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Journal articles on the topic 'Linear singularly perturbed problem'

Author: Grafiati

Published: 3 June 2025

Last updated: 19 July 2025

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1

Dauylbayev, Muratkhan, Marat Akhmet, and Aviltay Nauryzbay. "ASYMPTOTIC EXPANSION OF THE SOLUTION FOR SINGULARPERTURBED LINEAR IMPULSIVE SYSTEMS." Journal of Mathematics, Mechanics and Computer Science 122, no. 2 (2024): 14–26. http://dx.doi.org/10.26577/jmmcs2024-122-02-b2.

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In this study, a singularly perturbed linear impulsive system with singularly perturbed impulses is considered. Many books discuss different types of singular perturbation problems. In the present work, an impulse system is considered in which a small parameter is introduced into the impulse equation. This is the main novelty of our study, since other works [25] have only considered a small parameter in the differential equation. A necessary condition is also established to prevent the impulse function from bloating as the parameter approaches zero. As a result, the notion of singularity for discontinuous dynamics is greatly extended. An asymptotic expansion of the solution of a singularly perturbed initial problem with an arbitrary degree of accuracy for a small parameter is constructed. A theorem for estimating the residual term of the asymptotic expansion is formulated, which estimates the difference between the exact solution and its approximation. The results extend those of [32], which formulates an analogue of Tikhonov’s limit transition theorem. The theoretical results are confirmed by a modelling example.

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2

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 (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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3

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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4

Zhumanazarova, Assiya, and Young Im Cho. "Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem." Mathematics 8, no. 2 (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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5

Mirzakulova, A. E., and K. T. Konisbayeva. "Uniform asymptotic expansion of the solution for the initial value problem with a piecewise constant argument." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 116, no. 4 (2024): 138–48. https://doi.org/10.31489/2024m4/138-148.

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The article is devoted to the study of a singularly perturbed initial problem for a linear differential equation with a piecewise constant argument second-order for a small parameter. This paper is considered the asymptotic expansion of the solution to the Cauchy problem for singularly perturbed differential equations with piecewise-constant argument. The initial value problem for first order linear differential equations with piecewise-constant argument was obtained that determined the regular members. The Cauchy problems for linear nonhomogeneous differential equations with a constant coefficient were obtained, which determined the boundary layer terms. An asymptotic estimate for the remainder term of the solution of the Cauchy problem was obtained. Using the remainder term, we construct a uniform asymptotic solution with accuracy O(εN+1) on the θi ≤ t ≤ θi+1, i = 0, p segment of the singularly perturbed Cauchy problem with a piecewise constant argument.

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6

Vaculíková, Ľudmila, and Vladimír Liška. "Singularly Perturbed Linear Neumann Problem with the Characteristic Roots on the Imaginary Axis." Research Papers Faculty of Materials Science and Technology Slovak University of Technology 18, no. 28 (2010): 163–68. http://dx.doi.org/10.2478/v10186-010-0020-4.

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Singularly Perturbed Linear Neumann Problem with the Characteristic Roots on the Imaginary Axis We investigate the problem of existence and asymptotic behavior of solutions for the singularly perturbed linear Neumann problem <img src="/fulltext-image.asp?format=htmlnonpaginated&src=C6551P41673P4147_html\Journal10186_Volume18_Issue28_20_paper.gif" alt=""/> Our approach relies on the analysis of integral equation equivalent to the problem above.

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7

Mane, Shilpkala T., and Ram Kishun Lodhi. "Quartic B-Spline Technique for Third-Order Linear Singularly Perturbed Boundary Value Problem with Discontinuous Source Term." International Journal of Mathematical, Engineering and Management Sciences 10, no. 4 (2024): 1178–91. https://doi.org/10.33889/ijmems.2025.10.4.056.

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In this paper, we developed an effective computational technique for addressing third-order linear singularly perturbed problems having the source term discontinuous. Boundary or interior layers are frequently present in singular perturbation issues, making traditional numerical techniques more challenging. Here, we present a quartic B-spline method (QBSM) for the approximate solution of the third-order singularly perturbed boundary value problem, improving both the accuracy and efficiency of the solutions. In addition, the proposed method's convergence and error are investigated. The performance of the current technique is demonstrated through numerous numerical tests. The numerical findings are compared to other approaches reported in the literature.

