Difference Schemes for the Singularly Perturbed Sobolev Equations

Amiraliyev, Gabil M.; Amiraliyeva, I. G.

doi:10.1142/9789812770752_0003

Cited by 30 publications

(49 citation statements)

References 11 publications

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“…Using the appropriate interpolating quadrature rules with weight and remainder term in integral form (see, e.g., [1][2][3]), we have the relations…”

Section: Difference Schemementioning

confidence: 99%

“…Various numerical treatments of equations of this type in the regular cases have been considered in [2,8,9,11,12,20] (see also the references cited in them). The numerical investigation of singular perturbation cases has always been far from trivial because of the boundary layer behavior of the solution [1,3] (see, also monographs [10,15]). …”

Section: Introductionmentioning

confidence: 99%

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A parameter-uniform numerical method for a Sobolev problem with initial layer

2007

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The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.

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“…Using the appropriate interpolating quadrature rules with weight and remainder term in integral form (see, e.g., [1][2][3]), we have the relations…”

Section: Difference Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A parameter-uniform numerical method for a Sobolev problem with initial layer

2007

Self Cite

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“…methods that are uniformly convergent with respect to the perturbation parameter [8][9][10]. One of the simplest ways to derive such methods consists of using exponentially fitted difference schemes (see, e.g., [8][9][10][11][12] for motivation for this type of mesh). In the direction of numerical treatment for first order singularly perturbed delay differential equations, several can be seen in [13][14][15][16].…”

mentioning

confidence: 99%

“…The difference scheme is constructed by the method of integral identities with the use of exponentially basis functions and interpolating quadrature rules with weight and remainder terms integral form [11,12]. This method of approximation has the advantage that the schemes can also be effective in the case when the continuous problem is considered under certain restrictions.…”

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confidence: 99%

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Fitted finite difference method for singularly perturbed delay differential equations

Erdoğan

Amiraliyev

2011

Numer Algor

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This paper deals with singularly perturbed initial value problem for linear second-order delay differential equation. An exponentially fitted difference scheme is constructed in an equidistant mesh, which gives first order uniform convergence in the discrete maximum norm. The difference scheme is shown to be uniformly convergent to the continuous solution with respect to the perturbation parameter. A numerical example is solved using the presented method and compared the computed result with exact solution of the problem.

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Stability and convergence of spectral radial point interpolation method locally applied on two‐dimensional pseudoparabolic equation

Shivanian

Aslefallah

2016

Numerical Methods Partial

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In this article, we study a spectral meshless radial point interpolation of pseudoparabolic equations in two spatial dimensions. Shape functions, which are constructed through point interpolation method using the radial basis functions, help us to treat problem locally with the aim of high‐order convergence rate. The time derivatives are approximated by the finite difference time‐stepping method. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical results are presented to illustrate the theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 724–741, 2017

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Difference Schemes for the Singularly Perturbed Sobolev Equations

Cited by 30 publications

References 11 publications

A parameter-uniform numerical method for a Sobolev problem with initial layer

A parameter-uniform numerical method for a Sobolev problem with initial layer

Fitted finite difference method for singularly perturbed delay differential equations

Stability and convergence of spectral radial point interpolation method locally applied on two‐dimensional pseudoparabolic equation

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