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1
Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts
Sahu Subal Ranjan, Mohapatra Jugal
Journal of Applied Analysis
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2022
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Vol. 28, nr 1
121--134
EN
A time dependent singularly perturbed differential-difference equation is considered. The problem involves time delay and general small space shift terms. Taylor series approximation is used to expand the space shift term. A robust numerical scheme based on the backward Euler scheme for the time and classical upwind scheme for space is proposed. The convergence analysis is carried out. It is observed that the proposed scheme converges almost first order up to a logarithm term and optimal first order in space on the Shishkin and Bakhvalov-Shishkin mesh, respectively. Numerical results confirm the efficiency of the proposed scheme, which are in agreement with the theoretical bounds.
2
Content available An extended finite difference method for singular perturbation problems on a non-uniform mesh
Swarnakar D., Kumar V. Ganesh, Soujanya G. B. S. L.
International Journal of Applied Mechanics and Engineering
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2022
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Vol. 27, no. 1
203--214
EN
An extended second order finite difference method on a variable mesh is proposed for the solution of a singularly perturbed boundary value problem. A discrete equation is achieved on the non uniform mesh by extending the first and second order derivatives to the higher order finite differences. This equation is solved efficiently using a tridiagonal solver. The proposed method is analysed for convergence, and second order convergence is derived. Model examples are solved by the proposed scheme and compared with available methods in the literature to uphold the method.
3
Content available Numerical solution of singularly perturbed two parameter problems using exponential splines
Padmaja P., Aparna P., Gorla Rama Subba Reddy, Pothanna N.
International Journal of Applied Mechanics and Engineering
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2021
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Vol. 26, no. 2
160--172
EN
In this paper, we have studied a method based on exponential splines for numerical solution of singularly perturbed two parameter boundary value problems. The boundary value problem is solved on a Shishkin mesh by using exponential splines. Numerical results are tabulated for different values of the perturbation parameters. From the numerical results, it is found that the method approximates the exact solution very well.
4
Content available remote Exponential spline method for singularly perturbed third-order boundary value problems
Wakjira Yohannis Alemayehu, Duressa Gemechis File
Demonstratio Mathematica
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2020
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Vol. 53, nr 1
360--372
EN
The exponential spline function is presented to find the numerical solution of third-order singularly perturbed boundary value problems. Convergence analysis of the method is briefly discussed, and it is shown to be sixth order convergence. To validate the applicability of the method, some model problems are solved for different values of the perturbation parameter, and the numerical results are presented both in tables and graphs. Furthermore, the present method gives more accurate solution than some methods existing in the literature.
5
Content available Dynamic Contraction Method approach to digital longitudinal aircraft flight controller design
Czyba Roman, Stajer Lukasz
Archives of Control Sciences
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2019
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Vol. 29, No. 1
97--109
EN
This paper presents the design of digital controller for longitudinal aircraft model basedon the Dynamic Contraction Method. The control task is formulated as a tracking problem of velocity and flight path angle, where decoupled output transients are accomplished in spite of incomplete information about varying parameters of the system and external disturbances. The design of digital controller based on the pseudo-continuous approach is presented, where the digital controller is the result of continuous-time controller discretization. A resulting output feedback controller has a simple form of a combination of low-order linear dynamical systems and a matrix whose entries depend nonlinearly on certain known process variables. Simulation results for an aircraft model confirm theoretical expectations.
6
Content available remote The structural analysis of singularly perturbed state models of electronic circuits using formula templates
Rogoza V.
Przegląd Elektrotechniczny
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2015
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R. 91, nr 11
170-177
EN
The paper deals with the origin of small parameters in the singularly perturbed state models of electronic circuits. As is shown in the paper, there are at least two origins of small parameters - small values of circuit components and certain relationships between the values of circuit components which need not be small in magnitude. The approach is proposed to building and structural analysis of singularly perturbed mathematical models based on the formula templates, as they are called.