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8

Tsekhan, Olga. "Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation." Axioms 8, no. 2 (2019): 71. http://dx.doi.org/10.3390/axioms8020071.

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The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system.

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9

Grossmann, Christian, Lars Ludwig, and Hans-Görg Roos. "Layer-adapted methods for a singularly perturbed singular problem." Computational Methods in Applied Mathematics 11, no. 2 (2011): 192–205. http://dx.doi.org/10.2478/cmam-2011-0010.

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Abstract In the present paper we analyze linear finite elements on a layer adapted mesh for a boundary value problem characterized by the overlapping of a boundary layer with a singularity. Moreover, we compare this approach numerically with the use of adapted basis functions, in our case modified Bessel functions. It turns out that as well adapted meshes as adapted basis functions are suitable where for our one-dimensional problem adapted bases work slightly better.

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10

Akmatov, A. "Investigation of Solutions to a System of Singularly Perturbed Differential Equations." Bulletin of Science and Practice 8, no. 5 (2022): 15–23. http://dx.doi.org/10.33619/2414-2948/78/01.

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Solutions of linear systems of singularly perturbed differential equations are investigated in the work, in the case when the matrix function had multiple eigenvalues. And also in the study of solutions to a system of singularly perturbed differential equations, we apply the level line method. We define a stable and unstable interval. We take the starting point in stable intervals. Passing to the complex domain, we define the domain that we study for solutions of the problem under consideration. We divide the defined areas near the singular point into several areas. In each area, we estimate the solutions of the problem. To do this, we choose the integration path and prove the lemma and theorem. As a result, we will prove the asymptotic proximity of the solutions of the perturbed and unperturbed problems.

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11

Lou, Yuan, and Meijun Zhu. "A singularly perturbed linear eigenvalue problem in C1domains." Pacific Journal of Mathematics 214, no. 2 (2004): 323–34. http://dx.doi.org/10.2140/pjm.2004.214.323.

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12

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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13

Kathirkamanayagan, M., and G. S. Ladde. "Singularly perturbed linear boundary value problems." Journal of Mathematical Analysis and Applications 168, no. 2 (1992): 430–59. http://dx.doi.org/10.1016/0022-247x(92)90171-9.

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14

Zhang, Yan, Zhengfan Liu, Zhong Yang, and Haifei Si. "Robust Control of Wind Turbines by Using Singular Perturbation Method and Linear Parameter Varying Model." Journal of Control Science and Engineering 2016 (2016): 1–10. http://dx.doi.org/10.1155/2016/2830736.

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The maximum power point tracking problem of variable-speed wind turbine systems is studied in this paper. The wind conversion systems contain both mechanical part and electromagnetic part, which means the systems have time scale property. The wind turbine systems are modeled using singular perturbation methodology. A linear parameter varying (LPV) model is developed to approximate the nonlinear singularly perturbed model. Then stability and robust properties of the open-loop linear singularly perturbed system are analyzed using linear matrix inequalities (LMIs). An algorithm of designing a stabilizing state-feedback controller is proposed which can guarantee the robust property of the closed-loop system. Two numerical examples are provided to demonstrate the effectiveness of the control scheme proposed.

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15

Vrabel, Robert. "Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem." Axioms 11, no. 8 (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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16

Padmaja, P., P. Aparna, and R. S. R. Gorla. "An Intial-Value Technique for Self-Adjoint Singularly Perturbed Two-Point Boundary Value Problems." International Journal of Applied Mechanics and Engineering 25, no. 1 (2020): 106–26. http://dx.doi.org/10.2478/ijame-2020-0008.

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AbstractIn this paper, we present an initial value technique for solving self-adjoint singularly perturbed linear boundary value problems. The original problem is reduced to its normal form and the reduced problem is converted to first order initial value problems. This replacement is significant from the computational point of view. The classical fourth order Runge-Kutta method is used to solve these initial value problems. This approach to solve singularly perturbed boundary-value problems is numerically very appealing. To demonstrate the applicability of this method, we have applied it on several linear examples with left-end boundary layer and right-end layer. From the numerical results, the method seems accurate and solutions to problems with extremely thin boundary layers are obtained.

