A numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations

Shanthi, V.; Ramanujam, N.

doi:10.1016/s0096-3003(01)00040-6

Cited by 40 publications

(17 citation statements)

References 17 publications

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“…Shishkin meshes are used in the FEM by Sun and Stynes [60,61]. An exponentially tted nite di erence scheme is proposed by Shanthi and Ramanujam [53] and a high-order nite volume method is given by Chen, He and Wu [7]. For the two-dimensional case of problem (3.2), a nonconforming 2 -element FEM is obtained by Nilssen, Tai and…”

Section: Tfpm For Singular Perturbation Problemsmentioning

confidence: 99%

The Tailored Finite Point Method

Han

Huang

2014

Computational Methods in Applied Mathematics

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Abstract:In this paper, a brief review of tailored nite point methods (TFPM) is given. The TFPM is a new approach to construct the numerical solutions of partial di erential equations. The TFPM has been tailored based on the local properties of the solution for each given problem. Especially, the TFPM is very e cient for solutions which are not smooth enough, e.g., for solutions possessing boundary/interior layers or solutions being highly oscillated. Recently, the TFPM has been applied to singular perturbation problems, the Helmholtz equation with high wave numbers, the rst-order wave equation in high frequency cases, transport equations with interface, second-order elliptic equations with rough or highly oscillatory coe cients, etc.

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Section: Tfpm For Singular Perturbation Problemsmentioning

confidence: 99%

The Tailored Finite Point Method

Han

Huang

2014

Computational Methods in Applied Mathematics

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“…The difference scheme (15) to be of (ℎ 3 ), the coefficients of ℎ 4 in (16) must be zero, and hence we obtain…”

Section: Derivation Of the Numerical Schemementioning

confidence: 99%

“…In the recent past, the numerical solution of fourth-order differential equations has been developed using multiderivative, finite element method, Ritz method, spline collocation, and finite difference method [15][16][17][18]. The determination of eigen values of self adjoint fourthorder differential equations was developed in [19] using finite difference scheme.…”

Section: Introductionmentioning

confidence: 99%

Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System

Jha

Mohanty

Chauhan

2013

Advances in Numerical Analysis

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Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourthorder ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method.

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“…Many mathematicians have studied this type of problems. For one dimensional problems, Roos and Stynes proposed a uniformly convergent conforming finite element method (FEM) [2], Semper developed a kind of conforming FEM [3], Sun and Stynes studied the FEM using Shishkin meshes technique [4,5], Shanthi and Ramanujam proposed an exponentially fitted finite difference scheme [6], Chen et al studied a high order finite volume method [7], etc. For two dimensional problems, Nilssen et al proposed a nonconforming H 2 -element FEM [8], Wang et al developed a modified Morley element [9], both of these articles proved that their methods have a half order uniform convergence rate in energy norm.…”

Section: Introductionmentioning

confidence: 99%

An equation decomposition method for the numerical solution of a fourth‐order elliptic singular perturbation problem

Han

Huang

2011

Numerical Methods Partial

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In this article, we propose a tailored finite point method (TFPM) for the numerical solution of a type of fourth-order singular perturbation problem in two dimensions based on the equation decomposition technique. Our finite point method has been tailored based on the local exponential basis functions. Furthermore, our TFPM satisfies the discrete maximum principle automatically. Our numerical examples show that our method has second order convergence rate in energy norm as ε → 0.

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A numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations

Cited by 40 publications

References 17 publications

The Tailored Finite Point Method

The Tailored Finite Point Method

Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System

An equation decomposition method for the numerical solution of a fourth‐order elliptic singular perturbation problem

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