain that the solution to the perturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of $t=0.$
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10

PERJAN, ANDREI, and GALINA RUSU. "Abstract linear second order differential equations with two small parameters and depending on time operators." Carpathian Journal of Mathematics 33, no. 2 (2017): 233–46. http://dx.doi.org/10.37193/cjm.2017.02.10.

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In a real Hilbert space H consider the following singularly perturbed Cauchy problem. We study the behavior of solutions uεδ to this problem in two different cases: ε → 0 and δ ≥ δ0 > 0; ε → 0 and δ → 0, relative to solution to the corresponding unperturbed problem. We obtain some a priori estimates of solutions to the perturbed problem, which are uniform with respect to parameters, and a relationship between solutions to both problems. We establish that the solution to the perturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of t = 0. We show the boundary layer and boundary layer function in both cases.
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11

Zhumanazarova, Assiya, and Young Im Cho. "Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem." Mathematics 8, no. 2 (February 7, 2020): 213. http://dx.doi.org/10.3390/math8020213.

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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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12

Giani, Stefano, Luka Grubišić, Luca Heltai, and Ornela Mulita. "Smoothed-Adaptive Perturbed Inverse Iteration for Elliptic Eigenvalue Problems." Computational Methods in Applied Mathematics 21, no. 2 (March 12, 2021): 385–405. http://dx.doi.org/10.1515/cmam-2020-0027.

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Abstract We present a perturbed subspace iteration algorithm to approximate the lowermost eigenvalue cluster of an elliptic eigenvalue problem. As a prototype, we consider the Laplace eigenvalue problem posed in a polygonal domain. The algorithm is motivated by the analysis of inexact (perturbed) inverse iteration algorithms in numerical linear algebra. We couple the perturbed inverse iteration approach with mesh refinement strategy based on residual estimators. We demonstrate our approach on model problems in two and three dimensions.
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13

Abouelmagd, Elbaz I., Juan Luis García Guirao, and Jaume Llibre. "On the Periodic Orbits of the Perturbed Two- and Three-Body Problems." Galaxies 11, no. 2 (April 18, 2023): 58. http://dx.doi.org/10.3390/galaxies11020058.

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In this work, a perturbed system of the restricted three-body problem is derived when the perturbation forces are conservative alongside the corresponding mean motion of two primaries bodies. Thus, we have proved that the first and second types of periodic orbits of the rotating Kepler problem can persist for all perturbed two-body and circular restricted three-body problems when the perturbation forces are conservative or the perturbed motion has its own extended Jacobian integral.
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14

Daniyarova, Zh K. "Ingularly perturbed equations in critical cases." Bulletin of the Innovative University of Eurasia 84, no. 4 (December 23, 2021): 69–75. http://dx.doi.org/10.37788/2021-4/69-75.

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Singularly perturbed partial differential equations with small parameters with higher derivatives deserve special attention, which often arise in a variety of applied problems and are used in describing mathematical models of diffusion processes, absorption taking into account small diffusion, filtration of liquids in porous media, chemical kinetics, chromatography, heat and mass transfer, hydrodynamics and many other fields. It is necessary to consider the creation of an asymptotic classification of solutions of singularly perturbed equations using a well-known approach to solving the boundary value problem. In this case, the singular problem is understood as the problem of constructing the asymptotics of the solution of the Cauchy problem for a system of ordinary differential equations with a small parameter with a large derivative. The asymptotics of the solution in all cases is based on the last time interval or the construction of a boundary value problem for a system with a weak clot in an asymptotically large time interval. Purpose - to construct and substantiate the asymptotics of solving a singular initial problem for a system of two nonlinear ordinary differential equations with a small parameter; To date, a number of methods have been developed for constructing asymptotic expansions of solutions to various problems. This is the method of boundary functions developed in the works of A.B. Vasilyeva, M.I. Vishik, L.A. Lusternik, V.F. Butuzov; the regularization method of S. A. Lomov, methods of averaging, VKB, splicing of asymptotic decompositions of A.M. Ilyin and others. All the above methods allow us to obtain asymptotic expansions of solutions for wide classes of equations. At the same time, such singularly perturbed problems often arise, to which ready-made methods are not applicable or do not allow to obtain an effective result. Therefore, the development of methods for solving equations remains a very urgent problem. As a result of the study, an algorithm for constructing an asymptotic classification of the initial solution of the problem with a singular perturbation is given, and approaches to estimating the residual term are also shown.
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15

Bouaziz, Ferdaous, and Abdullah A. Ansari. "Perturbed Hill's problem with variable mass." Astronomische Nachrichten 342, no. 4 (April 29, 2021): 666–74. http://dx.doi.org/10.1002/asna.202113870.