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17

Dudkin, M. E., and O. Yu Dyuzhenkova. "Singularly perturbed rank one linear operators." Matematychni Studii 56, no. 2 (2021): 162–75. http://dx.doi.org/10.30970/ms.56.2.162-175.

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The basic principles of the theory of singularly perturbed self-adjoint operatorsare generalized to the case of closed linear operators with non-symmetric perturbation of rank one.Namely, firstly linear closed operators are considered that coincide with each other on a dense set in a Hilbert space.The theory of singularly perturbed self-adjoint operators arose from the need to consider differential expressions in such terms as the Dirac $\delta$-function.Since it is important to consider expressions given not only by symmetric operators, the generalization (transfer) of the basic principles of the theory of singularly perturbed self-adjoint operators in the case of non-symmetric ones is important problem. The main facts of the theory include the definition of a singularly perturbed linear operator and the resolvent formula in the cases of ${\mathcal H}_{-1}$-class and ${\mathcal H}_{-2}$-class.The paper additionally describes the possibility of the appearance a point of the point spectrum and the construction of a perturbation with a predetermined point.In comparison with self-adjoint perturbations, the description of perturbations by non-symmetric terms is unexpected.Namely, in some cases, when the perturbed by a vectors from ${\mathcal H}_{-2}$ operator can be conveniently described by methods of class ${\mathcal H}_{-1}$, that is impossible in the case of symmetric perturbations of a self-adjoint operator. The perturbation of self-adjoint operators in a non-symmetric manner fully fits into the proposed studies.Such operators, for example, generalize models with nonlocal interactions, perturbations of the harmonic oscillator by the $\delta$-potentials, and can be used to study perturbations generated by a delay or an anticipation.

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18

Bukanay, N. U., A. E. Mirzakulova, and A. T. Assanova. "Asymptotic estimates of the solution for a singularly perturbed Cauchy problem." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 118, no. 2 (2025): 44–51. https://doi.org/10.31489/2025m2/44-51.

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The article focuses on the initial problem for a third-order linear integro-differential equation with a small parameter at the higher derivatives, assuming that the roots of the additional characteristic equation have opposite signs. This paper presents a fundamental set of solutions and initial functions for a singularly perturbed homogeneous differential equation. The solution to the singularly perturbed initial integrodifferential problem employs analytical formulas. A theorem concerning asymptotic estimates of the solution is established.

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19

Karandjulov, Ljudmil Ivanov, and Yana Petrova Stoyanova. "Problem of Cauchy for linear singularly perturbed impulsive systems." Miskolc Mathematical Notes 3, no. 1 (2002): 25. http://dx.doi.org/10.18514/mmn.2002.48.

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20

Kim, Beom-Soo, Young-Joong Kim, and Myo-Taeg Lim. "LQG Control for Nonstandard Singularly Perturbed Discrete-Time Systems." Journal of Dynamic Systems, Measurement, and Control 126, no. 4 (2004): 860–64. http://dx.doi.org/10.1115/1.1850537.

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In this paper we present a control method and a high accuracy solution technique in solving the linear quadratic Gaussian problems for nonstandard singularly perturbed discrete time systems. The methodology that exists in the literature for the solution of the standard singularly perturbed discrete time linear quadratic Gaussian optimal control problem cannot be extended to the corresponding nonstandard counterpart. The solution of the linear quadratic Gaussian optimal control problem is obtained by solving the pure-slow and pure-fast reduced-order continuous-time algebraic Riccati equations and by implementing the pure-slow and pure-fast reduced-order Kalman filters. In order to show the effectiveness of the proposed method, we present the numerical result for a one-link flexible robot arm.

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21

YUZBASI, SUAYIP, EMRAH GOK, and MEHMET SEZER. "A MUNTZ-LEGENDRE APPROACH TO OBTAIN SOLUTIONS OF SINGULAR PERTURBED PROBLEMS." Journal of Science and Arts 20, no. 3 (2020): 537–44. http://dx.doi.org/10.46939/j.sci.arts-20.3-a04.

