Numerical analysis of boundary‐value problems for singularly perturbed differential‐difference equations: small shifts of mixed type with rapid oscillations

Kadalbajoo, Mohan K.; Sharma, Kapil K.

doi:10.1002/cnm.657

Cited by 21 publications

(9 citation statements)

References 16 publications

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“…Further studies of the effect of on the layer behaviour of the solution has been carried out by Kadalbajoo and Sharma [5]. Similar observations have been made by the same authors when is also present [6]. On the other hand, in this paper, we see that for the problems having solutions with layer behaviour, the presence of both delay and advance does not affect the location of the layer.…”

Section: Y (X) + A(x)y(x − ) + W(x)y(x) + B(x)y(xsupporting

confidence: 86%

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Uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance

Patidar

Sharma²

2006

Int. J. Numer. Meth. Engng

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SUMMARYA new class of fitted operator finite difference methods are constructed via non-standard finite difference methods ((NSFDM)s) for the numerical solution of singularly perturbed differential difference equations having both delay and advance arguments. The main idea behind the construction of our method(s) is to replace the denominator function of the classical second-order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results. Unlike other FOFDMs constructed in standard ways, the methods that we present in this paper are fairly simple to construct (and thus enrich the class of fitted operator methods by adding these new methods). These methods are shown to be -uniformly convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration. This paper further clarifies several doubts, e.g. why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies. Finally, we provide some numerical examples which illustrate the theoretical findings.

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Section: Y (X) + A(x)y(x − ) + W(x)y(x) + B(x)y(xsupporting

confidence: 86%

“…For occurrence and further motivation for solving such problems, the readers may refer to the introduction sections in References [5][6][7][8] and the references therein. Some of the other relevant references are [1,2,[9][10][11][12][13][14].…”

Section: Y (X) + A(x)y(x − ) + W(x)y(x) + B(x)y(xmentioning

confidence: 99%

Uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance

Patidar

Sharma²

2006

Int. J. Numer. Meth. Engng

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“…Numerical analysis of singularly perturbed differentialdifference turning point problems was initiated by Kadalbajoo and Sharma. In a series of papers, [8][9][10], they gave many robust numerical techniques for the solution of such type of problems. Kadalbajoo and Sharma [8] elucidate a numerical method to solve boundary value problems for singularly perturbed differential-difference equation with mixed shifts.…”

Section: Introductionmentioning

confidence: 99%

Solution of Singularly Perturbed Differential-Difference Equations with Mixed Shifts Using Galerkin Method with Exponential Fitting

Swamy

Phaneendra

Reddy

2016

Chinese Journal of Mathematics

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Galerkin method is presented to solve singularly perturbed differential-difference equations with delay and advanced shifts using fitting factor. In the numerical treatment of such type of problems, Taylor's approximation is used to tackle the terms containing small shifts. A fitting factor in the Galerkin scheme is introduced which takes care of the rapid changes that occur in the boundary layer. This fitting factor is obtained from the asymptotic solution of singular perturbations. Thomas algorithm is used to solve the tridiagonal system of the fitted Galerkin method. The method is analysed for convergence. Several numerical examples are solved and compared to demonstrate the applicability of the method. Graphs are plotted for the solutions of these problems to illustrate the effect of small shifts on the boundary layer solution.

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“…Assume that conditions (8)-(12) and(41)-(42) are satisfied and β > 0. Suppose A h ε (x) ≥ 0 for x ≥ 0 and A h ε (x) ≤ 0 for x ≤ 0.…”

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confidence: 99%

Parameter uniform numerical method for singularly perturbed differential–difference equations with interior layers

Rai

Sharma

2011

International Journal of Computer Mathematics

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The present study is devoted to the numerical study of boundary value problems for singularly perturbed linear second-order differential-difference equations with a turning point. The points of the domain where the coefficient of the convection term in the singularly perturbed differential equation vanishes are known as the turning points. The solution of such type of differential equations exhibits boundary layer(s) or interior layer(s) behaviour depending upon the nature of the coefficient of convection term and the reaction term. In particular, this paper focuses on problems whose solution exhibits interior layers. In the development of numerical schemes for singularly perturbed differential-difference equations with a turning point, we use El-Mistikawy-Werle exponential finite difference scheme with some modifications. Some priori estimates have been established and parameter uniform convergence analysis of the proposed scheme is also discussed. Several examples are considered to demonstrate the performance of the proposed scheme and effect of the size of the delay/advance arguments and coefficients of the delay/advance term on the layer behaviour of the solution.

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Numerical analysis of boundary‐value problems for singularly perturbed differential‐difference equations: small shifts of mixed type with rapid oscillations

Cited by 21 publications

References 16 publications

Uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance

Uniformly convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance

Solution of Singularly Perturbed Differential-Difference Equations with Mixed Shifts Using Galerkin Method with Exponential Fitting

Parameter uniform numerical method for singularly perturbed differential–difference equations with interior layers

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