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16

Zholtikov, Vitaliy P., and Vladislav V. Efendiev. "Singularly Perturbed Control with Delay Problem." Journal of Automation and Information Sciences 29, no. 1 (1997): 40–43. http://dx.doi.org/10.1615/jautomatinfscien.v29.i1.50.

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17

Armellin, Roberto, David Gondelach, and Juan Felix San Juan. "Multiple Revolution Perturbed Lambert Problem Solvers." Journal of Guidance, Control, and Dynamics 41, no. 9 (September 2018): 2019–32. http://dx.doi.org/10.2514/1.g003531.

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18

Gol’dberg, V. N. "Stability of a singularly perturbed problem." Differential Equations 48, no. 4 (April 2012): 524–37. http://dx.doi.org/10.1134/s0012266112040076.

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19

Stahlhofen, A. A. "Once more the perturbed Kepler problem." American Journal of Physics 62, no. 12 (December 1994): 1145–47. http://dx.doi.org/10.1119/1.17676.

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20

Waldvogel, Jörg. "Quaternions and the perturbed Kepler problem." Celestial Mechanics and Dynamical Astronomy 95, no. 1-4 (August 17, 2006): 201–12. http://dx.doi.org/10.1007/s10569-005-5663-7.

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21

Vrbik, J. "Perturbed Kepler problem in quaternionic form." Journal of Physics A: Mathematical and General 28, no. 21 (November 7, 1995): 6245–52. http://dx.doi.org/10.1088/0305-4470/28/21/027.

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22

Li, Gongbao, Peng Luo, Shuangjie Peng, Chunhua Wang, and Chang-Lin Xiang. "A singularly perturbed Kirchhoff problem revisited." Journal of Differential Equations 268, no. 2 (January 2020): 541–89. http://dx.doi.org/10.1016/j.jde.2019.08.016.

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23

Sharip, B., and А. Т. Yessimova. "ESTIMATION OF A BOUNDARY VALUE PROBLEM SOLUTION WITH INITIAL JUMP FOR LINEAR DIFFERENTIAL EQUATION." BULLETIN Series of Physics & Mathematical Sciences 69, no. 1 (March 10, 2020): 168–73. http://dx.doi.org/10.51889/2020-1.1728-7901.28.

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The paper considers a boundary value problem for a singularly perturbed linear differential equation with constant third-order coefficients. In this problem, a small parameter is indicated before the highest derivatives that are part of the differential equation and the boundary condition at t = 0.The fundamental system of solutions of a homogeneous singularly perturbed differential equation is constructed on the basis of asymptotic representations obtained for the roots of the corresponding characteristic equation. This system was used to construct the Cauchy function, special functions of boundary value problems, and also the Green function. With the help of these functions, an analytical formula is obtained for solving a singularly perturbed boundary value problem and it turns out that this solution has an initial zero-order jump at t = 0. It is proved that the solution to the considered singularly perturbed boundary value problem tends to the corresponding unperturbed problem obtained from it under .
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24

Littig, Samuel, and Friedemann Schuricht. "Perturbation results involving the 1-Laplace operator." Advances in Calculus of Variations 12, no. 3 (July 1, 2019): 277–302. http://dx.doi.org/10.1515/acv-2017-0006.

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AbstractWe consider perturbed eigenvalue problems of the 1-Laplace operator and verify the existence of a sequence of solutions. It is shown that the eigenvalues of the perturbed problem converge to the corresponding eigenvalue of the unperturbed problem as the perturbation becomes small. The results rely on nonsmooth critical point theory based on the weak slope.
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25

Asadi, S., and H. Mansouri. "A Full-Newton step infeasible-interior-point algorithm for P*(k)-horizontal linear complementarity problems." Yugoslav Journal of Operations Research 25, no. 1 (2015): 57–72. http://dx.doi.org/10.2298/yjor130515034a.