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Singularly perturbed differential equations are encountered in mathematical modelling of processes in physics and engineering. Aim of this study is to give a collocation approach for solutions of singularly perturbed two-point boundary value problems. The method provides obtaining the approximate solutions in the form of Müntz-Legendre polynomials by using collocation points and matrix relations. Singularly perturbed problem is transformed into a system of linear algebraic equations. By solving this system, the approximate solution is computed. Also, an error estimation is done using the residual function and the approximate solutions are improved by means of the estimated error function. Two numerical examples are given to show the applicability of the method.

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22

YUZBASI, SUAYIP, and NURCAN BAYKUS SAVASANERIL. "HERMITE POLYNOMIAL APPROACH FOR SOLVING SINGULAR PERTURBATED DELAY DIFFERENTIAL EQUATIONS." Journal of Science and Arts 20, no. 4 (2020): 845–54. http://dx.doi.org/10.46939/j.sci.arts-20.4-a06.

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In this study, a collocation approach based on the Hermite polyomials is applied to solve the singularly perturbated delay differential eqautions by boundary conditions. By means of the matix relations of the Hermite polynomials and the derivatives of them, main problem is reduced to a matrix equation. And then, collocation points are placed in equation of the matrix. Hence, the singular perturbed problem is transformed into an algebraic system of linear equations. This system is solved and thus the coefficients of the assumed approximate solution are determined. Numerical applications are made for various values of N.

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23

Mel'nik, T. A. "Linear singularly perturbed problems with pulse influence." Ukrainian Mathematical Journal 51, no. 1 (1999): 146–54. http://dx.doi.org/10.1007/bf02591924.

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24

Yüzbaşı, Şuayip, and Mehmet Sezer. "Exponential Collocation Method for Solutions of Singularly Perturbed Delay Differential Equations." Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/493204.

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This paper deals with the singularly perturbed delay differential equations under boundary conditions. A numerical approximation based on the exponential functions is proposed to solve the singularly perturbed delay differential equations. By aid of the collocation points and the matrix operations, the suggested scheme converts singularly perturbed problem into a matrix equation, and this matrix equation corresponds to a system of linear algebraic equations. Also, an error analysis technique based on the residual function is introduced for the method. Four examples are considered to demonstrate the performance of the proposed scheme, and the results are discussed.

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25

Kathirkamanayagan, M., and G. S. Ladde. "Large scale singularly perturbed boundary value problems." Journal of Applied Mathematics and Simulation 2, no. 3 (1989): 139–67. http://dx.doi.org/10.1155/s1048953389000122.

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In this paper an alternative approach to the method of asymptotic expansions for the study of a singularly perturbed linear system with multiparameters and multiple time scales is developed. The method consists of developing a linear non-singular transformation that transforms an arbitrary n—time scale system into a diagonal form. Furthermore, a dichotomy transformation is employed to decompose the faster subsystems into stable and unstable modes. Fast, slow, stable and unstable modes decomposition processes provide a modern technique to find an approximate solution of the original system in terms of the solution of an auxiliary system. This method yields a constructive and computationally attractive way to investigate the system.

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26

Melesse, Wondwosen Gebeyaw, Awoke Andargie Tiruneh, and Getachew Adamu Derese. "Uniform Hybrid Difference Scheme for Singularly Perturbed Differential-Difference Turning Point Problems Exhibiting Boundary Layers." Abstract and Applied Analysis 2020 (March 16, 2020): 1–14. http://dx.doi.org/10.1155/2020/7045756.

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In this paper, a class of linear second-order singularly perturbed differential-difference turning point problems with mixed shifts exhibiting two exponential boundary layers is considered. For the numerical treatment of these problems, first we employ a second-order Taylor’s series approximation on the terms containing shift parameters and obtain a modified singularly perturbed problem which approximates the original problem. Then a hybrid finite difference scheme on an appropriate piecewise-uniform Shishkin mesh is constructed to discretize the modified problem. Further, we proved that the method is almost second-order ɛ-uniformly convergent in the maximum norm. Numerical experiments are considered to illustrate the theoretical results. In addition, the effect of the shift parameters on the layer behavior of the solution is also examined.

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27

Munyakazi, Justin B., and Olawale O. Kehinde. "A New Parameter-Uniform Discretization of Semilinear Singularly Perturbed Problems." Mathematics 10, no. 13 (2022): 2254. http://dx.doi.org/10.3390/math10132254.

