A second-order hybrid finite difference scheme for a system of singularly perturbed initial value problems

Cen, Zhongdi; Xu, Aiqiang; Le, Anbo

doi:10.1016/j.cam.2010.05.006

Cited by 24 publications

(9 citation statements)

References 9 publications

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“…Schemes for the Problems (4.6) -(4.9). Applying the hybrid finite difference scheme given in [22,23] to the above singularly perturbed problems (4.6)-(4.9), we get…”

Section: Hybrid Finite Differencementioning

confidence: 99%

An Initial Value Method for Singularly Perturbed System of Reaction-Diffusion Type Delay Differential Equations

Subburayan¹,

Ramanujam²

2013

Journal of the Korea Society for Industrial and Applied Mathema

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In this paper an asymptotic numerical method named as Initial Value Method (IVM) is suggested to solve the singularly perturbed weakly coupled system of reaction-diffusion type second order ordinary differential equations with negative shift (delay) terms. In this method, the original problem of solving the second order system of equations is reduced to solving eight first order singularly perturbed differential equations without delay and one system of difference equations. These singularly perturbed problems are solved by the second order hybrid finite difference scheme. An error estimate for this method is derived by using supremum norm and it is of almost second order. Numerical results are provided to illustrate the theoretical results.

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“…Schemes for the Problems (4.6) -(4.9). Applying the hybrid finite difference scheme given in [22,23] to the above singularly perturbed problems (4.6)-(4.9), we get…”

Section: Hybrid Finite Differencementioning

confidence: 99%

An Initial Value Method for Singularly Perturbed System of Reaction-Diffusion Type Delay Differential Equations

Subburayan¹,

Ramanujam²

2013

Journal of the Korea Society for Industrial and Applied Mathema

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“…In recent years, there has been tremendous interest in developing some robust numerical methods for a system of nonlinear singularly perturbed initial value problems. Cen et al 7 considered the above problem (1)-( 3) and constructed a hybrid finite difference scheme on a piecewise-uniform Shishkin mesh, which was almost second-order accurate, uniformly in both small parameters. Das 8 developed an a posterior error estimation in maximum norm for a system of nonlinear singularly perturbed delay differential equations with one perturbation parameter 𝜀 on an adaptive mesh.…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of a system of semilinear singularly perturbed first‐order differential equations on an adaptive grid

Liu

Zhu

Long

2021

Math Methods in App Sciences

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An adaptive grid method based on the backward Euler formula for a system of semilinear singularly perturbed initial value problems is studied. Based on the a priori error analysis and mesh equidistribution principle, we prove that the convergence rate of our semidiscrete adaptive grid method is first order, which is robust with respect to the perturbation parameters. Then, in order to construct a fully discrete adaptive grid method, a standard residual-type a posterior error estimation is constructed by using the linear polynomial interpolation technique. A partly heuristic argument based on this a posteriori error estimator leads to an optimal monitor function, which is used to design an adaptive grid algorithm. Furthermore, we also extend our presented adaptive grid method to a nonlinear system of singularly perturbed problem arising in the modeling of enzyme kinetics and a system of singularly perturbed delay differential equations, respectively. Finally, numerical results are provided to illustrate the effectiveness of our presented adaptive grid method.

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“…In [9], first order (up to logarithmic factor) uniformly convergent numerical method was developed. A hybrid finite difference scheme on a piecewise-uniform Shishkin mesh was considered in [1] and the scheme was almost second-order accurate, uniformly in both small parameters. In all these works the source term is smooth.…”

mentioning

confidence: 99%

“…The case where the small parameter is associated with only one equation was considered in [4]. The most difficult and general case is that each component of the solution has its own initial layer that overlaps and interacts with others and this was considered in [1,5,9] . In [9], first order (up to logarithmic factor) uniformly convergent numerical method was developed.…”

mentioning

confidence: 99%

Numerical Solution for a Coupled System of Singularly Perturbed Initial Value Problems with Discontinuous Source Term

Rao

Chawla

2015

Springer Proceedings in Mathematics &Amp; Statistics

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In this work, we study a numerical method for a coupled system of singularly perturbed initial value problems having discontinuous source term. The leading term of each equation is multiplied by a distinct small positive parameter, due to which the overlapping initial and interior layers are generated in the solution. The problem is discretized using backward Euler difference scheme which involves an appropriate piecewise-uniform variant of Shishkin mesh that is fitted to both the initial and interior layers. The method is proved to be uniformly almost first-order accurate with respect to all the parameters. Numerical results are presented in support of the theory.

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A second-order hybrid finite difference scheme for a system of singularly perturbed initial value problems

Cited by 24 publications

References 9 publications

An Initial Value Method for Singularly Perturbed System of Reaction-Diffusion Type Delay Differential Equations

An Initial Value Method for Singularly Perturbed System of Reaction-Diffusion Type Delay Differential Equations

Numerical analysis of a system of semilinear singularly perturbed first‐order differential equations on an adaptive grid

Numerical Solution for a Coupled System of Singularly Perturbed Initial Value Problems with Discontinuous Source Term

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