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In this paper we generalize an infeasible interior-point method for linear optimization to horizontal linear complementarity problem (HLCP). This algorithm starts from strictly feasible iterates on the central path of a perturbed problem that is produced by suitable perturbation in HLCP problem. Then, we use so-called feasibility steps that serves to generate strictly feasible iterates for the next perturbed problem. After accomplishing a few centering steps for the new perturbed problem, we obtain strictly feasible iterates close enough to the central path of the new perturbed problem. The complexity of the algorithm coincides with the best known iteration complexity for infeasible interior-point methods.
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26

Sobolev, Vladimir. "Efficient decomposition of singularly perturbed systems." Mathematical Modelling of Natural Phenomena 14, no. 4 (2019): 410. http://dx.doi.org/10.1051/mmnp/2019023.

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The problem of the decomposition of singularly perturbed differential systems by the method of integral manifolds is studied and the application of the method to the problems of enzyme kinetics is considered.
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27

Vrabel, Robert. "Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem." Axioms 11, no. 8 (August 11, 2022): 394. http://dx.doi.org/10.3390/axioms11080394.

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In this brief note, we study the problem of asymptotic behavior of the solutions for non-resonant, singularly perturbed linear Neumann boundary value problems εy″+ky=f(t), y′(a)=0, y′(b)=0, k>0, with an indication of possible extension to more complex cases. Our approach is based on the analysis of an integral equation associated with this problem.
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28

Abouelmagd, Elbaz I., Juan Luis García Guirao, and Jaume Llibre. "Periodic Orbits of Quantised Restricted Three-Body Problem." Universe 9, no. 3 (March 15, 2023): 149. http://dx.doi.org/10.3390/universe9030149.

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In this paper, perturbed third-body motion is considered under quantum corrections to analyse the existence of periodic orbits. These orbits are studied through two approaches to identify the first (second) periodic-orbit types. The essential conditions are given in order to prove that the circular (elliptical) periodic orbits of the rotating Kepler problem (RKP) can continue to the perturbed motion of the third body under quantum corrections where a massive primary body has excessive gravitational force over the smaller primary body. The primaries moved around each other in circular (elliptical) orbits, and the mass ratio was assumed to be sufficiently small. We prove the existence of the two types of orbits by using the terminologies of Poincaré for quantised perturbed motion.
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29

Mo, Jia-qi. "Quasilinear singularly perturbed problem with boundary perturbation." Journal of Zhejiang University-SCIENCE A 5, no. 9 (September 2004): 1144–47. http://dx.doi.org/10.1631/jzus.2004.1144.

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30

Cordaro, Giuseppe. "Multiple solutions to a perturbed Neumann problem." Studia Mathematica 178, no. 2 (2007): 167–75. http://dx.doi.org/10.4064/sm178-2-3.

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31

Grecu, Andrei. "A perturbed eigenvalue problem in exterior domain." Mathematica Slovaca 72, no. 4 (August 1, 2022): 945–58. http://dx.doi.org/10.1515/ms-2022-0065.

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Abstract Let Ω ⊂ R N (N ≥ 2) be a simply connected bounded domain, containing the origin, with C 2 boundary denoted by ∂Ω. Denote by Ω e x t := R N ∖ Ω ¯ $\Omega^{\mathrm{ext}}:=\mathbb{R}^{N} \backslash \bar{\Omega}$ the exterior of Ω. We consider the perturbed eigenvalue problem − Δ p u − Δ q u = μ K ( x ) | u | p − 2 u for x ∈ Ω ext u ( x ) = 0 for x ∈ ∂ Ω u ( x ) → 0 , as | x | → ∞ , $$\left\{\begin{array}{lcl} -\Delta_{p} u-\Delta_{q} u=\mu K(x)|u|^{p-2} u & \text { for } & x \in \Omega^{\text {ext }} \\ u(x)=0 & \text { for } & x \in \partial \Omega \\ u(x) \rightarrow 0, & \text { as } & |x| \rightarrow \infty, \end{array}\right.$$ where p, q ∈ (1,N), p ≠ q $p \neq q$ and K is a positive weight function defined on Ωext having the property that K ∈ L∞ (Ωext) ∩ LN/p (Ωext) . We show that the set of parameters μ for which the above eigenvalue problem possesses nontrivial weak solutions is exactly an unbounded open interval.
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32

Tursunov, Dilmurat Abdillazhanovich, Gulbayra Abdimalikovna Omaralieva, Makhfuzakhon Ibrakhimzhanovna Mamatbuvaeva, and Shahzadakhan Adylzhanovna Ramankulova. "SINGULARLY PERTURBED PROBLEM WITH DOUBLE BOUNDARY LAYER." Bulletin of Osh State University 1, no. 1 (2021): 102–9. http://dx.doi.org/10.52754/16947452_2021_1_1_102.