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In this paper, we present a numerical approach to solving singularly perturbed semilinear convection-diffusion problems. The nonlinear part of the problem is linearized via the quasilinearization technique. We then design and implement a fitted operator finite difference method to solve the sequence of linear singularly perturbed problems that emerges from the quasilinearization process. We carry out a rigorous analysis to attest to the convergence of the proposed procedure and notice that the method is first-order uniformly convergent. Some numerical evaluations are implemented on model examples to confirm the proposed theoretical results and to show the efficiency of the method.

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28

Kurina, G. A., and N. V. Nekrasova. "Asymptotic solution of discrete periodic singularly perturbed linear-quadratic problem *." IFAC Proceedings Volumes 37, no. 17 (2004): 169–75. http://dx.doi.org/10.1016/s1474-6670(17)30813-3.

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29

Haiyun, Ding, Ni Mingkang, Lin Wuzhong, and Cao Yang. "Singularly perturbed semi-linear boundary value problem with discontinuous function." Acta Mathematica Scientia 32, no. 2 (2012): 793–99. http://dx.doi.org/10.1016/s0252-9602(12)60059-9.

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30

Doğan, Nurettin, Vedat Suat Ertürk, and Ömer Akın. "Numerical Treatment of Singularly Perturbed Two-Point Boundary Value Problems by Using Differential Transformation Method." Discrete Dynamics in Nature and Society 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/579431.

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Differential transform method is adopted, for the first time, for solving linear singularly perturbed two-point boundary value problems. Four numerical examples are given to demonstrate the effectiveness of the present method. Results show that the numerical scheme is very effective and convenient for solving a large number of linear singularly perturbed two-point boundary value problems with high accuracy.

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31

Rajashekhar, Reddy .Y. "NUMERICAL SOLUTION TO LINEAR SINGULARLY PERTURBED TWO POINT BOUNDARY VALUE PROBLEMS USING B-SPLINE COLLOCATION METHOD." International Journal of Research – Granthaalayah 4, no. 1 (2017): 158–64. https://doi.org/10.5281/zenodo.848519.

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A Recursive form cubic B-spline basis function is used as basis in B-spline collocation method to solve second linear singularly perturbed two point boundary value problem. The performance of the method is tested by considering the numerical examples with different boundary conditions. Results of numerical examples show the robustness of the method when compared with the analytical solution.

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32

Malek, Stephane. "On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities." Abstract and Applied Analysis 2020 (January 10, 2020): 1–32. http://dx.doi.org/10.1155/2020/7985298.

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A family of linear singularly perturbed difference differential equations is examined. These equations stand for an analog of singularly perturbed PDEs with irregular and Fuchsian singularities in the complex domain recently investigated by A. Lastra and the author. A finite set of sectorial holomorphic solutions is constructed by means of an enhanced version of a classical multisummability procedure due to W. Balser. These functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairs q-Gevrey and Gevrey bounds.

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33

Woldaregay, Mesfin, Worku Aniley, and Gemechis Duressa. "Fitted numerical scheme for singularly perturbed convection-diffusion reaction problems involving delays." Theoretical and Applied Mechanics, no. 00 (2021): 6. http://dx.doi.org/10.2298/tam201208006w.

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This paper deals with solution methods for singularly perturbed delay differential equations having delay on the convection and reaction terms. The considered problem exhibits an exponential boundary layer on the left or right side of the domain. The terms with the delay are treated using Taylor?s series approximation and the resulting singularly perturbed boundary value problem is solved using a specially designed exponentially finite difference method. The stability of the scheme is analysed and investigated using a comparison principle and solution bound. The formulated scheme converges uniformly with linear order of convergence. The theoretical findings are validated using three numerical test examples.

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34

O'Malley,, Robert E. "Singularly Perturbed Linear Two-Point Boundary Value Problems." SIAM Review 50, no. 3 (2008): 459–82. http://dx.doi.org/10.1137/060662058.

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35

Hou, Qianqian, Tai-Chia Lin, and Zhi-An Wang. "On a singularly perturbed semi-linear problem with Robin boundary conditions." Discrete & Continuous Dynamical Systems - B 26, no. 1 (2021): 401–14. http://dx.doi.org/10.3934/dcdsb.2020083.

