Uniform Convergence Method for a Delay Differential Problem with Layer Behaviour

Çimen, Erkan; Amiraliyev, Gabil M.

doi:10.1007/s00009-019-1335-9

Cited by 17 publications

(14 citation statements)

References 17 publications

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“…In [18,19], an initial value technique with an exponential tting factor is applied to solve convection-dominated delay differential equations. e di erence method on Shishkin for solving a singularly perturbed linear second-order delay di erential equation is developed in [20].…”

Section: Introductionmentioning

confidence: 99%

Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

Megiso

Woldaregay

Dinka

2022

Mathematical Problems in Engineering

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A uniformly convergent numerical method is presented for solving singularly perturbed time delay reaction-diffusion problems. Properties of the continuous solution are discussed. The Crank–Nicolson method is used for discretizing the temporal derivative, and an exponentially fitted tension spline method is applied for the spatial derivative. Using the comparison principle and solution bound, the stability of the method is analyzed. The proposed numerical method is second-order uniformly convergent. The theoretical analysis is supported by numerical test examples for various values of perturbation parameters and mesh size.

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Section: Introductionmentioning

confidence: 99%

Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

Megiso

Woldaregay

Dinka

2022

Mathematical Problems in Engineering

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“…Many other examples can be found in [2,3]. Extensive numerical methods have been developed for singularly perturbed delay di erential equations, such as [4][5][6][7] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Uniformly Convergent Scheme for Singularly Perturbed Space Delay Parabolic Differential Equation with Discontinuous Convection Coefficient and Source Term

Ayele

Tiruneh

Derese

2022

Journal of Mathematics

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A singularly perturbed delay parabolic problem of convection-diffusion type with a discontinuous convection coefficient and source term is examined. In the problem, strong interior layers and weak boundary layers are exhibited due to a large delay in the spatial variable and discontinuity of convection coefficient and source. The problem is discretized by a nonstandard finite difference scheme in the spatial variable and for the time derivative, we used the Crank–Nicolson scheme. To enhance the order of convergence of the spatial variable, the Richardson extrapolation technique is applied. The error analysis of the proposed scheme was carried out and proved that the scheme is uniformly convergent of second order in both spatial and temporal variables. Numerical experiments are performed to verify the theoretical estimates.

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“…Therefore, it is important to develop effective numerical methods to solve DDEs. In the past two decades, several numerical methods have been proposed for the second order DDEs problem including finite difference method [4,20], two-point block method [18], Adams-Moulton method [19], variational iteration method [15], Legendre-Gauss spectral collocation method [25], initial value method [21].…”

Section: Introductionmentioning

confidence: 99%

An efficient method for solving second-order delay differential equation

Çimen¹,

Uncu²

2021

MMN

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In this paper, the initial-value problem for a linear second order delay differential equation is considered. To solve this problem numerically, an appropriate difference scheme is constructed by using the method of integral identities which contains basis functions and interpolating quadrature rules with weight and remainder term in integral form. Besides, the method is proved to be first-order convergent in discrete maximum norm. The numerical illustration provided support the theoretical results. Finally, the proposed method is compared with the implicit Euler method.

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Uniform Convergence Method for a Delay Differential Problem with Layer Behaviour

Cited by 17 publications

References 17 publications

Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

Uniformly Convergent Scheme for Singularly Perturbed Space Delay Parabolic Differential Equation with Discontinuous Convection Coefficient and Source Term

An efficient method for solving second-order delay differential equation

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