Fourth-order stable central difference with Richardson extrapolation method for second-order self-adjoint singularly perturbed boundary value problems

Siraj, Muslima Kedir; Duressa, Gemechis File; Bullo, Tesfaye Aga

doi:10.1186/s42787-019-0047-4

Cited by 13 publications

(9 citation statements)

References 12 publications

(9 reference statements)

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“…A finite difference scheme is consistent if the limit of truncation error (TE) is equal to zero as the mesh size h goes to zero. Further, local truncation errors measure how well a finite difference discretization approximates the differential equation [19,21,[23][24][25][26].…”

Section: Consistency Of the Methodsmentioning

confidence: 99%

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Higher-Order Numerical Method for Singularly Perturbed Delay Reaction-Diffusion Problems

Duressa¹,

Bullo²

2021

PAMJ

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In this paper, a higher-order numerical method is presented for solving the singularly perturbed delay differential equations. Such kind of equations have a delay parameter on reaction term and exhibits twin boundary layers or oscillatory behavior. Recently, different numerical methods have been developed to solve the singularly perturbed delay reaction-diffusion problems. However, the obtained accuracy and its rate of convergence are satisfactory. Thus, to solve the considered problem with more satisfactory accuracy and a higher rate of convergence, the higher-order numerical method is presented. First, the given singularly perturbed delay differential equation is transformed to asymptotically equivalent singularly perturbed twopoint boundary value convection-diffusion differential equation by using Taylor series approximations. Then, the constructed singularly perturbed boundary value differential equation is replaced by three-term recurrence relation finite difference approximations. The Richardson extrapolation technique is applied to accelerate the fourth-order convergent of the developed method to the sixth-order convergent. The consistency and stability of the formulated method have been investigated very well to guarantee the convergence of the method. The rate of convergence for both the theoretical and numerical have been proven and are observed to be in accord with each other. To demonstrate the efficiency of the method, different model examples have been considered and simulation of numerical results have been presented by using MATLAB software. Numerical experimentation has been done and the results are presented for different values of the parameters. Further, The obtained numerical results described that the finding of the present method is more accurate than the findings of some methods discussed in the literature.

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Section: Consistency Of the Methodsmentioning

confidence: 99%

“…The purpose of this section is to improve the accuracy and the order of convergence by convergence acceleration technique which involves a combination of two computed approximate solutions. The linear combination turns out to be a better approximation, [24,26].…”

Section: Richardson Extrapolationmentioning

confidence: 96%

Higher-Order Numerical Method for Singularly Perturbed Delay Reaction-Diffusion Problems

Duressa¹,

Bullo²

2021

PAMJ

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“…Since A is real and symmetric it follows that A −1 is also real and symmetric so that, its eigenvalues are real and given by 1 λ s . Hence, as [19] the stability condition of the method will be satisfied when;…”

Section: Stability Analysismentioning

confidence: 99%

Robust numerical method for singularly perturbed differential equations having both large and small delay

Debela

2021

AJMS

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PurposeThe purpose of this study is to develop stable, convergent and accurate numerical method for solving singularly perturbed differential equations having both small and large delay.Design/methodology/approachThis study introduces a fitted nonpolynomial spline method for singularly perturbed differential equations having both small and large delay. The numerical scheme is developed on uniform mesh using fitted operator in the given differential equation.FindingsThe stability of the developed numerical method is established and its uniform convergence is proved. To validate the applicability of the method, one model problem is considered for numerical experimentation for different values of the perturbation parameter and mesh points.Originality/valueIn this paper, the authors consider a new governing problem having both small delay on convection term and large delay. As far as the researchers' knowledge is considered numerical solution of singularly perturbed boundary value problem containing both small delay and large delay is first being considered.

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“…Based on the parameters, the perturbation and/or delay parameters they involved, singularly perturbed problems can be categorized into the singularly perturbed differential equations or singularly perturbed differential-difference equations. Many researchers, like in, [7]- [16] have been providing different numerical methods for solving singularly perturbed differentialdifference equations. But, most of those author's considered the stated problem when it involves one perturbation parameter.…”

Section: Introductionmentioning

confidence: 99%

Quartic Non-Polynomial Spline Method for Singularly Perturbed Differential-difference Equation with Two Parameters

2021

Self Cite

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Quartic non-polynomial spline method is presented to solve the singularly perturbed differential-difference equation containing two parameters. The considered equation is transformed into an asymptotical equivalent differential equation, and the derivatives are replaced finite difference approximation using the quartic non-polynomial spline method. The convergence analysis of the method has been established. Numerical experimentation is carried out on model examples, and the results are presented both in tables and graphs. Furthermore, the present method gives a more accurate solution than some existing methods reported in the literature.

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Fourth-order stable central difference with Richardson extrapolation method for second-order self-adjoint singularly perturbed boundary value problems

Cited by 13 publications

References 12 publications

Higher-Order Numerical Method for Singularly Perturbed Delay Reaction-Diffusion Problems

Higher-Order Numerical Method for Singularly Perturbed Delay Reaction-Diffusion Problems

Robust numerical method for singularly perturbed differential equations having both large and small delay

Quartic Non-Polynomial Spline Method for Singularly Perturbed Differential-difference Equation with Two Parameters

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