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36

Kostyukova, O. I. "An Investigation of a Linear Hamiltonian Singularly Perturbed Boundary Value Problem." Differential Equations 40, no. 5 (2004): 652–61. http://dx.doi.org/10.1023/b:dieq.0000043523.61257.6c.

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37

Jingde, Du. "Singularly perturbed boundary value problem for linear equations with turning points." Journal of Mathematical Analysis and Applications 155, no. 2 (1991): 322–37. http://dx.doi.org/10.1016/0022-247x(91)90003-i.

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38

Ferguson, Jr., Warren E. "Analysis of a Singularly-Perturbed Linear Two-Point Boundary-Value Problem." SIAM Journal on Numerical Analysis 23, no. 5 (1986): 940–47. http://dx.doi.org/10.1137/0723062.

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39

Yakubu, Gulibur, Isah Abdullahi, Ali Musa, and Abdullahi M. Gamsha. "Attainable Order of Hybrid Methods from Polynomial Nodes for Non-Linear Singularly Perturbed Initial Value Problems." International Journal of Development Mathematics (IJDM) 2, no. 1 (2025): 001–21. https://doi.org/10.62054/ijdm/0201.01.

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We develop hybrid methods utilizing collocation at polynomial nodes for the numerical integration of singularly perturbed non-linear initial-value problems. The present methods are intended for solving nonlinear singularly perturbed initial value problems without linearization and provide third and fourth-order convergence results. We use piecewise-uniform meshes which resolve the difficulties arising from the steep gradient of the solution in the initial layer. Linear stability of these methods are studied. Numerical experiments are carried out to verify the efficiency and accuracy of the methods. The new hybrid integrators are effectively employed to achieve accurate representation of singularly perturbed systems, providing physical interpretation of what they represent in natural phenomena. These are illustrated through phase plots that exhibit unusual and novel behaviors. The resulting surface phase plot curves represent portions of the phase space of a perturbed system, frequently illustrating real world observed phenomena. These are depicted graphically as surface phase plot curves shown in Figures, while numerical values are presented side by side in Tables.

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Kovalenko, Savva, Evgenia Kirillova, Vladimir Chekanov, Aminat Uzdenova, and Mahamet Urtenov. "Analytical Solutions and Computer Modeling of a Boundary Value Problem for a Nonstationary System of Nernst–Planck–Poisson Equations in a Diffusion Layer." Mathematics 12, no. 24 (2024): 4040. https://doi.org/10.3390/math12244040.

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This article proposes various new approximate analytical solutions of the boundary value problem for the non-stationary system of Nernst–Planck–Poisson (NPP) equations in the diffusion layer of an ideally selective ion-exchange membrane at overlimiting current densities. As is known, the diffusion layer in the general case consists of a space charge region and a region of local electroneutrality. The proposed analytical solutions of the boundary value problems for the non-stationary system of Nernst–Planck–Poisson equations are based on the derivation of a new singularly perturbed nonlinear partial differential equation for the potential in the space charge region (SCR). This equation can be reduced to a singularly perturbed inhomogeneous Burgers equation, which, by the Hopf–Cole transformation, is reduced to an inhomogeneous singularly perturbed linear equation of parabolic type. Inside the extended SCR, there is a sufficiently accurate analytical approximation to the solution of the original boundary value problem. The electroneutrality region has a curvilinear boundary with the SCR, and with an unknown boundary condition on it. The article proposes a solution to this problem. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transfer in membrane systems. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transport in membrane systems.

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Bijura, Angelina. "Singularly perturbed Volterra integral equations with weakly singular kernels." International Journal of Mathematics and Mathematical Sciences 30, no. 3 (2002): 129–43. http://dx.doi.org/10.1155/s016117120201325x.

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We consider finding asymptotic solutions of the singularly perturbed linear Volterra integral equations with weakly singular kernels. An interesting aspect of these problems is that the discontinuity of the kernel causes layer solutions to decay algebraically rather than exponentially within the initial (boundary) layer. To analyse this phenomenon, the paper demonstrates the similarity that these solutions have to a special function called the Mittag-Leffler function.

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Besova, Margarita, and Vasiliy Kachalov. "Axiomatic Approach in the Analytic Theory of Singular Perturbations." Axioms 9, no. 1 (2020): 9. http://dx.doi.org/10.3390/axioms9010009.

