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Journal articles on the topic 'Singularly Perturbed Differential Equation'

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Published: 6 September 2023

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1

Kanth, A. S. V. Ravi, and P. Murali Mohan Kumar. "A Numerical Technique for Solving Nonlinear Singularly Perturbed Delay Differential Equations." Mathematical Modelling and Analysis 23, no. 1 (February 12, 2018): 64–78. http://dx.doi.org/10.3846/mma.2018.005.

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This paper presents a numerical technique for solving nonlinear singu- larly perturbed delay differential equations. Quasilinearization technique is applied to convert the nonlinear singularly perturbed delay differential equation into a se- quence of linear singularly perturbed delay differential equations. An exponentially fitted spline method is presented for solving sequence of linear singularly perturbed delay differential equations. Error estimates of the method is discussed. Numerical examples are solved to show the applicability and efficiency of the proposed scheme.

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2

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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3

Battelli, Flaviano, and Michal Fečkan. "Periodic Solutions in Slowly Varying Discontinuous Differential Equations: The Generic Case." Mathematics 9, no. 19 (October 2, 2021): 2449. http://dx.doi.org/10.3390/math9192449.

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We study persistence of periodic solutions of perturbed slowly varying discontinuous differential equations assuming that the unperturbed (frozen) equation has a non singular periodic solution. The results of this paper are motivated by a result of Holmes and Wiggins where the authors considered a two dimensional Hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation.

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4

YUZBASI, SUAYIP, and NURCAN BAYKUS SAVASANERIL. "HERMITE POLYNOMIAL APPROACH FOR SOLVING SINGULAR PERTURBATED DELAY DIFFERENTIAL EQUATIONS." Journal of Science and Arts 20, no. 4 (December 30, 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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5

Et. al., M. Adilaxmi ,. "Solution Of Singularly Perturbed Delay Differential Equations Using Liouville Green Transformation." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 4 (April 11, 2021): 325–35. http://dx.doi.org/10.17762/turcomat.v12i4.510.

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This paper envisages the use of Liouville Green Transformation to find the solution of singularly perturbed delay differential equations. First, using Taylor series, the given singularly perturbed delay differential equation is approximated by an asymptotically equivalent singularly perturbation problem. Then the Liouville Green Transformation is applied to get the solution. The method is demonstrated by implementing several model examples by taking various values for the delay parameter and perturbation parameter.

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6

Duressa, Gemechis File, Imiru Takele Daba, and Chernet Tuge Deressa. "A Systematic Review on the Solution Methodology of Singularly Perturbed Differential Difference Equations." Mathematics 11, no. 5 (February 22, 2023): 1108. http://dx.doi.org/10.3390/math11051108.

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This review paper contains computational methods or solution methodologies for singularly perturbed differential difference equations with negative and/or positive shifts in a spatial variable. This survey limits its coverage to singular perturbation equations arising in the modeling of neuronal activity and the methods developed by numerous researchers between 2012 and 2022. The review covered singularly perturbed ordinary delay differential equations with small or large negative shift(s), singularly perturbed ordinary differential–differential equations with mixed shift(s), singularly perturbed delay partial differential equations with small or large negative shift(s) and singularly perturbed partial differential–difference equations of the mixed type. The main aim of this review is to find out what numerical and asymptotic methods were developed in the last ten years to solve such problems. Further, it aims to stimulate researchers to develop new robust methods for solving families of the problems under consideration.

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7

Bobodzhanov, A., B. Kalimbetov, and N. Pardaeva. "Construction of a regularized asymptotic solution of an integro-differential equation with a rapidly oscillating cosine." Journal of Mathematics and Computer Science 32, no. 01 (July 21, 2023): 74–85. http://dx.doi.org/10.22436/jmcs.032.01.07.

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In this paper, we consider a singularly perturbed integro-differential equation with a rapidly oscillating right-hand side, which includes an integral operator with a slowly varying kernel. Earlier, singularly perturbed differential and integro-differential equations with rapidly oscillating coefficients were considered. The main goal of this work is to generalize the Lomov's regularization method and to identify the rapidly oscillating right-hand side to the asymptotics of the solution to the original problem.

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8

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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9

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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10

Vrábeľ, Róbert. "Asymptotic behavior of $T$-periodic solutions of singularly perturbed second-order differential equation." Mathematica Bohemica 121, no. 1 (1996): 73–76. http://dx.doi.org/10.21136/mb.1996.125946.