PL
W artykule jest przeanalizowane pochodzenie małych parametrów w syngularnie zaburzonych równaniach różniczkowych opisujących stany układu elektronicznego. Udowodniono, że pochodzenie omówionych parametrów może być dwóch rodzajów - albo wskutek obecności składników układu o małych wartościach, albo wskutek pewnych relacji liczbowych między wartościami tych składników. Proponowane jest podejście do budowy i analizy strukturalnej modeli matematycznych z zaburzeniami syngularnymi na podstawie budowy tzw. szablonów wzorów. Analiza pochodzenia małych parametrów w syngularnie zaburzonych równaniach różniczkowych opisujących stany układu elektronicznego
7
Content available Parametric Borel summability for some semilinear system of partial differential equations
Yamazawa H., Yoshino M.
Opuscula Mathematica
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2015
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Vol. 35, no. 5
825--845
EN
In this paper we study the Borel summability of formal solutions with a parameter of first order semilinear system of partial differential equations with n independent variables. In [Singular perturbation of linear systems with a regular singularity, J. Dynam. Control. Syst. 8 (2002), 313-322], Balser and Kostov proved the Borel summability of formal solutions with respect to a singular perturbation parameter for a linear equation with one independent variable. We shall extend their results to a semilinear system of equations with general independent variables.
8
Stiff and singular perturbation equations
Dikusar V. V., Wójtowicz M., Zemcova N.
Logistyka
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2014
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nr 6
3093--3101
EN
We introduce control variable in explicit difference scheme. For verification of result we consider local and global errors. Introduction of control variable opens the opportunity to solve the stiff and singular perturbation equations in the frame of explicit schemes. Thus, we extend the limitations of applicability of explicit difference schemes.
PL
Wprowadzamy zmienną sterującą do jawnego schematu różnicowego. W celu weryfikacji wyniku oceniamy błędy lokalne i globalne. Wprowadzenie zmiennej sterującej otwiera możliwość rozwiązywania równań ze sztywnymi i singularnymi perturbacjami z zastosowaniem jawnych schematów. A zatem, poszerzamy zakres stosowania jawnych schematów różnicowych.
9
Content available remote Shape sensitivity analysis of eigenvalues revisited
Nazarov S. A., Sokolowski J.
Control and Cybernetics
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2008
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Vol. 37, no 4
999-1012
EN
The paper can be considered as a complement to previous papers of the authors. An insight into applied asymptotic analysis of boundary value problems in singularly perturbed domains is presented. As a result, the asymptotic expansions of eigenvalues are obtained and discussed in terms of integral attributes of the geometrical perturbations including the virtual mass tensor, polarization tensor etc. The results are presented in such a way that can be easily employed in numerical methods for shape optimization and inverse problems.
10
Content available remote Correctness of a constrained control Mayer's problem for a class of singularly perturbed functional-differential systems
Glizer V. Y.
Control and Cybernetics
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2008
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Vol. 37, no 2
329-351
EN
A Mayer's problem for a singularly perturbed controlled system with the general type of a small state delay is considered. The control is subject to geometrical constraints. The cost functional is a function of the terminal value of the slow state variable. A simpler parameter-free optimal control problem (the reduced problem) is associated with the original problem. A convergence of the optimal value of the cost functional in the original problem to the optimal value of the cost functional in the reduced problem, as a parameter of singular perturbation tends to zero, is established. An asymptotic suboptimality of the optimal control of the reduced problem in the original problem is shown. These results are extended to some more general optimal control problems. An illustrative example is presented.
11
Content available remote Quasi-homoclinic solutions to a system of ODEs
Kaźmierczak B.
Archives of Mechanics
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1999
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Vol. 51, nr 1
105-120
EN
This paper considers the problem of existence of "quasi-homoclinic" solution to a system of three first order ODEs containing a small parameter. These equations describe travelling wave solutions to a one-temperature model of laser-sustained plasma with absorption. This solution is homoclinic with respect to the first two variables. We use the methods of geometric singular perturbation theory to prove the existence of strictly homoclinic trajectory and then prove that it implies the existence of the desired solution.
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