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33

Melenk, J. M., and C. Schwab. "Analytic Regularity for a Singularly Perturbed Problem." SIAM Journal on Mathematical Analysis 30, no. 2 (January 1999): 379–400. http://dx.doi.org/10.1137/s0036141097317542.

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34

Abouelmagd, Elbaz I., and Juan L. G. Guirao. "On the perturbed restricted three-body problem." Applied Mathematics and Nonlinear Sciences 1, no. 1 (January 28, 2016): 123–44. http://dx.doi.org/10.21042/amns.2016.1.00010.

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AbstractIn this survey paper we offer an analytical study regarding the perturbed planar restricted three-body problem in the case that the three involved bodies are oblate. The existence of libration points and their linear stability are explored under the effects of the perturbations in Coriolis and centrifugal forces. The periodic orbits around these points are also studied under these effects. Moreover, the elements of periodic orbits around these points are determined.
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35

Zorica, Uzelac, and Surla Katarina. "Discretization of the semilinear singularly perturbed problem." Nonlinear Analysis: Theory, Methods & Applications 30, no. 8 (December 1997): 4741–47. http://dx.doi.org/10.1016/s0362-546x(97)00411-2.

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36

Cordaro, Giuseppe, and Giuseppe Rao. "Three solutions for a perturbed Dirichlet problem." Nonlinear Analysis: Theory, Methods & Applications 68, no. 12 (June 2008): 3879–83. http://dx.doi.org/10.1016/j.na.2007.04.027.

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37

Mihăilescu, Mihai, and Denisa Stancu-Dumitru. "A perturbed eigenvalue problem on general domains." Annals of Functional Analysis 7, no. 4 (November 2016): 529–42. http://dx.doi.org/10.1215/20088752-3660738.

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38

Struckmeier, J., and A. Unterreiter. "A singular-perturbed two-phase Stefan problem." Applied Mathematics Letters 14, no. 2 (February 2001): 217–22. http://dx.doi.org/10.1016/s0893-9659(00)00139-7.

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39

Prashanth, S., Sanjiban Santra, and Abhishek Sarkar. "On the perturbed Q -curvature problem onS4." Journal of Differential Equations 255, no. 8 (October 2013): 2363–91. http://dx.doi.org/10.1016/j.jde.2013.06.015.

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40

Kuz'mina, L. K. "Solution of the singularly perturbed stability problem." Journal of Applied Mathematics and Mechanics 55, no. 4 (January 1991): 475–80. http://dx.doi.org/10.1016/0021-8928(91)90009-j.

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41

Vrábel’, Róbert. "Singularly perturbed anharmonic quartic potential oscillator problem." Zeitschrift für angewandte Mathematik und Physik 55, no. 4 (July 2004): 720–24. http://dx.doi.org/10.1007/s00033-004-1082-y.

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42

Li, Lin. "Three solutions for a perturbed Navier problem." Ricerche di Matematica 61, no. 1 (August 26, 2011): 117–23. http://dx.doi.org/10.1007/s11587-011-0118-9.

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43

Bates, Larry M. "Geometric quantization of a perturbed Kepler problem." Reports on Mathematical Physics 28, no. 2 (October 1989): 289–97. http://dx.doi.org/10.1016/0034-4877(89)90049-9.

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44

Mo, J. Q. "A Singularly Perturbed Nonlinear Boundary Value Problem." Journal of Mathematical Analysis and Applications 178, no. 1 (September 1993): 289–93. http://dx.doi.org/10.1006/jmaa.1993.1307.

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45

Melesse, Wondwosen Gebeyaw, Awoke Andargie Tiruneh, and Getachew Adamu Derese. "Solving Linear Second-Order Singularly Perturbed Differential Difference Equations via Initial Value Method." International Journal of Differential Equations 2019 (November 22, 2019): 1–10. http://dx.doi.org/10.1155/2019/5259130.

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In this paper, an initial value method for solving a class of linear second-order singularly perturbed differential difference equation containing mixed shifts is proposed. In doing so, first, the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay and advance parameters using Taylor series expansion. From the modified problem, two explicit initial value problems which are independent of the perturbation parameter are produced; namely, the reduced problem and the boundary layer correction problem. These problems are then solved analytically and/or numerically, and those solutions are combined to give an approximate solution to the original problem. An error estimate for this method is derived using maximum norm. Several test problems are considered to illustrate the theoretical results. It is observed that the present method approximates the exact solution very well.
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46

Craven, B. D. "Convergence of discrete approximations for constrained minimization." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 36, no. 1 (July 1994): 50–59. http://dx.doi.org/10.1017/s0334270000010237.

