A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts

Ranjan, Rakesh; Prasad, Hari

doi:10.1007/s12190-020-01397-6

Cited by 11 publications

(13 citation statements)

References 25 publications

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“…e scheme before the extrapolation gives first-order uniform convergence and the extrapolated scheme gives second-order uniform convergence. In Tables 9 and 10, we compare the maximum absolute error of the proposed scheme with recently published papers in [14,15,19]. As one observes, the proposed scheme gives more accurate result.…”

Section: Examples and Numerical Resultsmentioning

confidence: 94%

“…As observed in the figures, for ε going small, strong boundary layer is created. [14,19] for δ � 0.6ε and η � 0.5ε.…”

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

“…ε↓ N ⟶ 10 100 10 100 10 100 Result in [19] Result in [14] Proposed scheme 10 International Journal of Differential Equations…”

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

“…Melesse et al [18] applied initial value technique for treating the considered SPDDEs. Ranjan and Prasad [19] used modified fitted operator FDM for solving the problem. Sirisha et al [20] developed fitted operator finite difference scheme using the procedure of domain decomposition.…”

Section: Introductionmentioning

confidence: 99%

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Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts

Woldaregay

Duressa

2020

International Journal of Differential Equations

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This paper deals with numerical treatment of singularly perturbed differential difference equations involving mixed small shifts on the reaction terms. The highest-order derivative term in the equation is multiplied by a small perturbation parameter ε taking arbitrary values in the interval 0,1 . For small values of ε , the solution of the problem exhibits exponential boundary layer on the left or right side of the domain and the derivatives of the solution behave boundlessly large. The terms having the shifts are treated using Taylor’s series approximation. The resulting singularly perturbed boundary value problem is solved using exponentially fitted operator FDM. Uniform stability of the scheme is investigated and analysed using comparison principle and solution bound. The formulated scheme converges uniformly with linear order before Richardson extrapolation and quadratic order after Richardson extrapolation. The theoretical analysis of the scheme is validated using numerical test examples for different values of ε and mesh number N .

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Section: Examples and Numerical Resultsmentioning

confidence: 94%

“…As observed in the figures, for ε going small, strong boundary layer is created. [14,19] for δ � 0.6ε and η � 0.5ε.…”

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

“…ε↓ N ⟶ 10 100 10 100 10 100 Result in [19] Result in [14] Proposed scheme 10 International Journal of Differential Equations…”

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts

Woldaregay

Duressa

2020

International Journal of Differential Equations

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“…For example, population ecology, control theory, viscous elasticity, and materials with thermal memory, hybrid optical system, in models for physiological processes, red blood cell system, predator-prey models, and so on as the detailed descriptions given in ( [1] , [2] , [3] , [4] , [5] ). A series of papers developed ( [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] ), and many more to obtain an approximate solution for different classes of singularly perturbed differential-difference equations. A variety of different numerical approaches have been suggested in an attempt to obtain accurate and reliable schemes for the treatment of boundary value problems of singularly perturbed differential-difference equations with a small negative shift in the convection term [9, 12].…”

Section: Introductionmentioning

confidence: 99%

Novel approach to solve singularly perturbed boundary value problems with negative shift parameter

Duressa

2021

Heliyon

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Singularly perturbed boundary value problems with negative shift parameter are special types of differential difference equations whose solution exhibits boundary layer behaviour. A simple but novel numerical method is developed to approximate the numerical solution of the problems of these types. The method gives accurate solutions for ℎ ≥ in the inner region of the boundary layer where other classical numerical methods fail to give smooth solution. The present method is proved to be point-wise uniformly convergent with second order rate of convergence.

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A novel fitted spline method for the numerical treatment of singularly perturbed differential equations having small delays

Ranjan

2023

J. Appl. Math. Comput.

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A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts

Cited by 11 publications

References 25 publications

Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts

Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts

Novel approach to solve singularly perturbed boundary value problems with negative shift parameter

A novel fitted spline method for the numerical treatment of singularly perturbed differential equations having small delays

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