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11

Artstein, Zvi. "On singularly perturbed ordinary differential equations with measure-valued limits." Mathematica Bohemica 127, no. 2 (2002): 139–52. http://dx.doi.org/10.21136/mb.2002.134168.

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12

Cengizci, Süleyman. "An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations." International Journal of Differential Equations 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/7269450.

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In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

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13

Ravi Kanth, A. S. V., and P. Murali Mohan Kumar. "Numerical Method for a Class of Nonlinear Singularly Perturbed Delay Differential Equations Using Parametric Cubic Spline." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 3-4 (June 26, 2018): 357–65. http://dx.doi.org/10.1515/ijnsns-2017-0126.

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AbstractIn this paper, we study the numerical method for a class of nonlinear singularly perturbed delay differential equations using parametric cubic spline. Quasilinearization process is applied to convert the nonlinear singularly perturbed delay differential equations into a sequence of linear singularly perturbed delay differential equations. When the delay is not sufficiently smaller order of the singular perturbation parameter, the approach of expanding the delay term in Taylor’s series may lead to bad approximation. To handle the delay term, we construct a special type of mesh in such a way that the term containing delay lies on nodal points after discretization. The parametric cubic spline is presented for solving sequence of linear singularly perturbed delay differential equations. The error analysis of the method is presented and shows second-order convergence. The effect of delay parameter on the boundary layer behavior of the solution is discussed with two test examples.

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14

Artstein, Zvi, and Alexander Vigodner. "Singularly perturbed ordinary differential equations with dynamic limits." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 3 (1996): 541–69. http://dx.doi.org/10.1017/s0308210500022903.

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Coupled slow and fast motions generated by ordinary differential equations are examined. The qualitative limit behaviour of the trajectories as the small parameter tends to zero is sought after. Invariant measures of the parametrised fast flow are employed to describe the limit behaviour, rather than algebraic equations which are used in the standard reduced order approach. In the case of a unique invariant measure for each parameter, the limit of the slow motion is governed by a chattering type equation. Without the uniqueness, the limit of the slow motion solves a differential inclusion. The fast flow, in turn, converges in a statistical sense to the direct integral, respectively the set-valued direct integral, of the invariant measures.

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15

Fečkan, Michal. "Singularly perturbed ordinary differential equations." Journal of Mathematical Analysis and Applications 170, no. 1 (October 1992): 214–24. http://dx.doi.org/10.1016/0022-247x(92)90015-6.

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16

Chatterjee, Sabyasachi, Amit Acharya, and Zvi Artstein. "Computing singularly perturbed differential equations." Journal of Computational Physics 354 (February 2018): 417–46. http://dx.doi.org/10.1016/j.jcp.2017.10.025.

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17

Samoilenko, V. H., Yu I. Samoilenko, and V. S. Vovk. "Asymptotic analysis of the singularly perturbed Korteweg-de Vries equation." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 1 (2019): 194–97. http://dx.doi.org/10.17721/1812-5409.2019/1.45.

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The paper deals with the singularly perturbed Korteweg-de Vries equation with variable coefficients. An algorithm for constructing asymptotic one-phase soliton-like solutions of this equation is described. The algorithm is based on the nonlinear WKB technique. The constructed asymptotic soliton-like solutions contain a regular and singular part. The regular part of this solution is the background function and consists of terms, which are defined as solutions to the system of the first order partial differential equations. The singular part of the asymptotic solution characterizes the soliton properties of the asymptotic solution. These terms are defined as solutions to the system of the third order partial differential equations. Solutions of these equations are obtained in a special way. Firstly, solutions of these equations are considered on the so-called discontinuity curve, and then these solutions are prolongated into a neighborhood of this curve. The influence of the form of the coefficients of the considered equation on the form of the equation for the discontinuity curve is analyzed. It is noted that for a wide class of such coefficients the equation for the discontinuity curve has solution that is determined for all values of the time variable. In these cases, the constructed asymptotic solutions are determined for all values of the independent variables. Thus, in the case of a zero background, the asymptotic solutions are certain deformations of classical soliton solutions.

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18

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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19

Muratova, A. K. "Asymptotic behavior of the solution of the boundary value problem for a singularly perturbed system of the integro-differential equations." Bulletin of the National Engineering Academy of the Republic of Kazakhstan 88, no. 2 (June 25, 2023): 126–34. http://dx.doi.org/10.47533/2023.1606-146x.13.