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Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type vector spaces was developed. When considering nonlinear singularly perturbed problems, an algebraic approach is currently used. This approval is based on the properties of algebra homomorphisms for holomorphic functions with various numbers of variables, as a result of which it is possible to obtain pseudoholomorphic solutions. In this paper, formally singularly perturbed equations are considered in topological algebras, which allows the authors to formulate the main concepts of the singular perturbation analytical theory from the standpoint of maximal generality.

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LUBE, GERT, and BENJAMIN TEWS. "OPTIMAL CONTROL OF SINGULARLY PERTURBED ADVECTION-DIFFUSION-REACTION PROBLEMS." Mathematical Models and Methods in Applied Sciences 20, no. 03 (2010): 375–95. http://dx.doi.org/10.1142/s0218202510004271.

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In this paper, we consider the numerical analysis of quadratic optimal control problems governed by a linear advection-diffusion-reaction equation without control constraints. In the case of dominating advection, the Galerkin discretization is stabilized via the one- or two-level variant of the local projection approach which leads to a symmetric optimality system at the discrete level. The optimal control problem simultaneously covers distributed and Robin boundary control. In the singularly perturbed case, the boundary control at inflow and/or characteristic parts of the boundary can be seen as regularization of a Dirichlet boundary control. Some numerical tests illustrate the analytical results.

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Manjumari, M. "Decision Making Problem with Multiple Attribute Group Employing an Array of Singularly Perturbed Differential Equations." Indian Journal Of Science And Technology 18, no. 14 (2025): 1147–54. https://doi.org/10.17485/ijst/v18i14.2820.

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Objectives: This study deals with an ensemble of linear singularly perturbed differential equations that has been employed to solve Multiple Attribute Group Decision Making issues on that the weights of the Decision Makers (DMs) are unknown. Methods: Two sophisticated operators-the Intuitionistic Fuzzy Generalised Hybrid Weighted Averaging (IFGHWA) operator as well as the Intuitionistic Fuzzy Weighted Averaging (IFWA) operator—are employed to aid in the course of decision-making. Findings: In order to determine the best course of action, these operators are used to combine intuitionistic fuzzy decision matrices into a collective decision matrix. We can use the newly proposed correlation coefficient method and score function for ranking the best alternative from the available alternatives. Novelty: To demonstrate the efficiency and applicability of the suggested method in resolving MAGDM situations with ambiguous decision maker weights, a computational instance is provided. Numerical illustration is given to show the effectiveness of the proposed approach. Keywords: Intuitionistic Fuzzy Sets (IFSs), Singular Perturbation Problem, IFGHWA operator, IFWA operator, MAGDM

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Omaralieva, G. "Sufficient Condition for the Existence of an Additional Zone in Singularly Perturbated Second-Order Boundary Problem." Bulletin of Science and Practice, no. 2 (February 15, 2023): 10–16. http://dx.doi.org/10.33619/2414-2948/87/01.

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Studies the Dirichlet, Neman and Robin boundary value problems for a singularly perturbed linear inhomogeneous second order ordinary differential equation. The considered boundary value problems have three features: the singular presence of a small parameter; the solution of the corresponding unperturbed equation has a k order pole and an additional boundary layer. The singular presence of a small parameter generates the classical boundary layer, and the singular point of the corresponding unperturbed equation generates the second boundary layer. As a result, we get a double boundary layer. A sufficient condition for the existence of an additional boundary layer is found.

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Karandjulov, L. I. "Singularly perturbed linear boundary-value problems with pulse effects and the regular reduced problem." Ukrainian Mathematical Journal 47, no. 4 (1995): 537–42. http://dx.doi.org/10.1007/bf01056039.

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Kalinin, A. I., and L. I. Lavrinovich. "Asymptotic method for solving the problem of transition process optimization in a three-tempo singularly perturbed system." Doklady of the National Academy of Sciences of Belarus 68, no. 3 (2024): 183–87. http://dx.doi.org/10.29235/1561-8323-2024-68-3-183-187.

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The problem of constructing a transition process with minimal energy costs for a linear singularly perturbed system containing three groups of variables with significantly different rates of change is considered. Asymptotic approximations to solving this problem are constructed in the form of an open-loop and feedback controls. The main advantage of the proposed computational procedures is that the original problem is split into three unperturbed optimal control problems of lower dimension.