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AbstractIf a constrained minimization problem, under Lipschitz or uniformly continuous hypotheses on the functions, has a strict local minimum, then a small perturbation of the functions leads to a minimum of the perturbed problem, close to the unperturbed minimum. Conditions are given for the perturbed minimum point to be a Lipschitz function of a perturbation parameter. This is used to study convergence rate for a problem of continuous programming, when the variable is approximated by step-functions. Similar conclusions apply to computation of optimal control problems, approximating the control function by step-functions.
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47

Geng, Fazhan, Suping Qian, and Shuai Li. "Numerical solutions of singularly perturbed convection-diffusion problems." International Journal of Numerical Methods for Heat & Fluid Flow 24, no. 6 (July 29, 2014): 1268–74. http://dx.doi.org/10.1108/hff-01-2013-0033.

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Purpose – The purpose of this paper is to find an effective numerical method for solving singularly perturbed convection-diffusion problems. Design/methodology/approach – The present method is based on the asymptotic expansion method and the variational iteration method (VIM). First a zeroth order asymptotic expansion for the solution of the given singularly perturbed convection-diffusion problem is constructed. Then the reduced terminal value problem is solved by using the VIM. Findings – Two numerical examples are introduced to show the validity of the present method. Obtained numerical results show that the present method can provide very accurate analytical approximate solutions not only in the boundary layer, but also away from the layer. Originality/value – The combination of the asymptotic expansion method and the VIM is applied to singularly perturbed convection-diffusion problems.
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48

Li, Ye. "An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems." Communications in Computational Physics 19, no. 2 (February 2016): 442–72. http://dx.doi.org/10.4208/cicp.021114.140715a.

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Abstract:
AbstractIn this paper, we propose an uniformly convergent adaptive finite element method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation (BEC) and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations for the problem are reviewed to confirm the asymptotic behaviors of the solutions in the boundary/interior layer regions. By using the normalized gradient flow, we propose an adaptive finite element with hybrid basis to solve the singularly perturbed nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively to the small parameter ε. Extensive numerical results are reported to show the uniform convergence property of our method. We also apply the AFEM-HB to compute the ground and excited states of BEC with box/harmonic/optical lattice potential in the semiclassical regime (0 <ε≪C 1). In addition, we give a detailed error analysis of our AFEM-HB to a simpler singularly perturbed two point boundary value problem, show that our method has a minimum uniform convergence order
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49

Nhan, Tran, Kien Nguyen, Nguyen Hung, and Nguyen Toan. "The inverse k-max combinatorial optimization problem." Yugoslav Journal of Operations Research, no. 00 (2022): 37. http://dx.doi.org/10.2298/yjor220516037n.

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Abstract:
Classical combinatorial optimization concerns finding a feasible subset of a ground set in order to optimize an objective function. We address in this article the inverse optimization problem with the k-max function. In other words, we attempt to perturb the weights of elements in the ground set at minimum total cost to make a predetermined subset optimal in the fashion of the k-max objective with respect to the perturbed weights. We first show that the problem is in general NP-hard. Regarding the case of independent feasible subsets, a combinatorial O(n2 log n) time algorithm is developed, where n is the number of elements in E. Special cases with improved complexity are also discussed.
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50

Yonchev, A. "Perturbation Analysis of the Continuous-time Regional Pole Assignment and H2 Performance Control Problems: an LMI Approach." Information Technologies and Control 12, no. 3-4 (December 1, 2014): 28–35. http://dx.doi.org/10.1515/itc-2016-0004.

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Abstract:
Abstract In the paper a method to conduct perturbation analysis of regional pole assignment and H2 performance control problems for linear continuous-time systems are investigated. The studied control problems are based on solving LMIs (Linear Matrix Inequalities) and applying Lyapunov functions. The problem of performing sensitivity analysis of the perturbed matrix inequalities is done in a similar way as for perturbed matrix equations, after introducing a slightly perturbed right hand part. The calculated perturbation bounds can be used to analyze the feasibility and performance of the considered control problems in presence of perturbations in the system and the controller. An illustrative numerical example is also discussed in this paper.
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