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In this paper, we study the asymptotic behavior of solutions to the boundary value problem for singularly perturbed systems of integro-differential equations. The aim of the work is to obtain an analytical formula, an asymptotic estimate of the solution of a boundary value problem, and to determine the asymptotic behavior of the solution by a smaller parameter at the starting point. The boundary value problem given in the paper is reduced to a boundary value problem posed in a singularly perturbed integral-differential equation of mixed type with respect to a fast variable. The Cauchy function, boundary functions and Green’s function of a singularly perturbed homogeneous differential equation are obtained, and their asymptotic estimates are also determined. With the help of these constructed functions, an analytical formula and an asymptotic estimate of this solution of the boundary value problem are obtained. The asymptotic behavior of the solution with respect to a small parameter is determined and the order of growth of its derivatives at the left point of a given segment is shown. It is established that the solution of the boundary value problem under consideration has an initial jump of zero order at the initial point.

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20

Adhikari, Mohit H., Evangelos A. Coutsias, and John K. McIver. "Periodic solutions of a singularly perturbed delay differential equation." Physica D: Nonlinear Phenomena 237, no. 24 (December 2008): 3307–21. http://dx.doi.org/10.1016/j.physd.2008.07.019.

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21

Bijura, A. M. "Singularly Perturbed Volterra Integro-differential Equations." Quaestiones Mathematicae 25, no. 2 (June 2002): 229–48. http://dx.doi.org/10.2989/16073600209486011.

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22

Vaid, Mandeep Kaur, and Geeta Arora. "Quintic B-Spline Technique for Numerical Treatment of Third Order Singular Perturbed Delay Differential Equation." International Journal of Mathematical, Engineering and Management Sciences 4, no. 6 (December 1, 2019): 1471–82. http://dx.doi.org/10.33889/ijmems.2019.4.6-116.

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In this paper, a class of third order singularly perturbed delay differential equation with large delay is considered for numerical treatment. The considered equation has discontinuous convection-diffusion coefficient and source term. A quintic trigonometric B-spline collocation technique is used for numerical simulation of the considered singularly perturbed delay differential equation by dividing the domain into the uniform mesh. Further, uniform convergence of the solution is discussed by using the concept of Hall error estimation and the method is found to be of first-order convergent. The existence of the solution is also established. Computation work is carried out to validate the theoretical results.

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23

Akmatov, A. "Solutions Asymptotics of a Homogeneous Bisingularly Perturbed Differential Equation in the Generalized Functions Theory." Bulletin of Science and Practice 8, no. 2 (February 15, 2022): 18–25. http://dx.doi.org/10.33619/2414-2948/75/02.

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In the space of generalized functions, a homogeneous system of singularly perturbed differential equations in the case of stability change is considered. A theorem on generalized solutions of the corresponding degenerate system of the equation is proved. At special points, the asymptotic closeness of the solutions of the perturbed and unperturbed problems in the singular domain is established. The novelty of the work lies in the fact that, for the first time, an estimate for the singular region was obtained. A degenerate system has a special point. At this point, we solve the equation in generalized functions. In turn, this is also a novelty, because previously performed works only considered the classical solution. The following novelty of the work lies in the fact that we take the starting point in an unstable interval and also head towards the unstable interval. This property is not characteristic of previously published works.

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24

Samusenko, P. F., and M. B. Vira. "Asymptotic solutions of boundary value problem for singularly perturbed system of differential-algebraic equations." Carpathian Mathematical Publications 14, no. 1 (April 25, 2022): 49–60. http://dx.doi.org/10.15330/cmp.14.1.49-60.

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This paper deals with the boundary value problem for a singularly perturbed system of differential algebraic equations of the second order. The case of simple roots of the characteristic equation is studied. The sufficient conditions for existence and uniqueness of a solution of the boundary value problem for system of differential algebraic equations are found. Technique of constructing the asymptotic solutions is developed.

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25

Dmitriev, M. G., A. A. Pavlov, and A. P. Petrov. "Nonstationary Fronts in the Singularly Perturbed Power-Society Model." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/172654.

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The theory of contrasting structures in singularly perturbed boundary problems for nonlinear parabolic partial differential equations is applied to the research of formation of steady state distributions of power within the nonlinear “power-society” model. The interpretations of the solutions to the equation are presented in terms of applied model. The possibility theorem for the problem of getting the solution having some preassigned properties by means of parametric control is proved.

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26

WOLDAREGAY, MESFIN MEKURIA, and GEMECHIS FILE DURESSA. "UNIFORMLY CONVERGENT NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC DIFFERENTIAL EQUATIONS ARISING IN COMPUTATIONAL NEUROSCIENCE." Kragujevac Journal of Mathematics 46, no. 1 (February 2022): 65–54. http://dx.doi.org/10.46793/kgjmat2201.065w.