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Popescu, Luminiţa, and Aníbal Rodriguez-Bernal. "On a singularly perturbed wave equation with dynamic boundary conditions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 2 (2004): 389–413. http://dx.doi.org/10.1017/s0308210500003279.

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In this paper we analyse a singular perturbation problem for linear wave equations with interior and boundary damping. We show how the solutions converge to the formal parabolic limit problem with dynamic boundary conditions. Conditions are given for uniform convergence in the energy space.

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Normuradov, Chori, Nasiba Djurayeva, Fatanah Deraman, Anuar Mat Safar, and Salina Mohd Asi. "One Effective Method for Solving Singularly Perturbed Equations." Malaysian Journal of Science 44, no. 1 (2025): 63–69. https://doi.org/10.22452/mjs.vol44no1.8.

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Numerical methods are widely used to study the solution of singularly perturbed equations. At the same time, their application to the solution of such equations encounters serious difficulties; they are associated with the presence of a small parameter at the highest derivative and the appearance in the solution area of areas with high frequency-amplitude sawtooth jumps. In this case, the requirements for the efficiency and accuracy of numerical methods increase sharply. Although numerous methods have been developed to date, the question of the effectiveness and accuracy of numerical methods remains open. Until now, different methods with uniform and non-uniform steps have been mainly used to solve singularly perturbed equations. As the value of the small parameter decreases, to increase the accuracy, it is necessary to refine the step of the difference grid. This, in turn, leads to a strong increase in the order of the matrix in the linear algebraic system being solved. Along with difference methods, spectral methods can be used to solve problems. In spectral methods, the solution to the equation is sought in the form of finite series in Chebyshev polynomials. The derivatives present in the equation are determined by differentiating the selected final series. When differentiating series, the order of the approximating polynomials is reduced, and this, in turn, affects the accuracy of the method used. In this paper, it is proposed to use the preliminary integration method to solve singularly perturbed equations. The essence of this method is as follows. The highest derivative and the right-hand side of the differential equation are expanded into finite series in Chebyshev polynomials of the first kind. Unlike spectral methods, in the preliminary integration method the highest derivative is expanded into a finite series. Before solving the problem, the series for the highest derivative is preliminarily integrated until an expression for solving the problem is found in the form of a finite series. When integrating series, unknown integration constants appear; they are determined from additional conditions of the problem. Only after this, the series for solving the derivatives of the right side are put into a singularly perturbed equation and a system of linear algebraic equations is obtained for determining the unknown expansion coefficients. It should be noted that when integrating series, the smoothness of the approximating polynomials improves, and this, in turn, increases the accuracy of the proposed method. At the same time, the order of the matrix of the algebraic system being solved does not increase. This ensures, at the same costs required in the spectral method, that the proposed method can solve a singularly perturbed equation even for small values of the small parameter of the problem. The high accuracy and efficiency of the preliminary integration method are demonstrated when solving a specific inhomogeneous singularly perturbed equation. The results of calculations are presented by comparing the approximate solution with the exact solution of the problem and with approximate solutions obtained by the spectral method.

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Yousaf, Muhammad Zain, Hari Mohan Srivastava, Muhammad Abbas, et al. "A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems." Symmetry 15, no. 10 (2023): 1929. http://dx.doi.org/10.3390/sym15101929.

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Singular singularly-perturbed problems (SSPPs) are a powerful mathematical tool for modelling a variety of real phenomena, such as nuclear reactions, heat explosions, mechanics, and hydrodynamics. In this paper, the numerical solutions to fourth-order singular singularly-perturbed boundary and initial value problems are presented using a novel quintic B-spline (QBS) approximation approach. This method uses a quasi-linearization approach to solve SSPNL initial/boundary value problems. And the non-linear problems are transformed into a sequence of linear problems by applying the quasi-linearization approach. The QBS functions produce more accurate results when compared to other existing approaches because of their local support, symmetry, and partition of unity features. This method can be applied to immediately solve the SSPPs without reducing the order in which they are presented. It has been demonstrated that the suggested numerical approach converges uniformly over the whole domain. The proposed approach is implemented on a few problems to validate the scheme. The computational results are compared, and they illustrate that the proposed approach performs better.

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