A parameter robust higher order numerical method for singularly perturbed two parameter problems with non-smooth data

Chandru, M.; Prabha, Thara V.; Shanthi, V.

doi:10.1016/j.cam.2016.06.009

Cited by 23 publications

(3 citation statements)

References 17 publications

(25 reference statements)

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“…(Doolan et al, 1980;Roos et al, 2008). Over the past two decades, to solve the SPBVP with DST, researchers have presented a large number of numerical techniques, such as Finite Difference Method (FDM) (Cen et al, 2017;Chandru and Shanti, 2015;Chandru et al, 2017;Clavero et al, 2017;Farrell et al, 2004a;Farrell et al, 2004b;Munyakazi, 2015;Shanthi et al, 2006), Finite Element Method (Chin and Krasny, 1983;Roos and Zarin, 2010), Boundary Value Technique (Chandru and Shanti, 2014;Shanti and Ramanujam, 2004) and spline approach (Kadalbajoo and Jha, 2012;Mane and Lodhi, 2023;Pathan and Vembu, 2017). A fourth-order convergent approach on a uniform mesh has been developed by Rashidinia et al (2010) using a NPS of degree five for SPBVP.…”

Section: Introductionmentioning

confidence: 99%

Nonpolynomial Spline for Numerical Solution of Singularly Perturbed Convection-Diffusion Equations with Discontinuous Source Term

T. Mane,

Lodhi

2024

Int. j. math. eng. manag. sci.

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This research addresses the numerical solution of singularly perturbed convection-diffusion kind boundary value problem of second-order with a discontinuity term. Due to the perturbation parameter and discontinuity term, the problem solution has a boundary layer and an interior layer. A nonpolynomial cubic spline method is utilized to solve the boundary value problem. A specific set of parameters associated with nonpolynomial spline is used to tailor the method. A comprehensive analysis of the stability and convergence of the recommended method is presented which gives second-order convergence results. The suggested method is implemented on two examples, and the obtained results are contrasted with an existing method, highlighting the precision and efficacy of the proposed method, which would enhance the method's novelty.

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Section: Introductionmentioning

confidence: 99%

Nonpolynomial Spline for Numerical Solution of Singularly Perturbed Convection-Diffusion Equations with Discontinuous Source Term

T. Mane,

Lodhi

2024

Int. j. math. eng. manag. sci.

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“…Te two-parameter singularly perturbed parabolic differential equation is the focus of some recent studies [4][5][6][7][8][9][10][11]. Some others considered this problem with nonsmooth data, which implies boundary and interior layers in their solution [12][13][14][15][16]. However, they did not consider the delay condition.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Approach for Diffusion‐Dominant Two‐Parameter Singularly Perturbed Delay Parabolic Differential Equations

Worku,

Duressa

2023

International Journal of Mathematics and Mathematical Sciences

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A numerical scheme is developed to solve a large time delay two-parameter singularly perturbed one-dimensional parabolic problem in a rectangular domain. Two small parameters multiply the convective and diffusive terms, which determine the width of the left and right lateral surface boundary layers. Uniform mesh and piece-wise uniform Shishkin mesh discretization are applied in time and spatial dimensions, respectively. The numerical scheme is formulated by using the Crank–Nicolson method on two consecutive time steps and the average central finite difference approximates in spatial derivatives. It is proved that the method is uniformly convergent, independent of the perturbation parameters, where the convection term is dominated by the diffusion term. The resulting scheme is almost second-order convergent in the spatial direction and second-order convergent in the temporal direction. Numerical experiments illustrate theoretical findings, and the method provides more accurate numerical solutions than prior literature.

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“…The authors in papers [21,22] developed uniform numerical methods for a singularly perturbed reaction-convectiondiffusion equation in one dimension with a discontinuous source term. In articles [23][24][25], the authors have developed robust numerical methods for singularly perturbed time delay parabolic problems with two parameters based on adaptive layer mesh methods.…”

Section: Introductionmentioning

confidence: 99%

A Fitted Numerical Approach for Singularly Perturbed Two-Parameter Parabolic Problem with Time Delay

Daba,

Melesse,

Kebede

2023

Computational and Mathematical Methods

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This paper is aimed at constructing and analyzing a fitted approach for singularly perturbed time delay parabolic problems with two small parameters. The proposed computational scheme comprises the implicit Euler and especially finite difference method for the time and space variable discretization, respectively, on uniform step size. The stability and convergence analysis of the method is provided and is first-order parameter uniform convergent. Further, the numerical results depict that the present method is more convergent than some methods available in the literature.

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A parameter robust higher order numerical method for singularly perturbed two parameter problems with non-smooth data

Cited by 23 publications

References 17 publications

Nonpolynomial Spline for Numerical Solution of Singularly Perturbed Convection-Diffusion Equations with Discontinuous Source Term

Nonpolynomial Spline for Numerical Solution of Singularly Perturbed Convection-Diffusion Equations with Discontinuous Source Term

A Numerical Approach for Diffusion‐Dominant Two‐Parameter Singularly Perturbed Delay Parabolic Differential Equations

A Fitted Numerical Approach for Singularly Perturbed Two-Parameter Parabolic Problem with Time Delay

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