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The motive of this work is to develop ε-uniform numerical method for solving singularly perturbed parabolic delay differential equation with small delay. To approximate the term with the delay, Taylor series expansion is used. The resulting singularly perturbed parabolic differential equation is solved by using non-standard finite difference method in spatial direction and implicit Runge-Kutta method for the resulting system of IVPs in temporal direction. Theoretically the developed method is shown to be accurate of order O(N −1 + (∆t) 2 ) by preserving ε-uniform convergence. Two numerical examples are considered to investigate εuniform convergence of the proposed scheme and the result obtained agreed with the theoretical one

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27

Daba, Imiru Takele, and Gemechis File Duressa. "An Efficient Computational Method for Singularly Perturbed Delay Parabolic Partial Differential Equations." International Journal of Mathematical Models and Methods in Applied Sciences 15 (July 21, 2021): 105–17. http://dx.doi.org/10.46300/9101.2021.15.14.

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In this communication, a parameter uniform numerical scheme is proposed to solve singularly perturbed delay parabolic convection-diffusion equations. Taylor’s series expansion is applied to approximate the shift term. Then the resulting singularly perturbed parabolic convection-diffusion equation is solved by utilizing the implicit Euler method for temporal discretization on uniform mesh and hybrid numerical scheme based on a midpoint upwind scheme in the coarse mesh regions and a cubic spline method in the fine mesh regions on a piecewise uniform Shishkin mesh for the spatial discretization. The proposed numerical scheme is shown to be an ε−uniformly convergent accuracy of first-order in time and almost second-order in space directions. Some test examples are considered to testify the theoretical predictions.

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Bouatta, Mohamed A., Sergey A. Vasilyev, and Sergey I. Vinitsky. "The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation." Discrete and Continuous Models and Applied Computational Science 29, no. 2 (December 15, 2021): 126–45. http://dx.doi.org/10.22363/2658-4670-2021-29-2-126-145.

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The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.

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Shishkin, Grigorii. "Approximation of Singularly Perturbed Parabolic Reaction-Diffusion Equations with Nonsmooth Data." Computational Methods in Applied Mathematics 1, no. 3 (2001): 298–315. http://dx.doi.org/10.2478/cmam-2001-0020.

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AbstractIn this paper we consider the Dirichlet problem on a rectangle for singularly perturbed parabolic equations of reaction-diffusion type. The reduced (for ε = 0) equation is an ordinary differential equation with respect to the time variable; the singular perturbation parameter ε may take arbitrary values from the half-interval (0,1]. Assume that sufficiently weak conditions are imposed upon the coefficients and the right-hand side of the equation, and also the boundary function. More precisely, the data satisfy the Hölder continuity condition with a small exponent α and α/2 with respect to the space and time variables. To solve the problem, we use the known ε-uniform numerical method which was developed previously for problems with sufficiently smooth and compatible data. It is shown that the numerical solution converges ε-uniformly. We discuss also the behavior of local accuracy of the scheme in the case where the data of the boundary-value problem are smoother on a part of the domain of definition.

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Akmatov, A. "Investigation of Solutions to a System of Singularly Perturbed Differential Equations." Bulletin of Science and Practice 8, no. 5 (May 15, 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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31

Cai, X., and F. Liu. "A Reynolds uniform scheme for singularly perturbed parabolic differential equation." ANZIAM Journal 47 (April 9, 2007): 633. http://dx.doi.org/10.21914/anziamj.v47i0.1067.

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Mallet-Paret, John, and Roger D. Nussbaum. "Multiple transition layers in a singularly perturbed differential-delay equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 123, no. 6 (1993): 1119–34. http://dx.doi.org/10.1017/s0308210500029772.

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SynopsisThe singularly perturbed differential-delay equationis studied for a class of step-function nonlinearities f. We show that in general the discrete systemdoes not mirror the dynamics of (*), even for small ε, but that rather a different systemdoes. Here F is related to, but different from, f, and describes the evolution of transition layers. In this context, we also study the effects of smoothing out the discontinuities of f.

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Abdulla, Murad Ibrahim, Gemechis File Duressa, and Habtamu Garoma Debela. "Robust numerical method for singularly perturbed differential equations with large delay." Demonstratio Mathematica 54, no. 1 (January 1, 2021): 576–89. http://dx.doi.org/10.1515/dema-2021-0020.

