Reproducing Kernel Method for Singularly Perturbed One-Dimensional Initial-Boundary Value Problems with Exponential Initial Layers

Geng, Fazhan; Tang, Z.Q.; Zhou, Yan

doi:10.1007/s12346-017-0242-3

Cited by 3 publications

(2 citation statements)

References 14 publications

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“…The PDEs with singular perturbation have been broadly studied by many scholars, including least squares method in [2], finite difference scheme in [4,26], Galerkin finite element method in [19], domain decomposition scheme in [20], reproducing kernel method(RKM) in [12] and others [6,27,30]. There are many references on numerical methods and numerical stability for delay differential equations, such as [5,15,17] to just list a few.…”

Section: Introductionmentioning

confidence: 99%

A Reproducing Kernel Method for Solving Singularly Perturbed Delay Parabolic Partial Differential Equations

Xie,

Zhang,

Niu

et al. 2023

Mathematical Modelling and Analysis

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In this article, we put forward an efficient method on the foundation of a few reproducing kernel spaces(RK-spaces) and the collocation method to seek the solution of delay parabolic partial differential equations(PDEs) with singular perturbation. The approximated solution to the equations is formulated and proved the exact solution is uniformly convergent by the solution. Furthermore, the partial differentiation of the approximated solution is also proved the partial derivatives of the exact solution is uniformly convergent by the solution. Meanwhile, we show that the accuracy of our method is in the order of T/n where T is the final time and n is the number of spatial (and time) discretization in the domain of interests. Three numerical examples are put forward to demonstrate the effectiveness of our presented scheme.

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Section: Introductionmentioning

confidence: 99%

A Reproducing Kernel Method for Solving Singularly Perturbed Delay Parabolic Partial Differential Equations

Xie,

Zhang,

Niu

et al. 2023

Mathematical Modelling and Analysis

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“…Aronszajn has given a systematic reproducing kernel theory containing the Bergman kernel function [10]. For more details, see [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

New Numerical Method for Solving Tenth Order Boundary Value Problems

Akgül

Băleanu

et al. 2018

Mathematics

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In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate solutions. We obtain very efficient results by this method. We show our numerical results by tables.

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Absolutely stable difference scheme for a general class of singular perturbation problems

et al. 2020

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This paper presents an absolutely stable noniterative difference scheme for solving a general class of singular perturbation problems having left, right, internal, or twin boundary layers. The original two-point second-order singular perturbation problem is approximated by a first-order delay differential equation with a variable deviating argument. This delay differential equation is transformed into a three-term difference equation that can be solved using the Thomas algorithm. The uniqueness and stability analysis are discussed, showing that the method is absolutely stable. An optimal estimate for the deviating argument is obtained to take advantage of the second-order accuracy of the central finite difference method in addition to the absolute stability property. Several problems having left, right, interior, or twin boundary layers are considered to validate and illustrate the method. The numerical results confirm that the deviating argument can stabilize the unstable discretized differential equation and that the new approach is effective in solving the considered class of singular perturbation problems.

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Reproducing Kernel Method for Singularly Perturbed One-Dimensional Initial-Boundary Value Problems with Exponential Initial Layers

Cited by 3 publications

References 14 publications

A Reproducing Kernel Method for Solving Singularly Perturbed Delay Parabolic Partial Differential Equations

A Reproducing Kernel Method for Solving Singularly Perturbed Delay Parabolic Partial Differential Equations

New Numerical Method for Solving Tenth Order Boundary Value Problems

Absolutely stable difference scheme for a general class of singular perturbation problems

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