Solving singularly perturbed delay differential equations via fitted mesh and exact difference method
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TL;DR: In this paper , different numerical schemes are developed and investigated for solving singularly perturbed delay differential equations (DDEs), which exhibits a strong boundary layer when the perturbation parameter approaches zero.
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Abstract: In this paper, different numerical schemes are developed and investigated for solving singularly perturbed delay differential equations (DDEs). The solution of the considered equations exhibits a strong boundary layer when the perturbation parameter approaches zero. Standard numerical methods developed for solving regular problems fail to treat accurately the considered equations. We developed three numerical schemes for treating the considered equations. The first scheme (Scheme I) uses a non-standard mid-point upwind finite difference method on uniform mesh; the second scheme (Scheme II) uses a standard mid-point upwind finite difference method on Shiskin mesh and the third scheme (Scheme III) uses a non-standard mid-point upwind finite difference method on Shiskin mesh. The existence of a unique discrete solution is investigated using the discrete maximum principle. The stability and uniform convergence of the schemes are investigated and proved. Scheme III gives better accuracy and order of convergence than Scheme I. It has a boundary layer resolving behaviour which is the main drawback of Scheme II. Numerical examples are considered and treated to validate the theoretical finding of the schemes for different values of the perturbation parameter and delay parameter.
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Citations
Fitted exact difference method for solutions of a singularly perturbed time delay parabolic PDE
Mesfin Mekuria Woldaregay,Tibebu Worrku Hunde,Vishnu Narayan Mishra +2 more
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Boundary Layer Resolving Exact Difference Scheme for Solving Singularly Perturbed Convection-Diffusion-Reaction Equation
TL;DR: In this article , the numerical treatment of singularly perturbed time-dependent convection-diffusion-reaction equation is considered and the existence of unique discrete solutions and the stability of the schemes are discussed and proved.
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Hybrid Fitted Numerical Scheme for Singularly Perturbed Convection-Diffusion Problem with a Small Time Lag
TL;DR: In this paper , a hybrid scheme has been constructed, which is a combination of the cubic spline method in the boundary layer region and the midpoint upwind scheme in the outer layer region on a piecewise Shishkin mesh in the spatial direction.
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Yang Kuang
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Alfredo Bellen,Marino Zennaro +1 more
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Analysis of some difference approximations for a singular perturbation problem without turning points
R. Bruce Kellogg,Alice Tsan +1 more
TL;DR: Some three point difference schemes are considered for a singular perturbation problem without turning points in this article, and bounds for the discretization error are obtained which are uniformly valid for all h and e > 0.
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Ronald E. Mickens
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TL;DR: The theory of nonstandard finite difference methods and applications to singular perturbation problems have been studied in the literature as mentioned in this paper, with a focus on the application of finite difference in non-smooth mechanics.
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