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Abstract In this paper, a singularly perturbed differential equation with a large delay is considered. The considered problem contains a large delay parameter on the reaction term. The solution of the problem exhibits the interior layer due to the delay parameter and the strong right boundary layer due to the small perturbation parameter ε. The resulting singularly perturbed problem is solved using the fitted non-polynomial spline method. The stability and parameter uniform convergence of the proposed method is proved. To validate the applicability of the scheme, two model problems of the variable coefficient are considered for numerical experimentation.

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Woldaregay, Mesﬁn Mekuria, and Gemechis File Duressa. "Uniformly convergent numerical scheme for singularly perturbed parabolic delay differential equations." ITM Web of Conferences 34 (2020): 02011. http://dx.doi.org/10.1051/itmconf/20203402011.

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This paper deals with numerical treatment of singularly perturbed parabolic diﬀerential diﬀerence equations having small shifts on the spatial variable. The considered problem contain small perturbation parameter (ε) multiplied on the diﬀusion term of the equation. For small values of ε the solution of the problem exhibits a boundary layer on the left or right side of the spatial domain depending on the sign of the convective term. The terms involving the shifts are approximated using Taylor’s series approximation. The resulting singularly perturbed parabolic partial diﬀerential equation is solved using implicit Euler method in the temporal discretization with exponentially ﬁtted operator ﬁnite diﬀerence method in the spatial discretization. The uniform stability of the scheme investigated using comparison principle and discrete solution bound by constructing barrier function. Uniform convergence analysis has been carried out. The scheme gives second order convergence for the case ε > N−1 and first order convergence for the case ε « N−1, where N is number of mesh interval. Test examples and numerical results are considered for validating the theoretical analysis of the scheme.

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35

Pasekov, V. P. "To the analysis of weak two-locus viability selection and quasi linkage equilibrium." Доклады Академии наук 484, no. 6 (May 23, 2019): 781–85. http://dx.doi.org/10.31857/s0869-56524846781-785.

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A model of weak viability selection at two multi-allele loci with standardization of approaches through the use of perturbation theory is examined. The estimate of the quasi-equilibrium value for the linkage disequilibrium coefficient D is analyzed, and results in terms of average effects in quantitative genetics and in terms of the theory of singular perturbations in mathematics are obtained. The approximation of a discrete-time model of a random mating population with non-overlapping generations under weak selection by ordinary differential equations is considered. Weak selection is considered as a perturbation of the model without selection. The resulting model is singularly perturbed; that is, fast (D) and slow (allele frequencies) variables can be distinguished. The first approximation equation for quasi-equilibrium of D is obtained using the first terms of the Taylor series expansion of the model functions. It coincides with the corresponding part of the system of the first approximation of the asymptotic series for solving singularly perturbed equations.

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Chen, Xiangyi, and Asok Ray. "On Singular Perturbation of Neutron Point Kinetics in the Dynamic Model of a PWR Nuclear Power Plant." Sci 2, no. 2 (April 26, 2020): 30. http://dx.doi.org/10.3390/sci2020030.

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This short communication makes use of the principle of singular perturbation to approximate the ordinary differential equation (ODE) of prompt neutron (in the point kinetics model) as an algebraic equation. This approximation is shown to yield a large gain in computational efficiency without compromising any significant accuracy in the numerical simulation of primary coolant system dynamics in a PWR nuclear power plant. The approximate (i.e., singularly perturbed) model has been validated with a numerical solution of the original set of neutron point-kinetic and thermal–hydraulic equations. Both models use variable-step Runge–Kutta numerical integration.

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37

Chen, Xiangyi, and Asok Ray. "On Singular Perturbation of Neutron Point Kinetics in the Dynamic Model of a PWR Nuclear Power Plant." Sci 2, no. 2 (May 27, 2020): 36. http://dx.doi.org/10.3390/sci2020036.

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Abstract:

This short communication makes use of the principle of singular perturbation to approximate the ordinary differential equation (ODE) of prompt neutron (in the point kinetics model) as an algebraic equation. This approximation is shown to yield a large gain in computational efficiency without compromising any significant accuracy in the numerical simulation of primary coolant system dynamics in a PWR nuclear power plant. The approximate (i.e., singularly perturbed) model has been validated with a numerical solution of the original set of neutron point-kinetic and thermal–hydraulic equations. Both models use variable-step Runge–Kutta numerical integration.

