Uniform difference method for singularly pertubated delay Sobolev problems

Chiyaneh, Akbar Barati; Durus, Hakkı

doi:10.2989/16073606.2019.1653395

Cited by 9 publications

(7 citation statements)

References 16 publications

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“…A. B. Chiyaneh and H. Duru [17,18] have formulated difference schemes to resolve singularly perturbed Sobolev initial-boundary value problems with time-delay parameter. S. Elango et.…”

Section: Introductionmentioning

confidence: 99%

A second-order numerical method for pseudo-parabolic equations having both layer behavior and delay argument

Güneş,

Duru

2024

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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In this paper, singularly perturbed pseudo-parabolic initial-boundary value problems with time-delay parameter are considered by numerically. Initially, the asymptotic properties of the analytical solution are investigated. Then, a discretization with exponential coefficient is suggested on a uniform mesh. The error approximations and uniform convergence of the presented method are estimated in the discrete energy norm. Finally, some numerical experiments are given to clarify the theory.

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“…A. B. Chiyaneh and H. Duru [17,18] have formulated difference schemes to resolve singularly perturbed Sobolev initial-boundary value problems with time-delay parameter. S. Elango et.…”

Section: Introductionmentioning

confidence: 99%

A second-order numerical method for pseudo-parabolic equations having both layer behavior and delay argument

Güneş,

Duru

2024

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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“…This fact has aroused the interest of researchers to construct numerical methods for solving DSEs. For example, Okcu and Amiraliyev [20] derived a fourth‐order differential‐difference scheme for DSEs and obtained the error estimate of the scheme, Amiraliyev et al [21] considered a finite difference method for solving linear DSEs, Chiyaneh and Duru [22, 23] dealt with two finite difference methods based on the adaptive and uniform meshes, respectively, for solving linear singularly perturbed DSEs, Amirali [24] gave the error estimate of a higher order difference method for a class of linear DSEs, and Zhang and Tan [25] studied linearized compact difference methods (LCDMs) combined with Richardson extrapolation for solving one‐dimensional (1D) and two‐dimensional (2D) nonlinear DSEs.…”

Section: Introductionmentioning

confidence: 99%

“…This fact has aroused the interest of researchers to construct numerical methods for solving DSEs. For example, Okcu and Amiraliyev [20] derived a fourth-order differential-difference scheme for DSEs and obtained the error estimate of the scheme, Amiraliyev et al [21] considered a finite difference method for solving linear DSEs, Chiyaneh and Duru [22,23] Numer Methods Partial Differential Eq. 2023;39:2141-2162. wileyonlinelibrary.com/journal/num © 2022 Wiley Periodicals LLC.…”

Section: Introductionmentioning

confidence: 99%

Linearized compact difference methods for solving nonlinear Sobolev equations with distributed delay

Tan

Ran

2022

Numerical Methods Partial

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This paper deals with the numerical computation and analysis for nonlinear Sobolev equations with distributed delay. We present linearized compact difference methods for solving one-and two-dimensional cases, respectively. The solvability, convergence and stability of the methods are derived and it is shown under some suitable conditions that the methods have second-order accuracy in time and fourth-order accuracy in space. Moreover, with several numerical examples, the theoretical accuracy and computational effectiveness of the proposed methods are confirmed.

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“…The authors proposed a finite difference scheme, which is proved to be ε -uniformly convergent of order two. The singularly perturbed time delay problems have been extensively studied in Ansari et al (2007), Chiyaneh and Durus (2019), Chiyaneh and Duru (2020), Das and Natesan (2015), Das and Natesan (2018), Govindarao et al (2019), Govindarao and Mohapatra (2019), Gowrisankar and Natesan (2017), Kaushik and Sharma (2012), Kumar and Kumar (2014), Kumar and Kumari (2019) and Rai and Yadav (2020). The differential equation arisen in modeling a furnace, which is used for processing metal sheets, is a striking example of singularly perturbed problems with a time delay (Ansari et al , 2007): For wide range of examples of time delay partial differential equation models, one can refer (Wu, 1996).…”

Section: Introductionmentioning

confidence: 99%

“…The authors also proved that the proposed method was better than the existing methods. Chiyaneh and Durus (2019) approximated the singularly perturbed delay Sobolov equations using a finite difference scheme on a B-mesh. The proposed method has ε -uniform convergence of order two in both space and time variable.…”

Section: Introductionmentioning

confidence: 99%

A higher order scheme for singularly perturbed delay parabolic turning point problem

Yadav

Rai

2020

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Purpose The purpose of this study is to construct and analyze a parameter uniform higher-order scheme for singularly perturbed delay parabolic problem (SPDPP) of convection-diffusion type with a multiple interior turning point. Design/methodology/approach The authors construct a higher-order numerical method comprised of a hybrid scheme on a generalized Shishkin mesh in space variable and the implicit Euler method on a uniform mesh in the time variable. The hybrid scheme is a combination of simple upwind scheme and the central difference scheme. Findings The proposed method has a convergence rate of order O(N−2L2+Δt). Further, Richardson extrapolation is used to obtain convergence rate of order two in the time variable. The hybrid scheme accompanied with extrapolation is second-order convergent in time and almost second-order convergent in space up to a logarithmic factor. Originality/value A class of SPDPPs of convection-diffusion type with a multiple interior turning point is studied in this paper. The exact solution of the considered class of problems exhibit two exponential boundary layers. The theoretical results are supported via conducting numerical experiments. The results obtained using the proposed scheme are also compared with the simple upwind scheme.

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Uniform difference method for singularly pertubated delay Sobolev problems

Cited by 9 publications

References 16 publications

A second-order numerical method for pseudo-parabolic equations having both layer behavior and delay argument

A second-order numerical method for pseudo-parabolic equations having both layer behavior and delay argument

Linearized compact difference methods for solving nonlinear Sobolev equations with distributed delay

A higher order scheme for singularly perturbed delay parabolic turning point problem

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