A parameter-uniform collocation scheme for singularly perturbed delay problems with integral boundary condition

Kumar, Devendra; Kumari, Parvin

doi:10.1007/s12190-020-01340-9

Cited by 18 publications

(11 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…• There are other additional ways [31][32][33][34] to solve the system of ordinary differential equations, however this article uses further discretization in space variables.…”

Section: Applicationmentioning

confidence: 99%

Approximation of solution for generalized Basset equation with finite delay using Rothe's approach

Devi

Pandey

2023

Malaya J. Mat.

View full text Add to dashboard Cite

This study focuses on the use of the Riemann-Liouville fractional (R-L) derivative to address an initial boundary value problem for a fractional order differential equation with finite delay (FDDE). Rothe's methodology is used to prove the existence and uniqueness of the strong solution and classical solution to the restated abstract FDDE. Some examples based on abstract theory and numerical solutions of FDDEs arising in fluid dynamics are presented.

show abstract

“…• There are other additional ways [31][32][33][34] to solve the system of ordinary differential equations, however this article uses further discretization in space variables.…”

Section: Applicationmentioning

confidence: 99%

Approximation of solution for generalized Basset equation with finite delay using Rothe's approach

Devi

Pandey

2023

Malaya J. Mat.

View full text Add to dashboard Cite

show abstract

“…For m = 1 , let x ∈ Γ and consider a neighborhood I = (a, a + √ ) , ∀x ∈ I . Then, applying the mean value theorem for some a ∈ ̄I and t ∈ (0, T ] , we can get Now, for any x in ̄I , we can get Using (11), we can get �u…”

Section: P R O O F D E F I N E B a R R I E R F U N C T I O N S A Smentioning

confidence: 99%

“…For instance, Kumar [10] proposed a collocation method for singularly perturbed turning point problems involving boundary/interior layers. Kumar and Kumari [11] solved singularly perturbed problems with integral boundary condition by constructing a parameteruniform collocation scheme. Cimen and Amiraliyev [12] solved singularly perturbed problems based on a piecewise uniform mesh of Shishkin type.…”

Section: Introductionmentioning

confidence: 99%

A uniformly convergent numerical scheme for solving singularly perturbed differential equations with large spatial delay

et al. 2022

View full text Add to dashboard Cite

In this study, a parameter-uniform numerical scheme is built and analyzed to treat a singularly perturbed parabolic differential equation involving large spatial delay. The solution of the considered problem has two strong boundary layers due to the effect of the perturbation parameter, and the large delay causes a strong interior layer. The behavior of the layers makes it difficult to solve such problem analytically. To treat the problem, we developed a numerical scheme using the weighted average ($$\theta$$ θ -method) difference approximation on a uniform time mesh and the central difference method on a piece-wise uniform spatial mesh. We established the Stability and convergence analysis for the proposed scheme and obtained that the method is uniformly convergent of order two in the temporal direction and almost second order in the spatial direction. To validate the applicability of the proposed numerical scheme, two model examples are treated and confirmed with the theoretical findings.

show abstract

“…Debela and Duressa [19] improved the order of accuracy for the method proposed in Debela and Duressa [18]. Kumar and Kumari [20] developed the method based on the idea of B-spline functions and an efficient numerical method on a piecewise-uniform mesh was recommended to approximate the solutions of SPPs having a delay of unit magnitude with an integral boundary condition. In the literature, only few authors considered a class of singularly perturbed parabolic partial differential equations (SPPPDEs) with integral boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Numerical treatment of singularly perturbed parabolic partial differential equations with nonlocal boundary condition

Wondimu

Woldaregay

Dinka

et al. 2022

Front. Appl. Math. Stat.

View full text Add to dashboard Cite

This paper presents numerical treatments for a class of singularly perturbed parabolic partial differential equations with nonlocal boundary conditions. The problem has strong boundary layers at x = 0 and x = 1. The nonstandard finite difference method was developed to solve the considered problem in the spatial direction, and the implicit Euler method was proposed to solve the resulting system of IVPs in the temporal direction. The nonlocal boundary condition is approximated by Simpsons 13 rule. The stability and uniform convergence analysis of the scheme are studied. The developed scheme is second-order uniformly convergent in the spatial direction and first-order in the temporal direction. Two test examples are carried out to validate the applicability of the developed numerical scheme. The obtained numerical results reflect the theoretical estimate.

show abstract

A parameter-uniform collocation scheme for singularly perturbed delay problems with integral boundary condition

Cited by 18 publications

References 28 publications

Approximation of solution for generalized Basset equation with finite delay using Rothe's approach

Approximation of solution for generalized Basset equation with finite delay using Rothe's approach

A uniformly convergent numerical scheme for solving singularly perturbed differential equations with large spatial delay

Numerical treatment of singularly perturbed parabolic partial differential equations with nonlocal boundary condition

Contact Info

Product

Resources

About