• Title/Summary/Keyword: perturbed linear differential equations

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A NON-ASYMPTOTIC METHOD FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS

  • File, Gemechis;Reddy, Y.N.
    • Journal of applied mathematics & informatics
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    • v.32 no.1_2
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    • pp.39-53
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    • 2014
  • In this paper, a non-asymptotic method is presented for solving singularly perturbed delay differential equations whose solution exhibits a boundary layer behavior. The second order singularly perturbed delay differential equation is replaced by an asymptotically equivalent first order neutral type delay differential equation. Then, Simpson's integration formula and linear interpolation are employed to get three term recurrence relation which is solved easily by Discrete Invariant Imbedding Algorithm. Some numerical examples are given to validate the computational efficiency of the proposed numerical scheme for various values of the delay and perturbation parameters.

SOLVING SECOND ORDER SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH LAYER BEHAVIOR VIA INITIAL VALUE METHOD

  • GEBEYAW, WONDWOSEN;ANDARGIE, AWOKE;ADAMU, GETACHEW
    • Journal of applied mathematics & informatics
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    • v.36 no.3_4
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    • pp.331-348
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    • 2018
  • In this paper, an initial value method for solving a class of singularly perturbed delay differential equations with layer behavior is proposed. In this approach, first the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay using Taylor series expansion. Then from the modified problem, two explicit Initial Value Problems which are independent of the perturbation parameter, ${\varepsilon}$, are produced: the reduced problem and boundary layer correction problem. Finally, these problems are solved analytically and combined to give an approximate asymptotic solution to the original problem. To demonstrate the efficiency and applicability of the proposed method three linear and one nonlinear test problems are considered. The effect of the delay on the layer behavior of the solution is also examined. It is observed that for very small ${\varepsilon}$ the present method approximates the exact solution very well.

Wavelet-based Analysis for Singularly Perturbed Linear Systems Via Decomposition Method (웨이블릿 및 시스템 분할을 이용한 특이섭동 선형 시스템 해석)

  • Kim, Beom-Soo;Shim, Il-Joo
    • Journal of Institute of Control, Robotics and Systems
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    • v.14 no.12
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    • pp.1270-1277
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    • 2008
  • A Haar wavelet based numerical method for solving singularly perturbed linear time invariant system is presented in this paper. The reduced pure slow and pure fast subsystems are obtained by decoupling the singularly perturbed system and differential matrix equations are converted into algebraic Sylvester matrix equations via Haar wavelet technique. The operational matrix of integration and its inverse matrix are utilized to reduce the computational time to the solution of algebraic matrix equations. Finally a numerical example is given to demonstrate the validity and applicability of the proposed method.

FITTED MESH METHOD FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL TURNING POINT PROBLEMS EXHIBITING TWIN BOUNDARY LAYERS

  • MELESSE, WONDWOSEN GEBEYAW;TIRUNEH, AWOKE ANDARGIE;DERESE, GETACHEW ADAMU
    • Journal of applied mathematics & informatics
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    • v.38 no.1_2
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    • pp.113-132
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    • 2020
  • In this paper, a class of linear second order singularly perturbed delay differential turning point problems containing a small delay (or negative shift) on the reaction term and when the solution of the problem exhibits twin boundary layers are examined. A hybrid finite difference scheme on an appropriate piecewise-uniform Shishkin mesh is constructed to discretize the problem. We proved that the method is almost second order ε-uniformly convergent in the maximum norm. Numerical experiments are considered to illustrate the theoretical results.

A FIFTH ORDER NUMERICAL METHOD FOR SINGULAR PERTURBATION PROBLEMS

  • Chakravarthy, P. Pramod;Phaneendra, K.;Reddy, Y.N.
    • Journal of applied mathematics & informatics
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    • v.26 no.3_4
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    • pp.689-706
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    • 2008
  • In this paper, a fifth order numerical method is presented for solving singularly perturbed two point boundary value problems with a boundary layer at one end point. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two point boundary value problem is obtained from the theory of singular perturbations. It is used in the fifth order compact difference scheme to get a two term recurrence relation and is solved. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory. It is observed that the present method approximates the exact solution very well.

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A ROBUST NUMERICAL TECHNIQUE FOR SOLVING NON-LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH BOUNDARY LAYER

  • Cakir, Firat;Cakir, Musa;Cakir, Hayriye Guckir
    • Communications of the Korean Mathematical Society
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    • v.37 no.3
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    • pp.939-955
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    • 2022
  • In this paper, we study a first-order non-linear singularly perturbed Volterra integro-differential equation (SPVIDE). We discretize the problem by a uniform difference scheme on a Bakhvalov-Shishkin mesh. The scheme is constructed by the method of integral identities with exponential basis functions and integral terms are handled with interpolating quadrature rules with remainder terms. An effective quasi-linearization technique is employed for the algorithm. We establish the error estimates and demonstrate that the scheme on Bakhvalov-Shishkin mesh is O(N-1) uniformly convergent, where N is the mesh parameter. The numerical results on a couple of examples are also provided to confirm the theoretical analysis.