Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations. VI. Small Shifts with Rapid Oscillations

Lange, Charles G.; Miura, Robert M.

doi:10.1137/s0036139992228119

Cited by 129 publications

(66 citation statements)

References 13 publications

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“…The singular perturbation analysis of boundary-value problem for di erential-di erence equations with small shifts has been given in a series of papers by Lange and Miura [8][9][10][11][12]. In the papers [8] and [12], an asymptotic approach to approximate the solution of BVPs for the two classes of singularly perturbed problems for DDEs with small shifts has been given. Also in Reference [12], they have shown the e ect of small shifts on the oscillatory behaviour of the solution and pointed out that shifts a ect oscillatory solutions more than boundary layer solutions.…”

Section: Introductionmentioning

confidence: 99%

“…In the papers [8] and [12], an asymptotic approach to approximate the solution of BVPs for the two classes of singularly perturbed problems for DDEs with small shifts has been given. Also in Reference [12], they have shown the e ect of small shifts on the oscillatory behaviour of the solution and pointed out that shifts a ect oscillatory solutions more than boundary layer solutions.…”

Section: Introductionmentioning

confidence: 99%

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Numerical analysis of boundary‐value problems for singularly perturbed differential‐difference equations: small shifts of mixed type with rapid oscillations

Kadalbajoo

Sharma

2004

Commun. Numer. Meth. Engng.

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SUMMARYWe study the boundary-value problems for singularly perturbed di erential-di erence equations with small shifts. Similar boundary-value problems are associated with expected ÿrst-exit time problems of the membrane potential in models for activity of neurons (SIAM J. Appl. Math. 1994; 54:249-283; 42:502-531; 45:687-734) and in variational problems in control theory. In this paper, we present a numerical method to solve boundary-value problems for a singularly perturbed di erentialdi erence equation of mixed type, i.e. which contains both type of terms having negative shifts as well as positive shifts, and consider the case in which the solution of the problem exhibits rapid oscillations. The stability and convergence analysis of the method is given. The e ect of small shift on the oscillatory solution is shown by considering the numerical experiments. The numerical results for several test examples demonstrate the e ciency of the method.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical analysis of boundary‐value problems for singularly perturbed differential‐difference equations: small shifts of mixed type with rapid oscillations

Kadalbajoo

Sharma

2004

Commun. Numer. Meth. Engng.

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“…The boundary value problems for such a class of delay differential equations are ubiquitous in the modeling of several physical and biological phenomena like first exit time problem in modeling of activation of neuronal variability [8], in the study of bistable devices [1] and evolutionary biology [14], in a variety of models for physiological processes or diseases [11,12,14], to describe the human pupillight reflex [10], variational problems in control theory [4,5], and in describing the motion of the sunflower [13].…”

Section: Numerical Treatment Of Boundary Value Problemsmentioning

confidence: 99%

<![CDATA[<B>Numerical treatment of boundary value problems for second order singularly perturbed delay differential equations</B>]]>

Kadalbajoo¹,

Sharma²

2005

Mat. apl. comput.

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Abstract. In this paper, the boundary value problems for second order singularly perturbed delay differential equations are treated. A generic numerical approach based on finite difference is presented to solve such boundary value problems. The stability and convergence analysis of the method is studied. The solution of the boundary value problems when delay is zero, exhibits layer behavior. Here, the study focuses on the effect of delay on the boundary layer behavior of the solution via numerical approach. The effect of the delay on the boundary layer behavior of the solution is shown by carrying out some numerical experiments.Mathematical subject classification: G.1.7.

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“…where 0 < ε ≪ 1 is the singular perturbation parameter, δ and η are called the delay and the advance parameters, sometimes referred to as negative shift and positive shift, respectively, as in [16,17]; precise assumptions will be given in the next section. This type of differential equation plays an important role in the mathematical modeling of various practical phenomena, for instance, variational problems in control theory [6], description of the so-called human pupil-light reflex [19], evolutionary biology [34], and a variety of models for physiological processes or diseases [21].…”

Section: Introductionmentioning

confidence: 99%

High-order uniformly convergent method for nonlinear singularly perturbed delay differential equations with small shifts

Salama¹,

Alamery²

2016

Turk J Math

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In this paper, we propose and analyze a high-order uniform method for solving boundary value problems (BVPs) for singularly perturbed nonlinear delay differential equations with small shifts (delay and advance). Such types of BVPs play an important role in the modeling of various real life phenomena, such as the variational problem in control theory and in the determination of the expected time for the generation of action potentials in nerve cells. To obtain parameter-uniform convergence, the present method is constructed on a piecewise-uniform Shishkin mesh. The error estimate is discussed and it is shown that the method is uniformly convergent with respect to the singular perturbation parameter. Moreover, a bound of the global error is also derived. The effect of small shifts on the solution behavior is shown by numerical computations. Several numerical examples are presented to support the theoretical results, and to demonstrate the efficiency and the high-order accuracy of the proposed method.

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Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations. VI. Small Shifts with Rapid Oscillations

Cited by 129 publications

References 13 publications

Numerical analysis of boundary‐value problems for singularly perturbed differential‐difference equations: small shifts of mixed type with rapid oscillations

Numerical analysis of boundary‐value problems for singularly perturbed differential‐difference equations: small shifts of mixed type with rapid oscillations

<![CDATA[<B>Numerical treatment of boundary value problems for second order singularly perturbed delay differential equations</B>]]>

High-order uniformly convergent method for nonlinear singularly perturbed delay differential equations with small shifts

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