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38

O'Riordan, E. "Numerical Methods for Singularly Perturbed Differential Equations." Irish Mathematical Society Bulletin 0016 (1986): 14–24. http://dx.doi.org/10.33232/bims.0016.14.24.

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39

Nhan, T. A. "Preconditioning techniques for singularly perturbed differential equations." Irish Mathematical Society Bulletin 0076 (2015): 35–36. http://dx.doi.org/10.33232/bims.0076.35.36.

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Artstein, Zvi, Ioannis G. Kevrekidis, Marshall Slemrod, and Edriss S. Titi. "Slow observables of singularly perturbed differential equations." Nonlinearity 20, no. 11 (September 28, 2007): 2463–81. http://dx.doi.org/10.1088/0951-7715/20/11/001.

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Artstein, Zvi. "Asymptotic stability of singularly perturbed differential equations." Journal of Differential Equations 262, no. 3 (February 2017): 1603–16. http://dx.doi.org/10.1016/j.jde.2016.10.023.

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Artstein, Zvi, and Marshall Slemrod. "On Singularly Perturbed Retarded Functional Differential Equations." Journal of Differential Equations 171, no. 1 (March 2001): 88–109. http://dx.doi.org/10.1006/jdeq.2000.3840.

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43

Koliha, J. J., and Trung Dinh Tran. "Semistable Operators and Singularly Perturbed Differential Equations." Journal of Mathematical Analysis and Applications 231, no. 2 (March 1999): 446–58. http://dx.doi.org/10.1006/jmaa.1998.6235.

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Slavova, Angela. "Nonlinear singularly perturbed systems of differential equations: A survey." Mathematical Problems in Engineering 1, no. 4 (1995): 275–301. http://dx.doi.org/10.1155/s1024123x95000172.

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In this paper a survey of the most effective methods in singular perturbations is presented. Many applied problems can be modeled by nonlinear singularly perturbed systems, as, for example, problems in kinetics, biochemistry, semiconductors theory, theory of electrical chains, economics, and so on. In this survey we consider averaging and constructive methods that are very useful from the point of view of their numerical and computer realizations.

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45

Govindarao, Lolugu, and Jugal Mohapatra. "A second order numerical method for singularly perturbed delay parabolic partial differential equation." Engineering Computations 36, no. 2 (March 11, 2019): 420–44. http://dx.doi.org/10.1108/ec-08-2018-0337.

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Purpose The purpose of this paper is to provide a robust second-order numerical scheme for singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Design/methodology/approach For the parabolic convection-diffusion initial boundary value problem, the authors solve the problem numerically by discretizing the domain in the spatial direction using the Shishkin-type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh) and in temporal direction using the uniform mesh. The time derivative is discretized by the implicit-trapezoidal scheme, and the spatial derivatives are discretized by the hybrid scheme, which is a combination of the midpoint upwind scheme and central difference scheme. Findings The authors find a parameter-uniform convergent scheme which is of second-order accurate globally with respect to space and time for the singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Also, the Thomas algorithm is used which takes much less computational time. Originality/value A singularly perturbed delay parabolic convection–diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a hybrid scheme. The method is parameter-uniform convergent and is of second order accurate globally with respect to space and time. Numerical results are carried out to verify the theoretical estimates.

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Zavizion, G. V. "Singularly perturbed system of differential equations with a rational singularity." Differential Equations 43, no. 7 (July 2007): 885–97. http://dx.doi.org/10.1134/s0012266107070014.

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Malek, Stéphane. "On Singularly Perturbed Partial Integro-Differential Equations with Irregular Singularity." Journal of Dynamical and Control Systems 13, no. 3 (July 20, 2007): 419–49. http://dx.doi.org/10.1007/s10883-007-9018-4.

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Malek, S. "On singularly perturbed q-difference-differential equations with irregular singularity." Journal of Dynamical and Control Systems 17, no. 2 (April 2011): 243–71. http://dx.doi.org/10.1007/s10883-011-9118-z.

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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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Min, Chao, and Liwei Wang. "Orthogonal Polynomials with Singularly Perturbed Freud Weights." Entropy 25, no. 5 (May 22, 2023): 829. http://dx.doi.org/10.3390/e25050829.

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In this paper, we are concerned with polynomials that are orthogonal with respect to the singularly perturbed Freud weight functions. By using Chen and Ismail’s ladder operator approach, we derive the difference equations and differential-difference equations satisfied by the recurrence coefficients. We also obtain the differential-difference equations and the second-order differential equations for the orthogonal polynomials, with the coefficients all expressed in terms of the recurrence coefficients.

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