Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions

Liu, Li-Bin; Liu, Huan-Wen; Chen, Yanping

doi:10.1016/j.amc.2011.01.047

Cited by 18 publications

(9 citation statements)

References 17 publications

(16 reference statements)

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“…For different number of elements = 8, 16, 32, 64, and 128, Examples 4 and 5 are solved using the present method. Maximum absolute errors of the numerical solutions are calculated and compared with those reported by [13,14], for different methods, in Tables 6 and 7. The present approach is giving much more accurate results than the others.…”

Section: Numerical Examples and Discussionmentioning

confidence: 99%

“…A system of (2 + 2) equations is obtained from (9) to (13). It includes all parameters of the system and the dependent variables at the boundaries.…”

Section: Symbolic Solutionmentioning

confidence: 99%

“…Such BVPs have wide application in astronomy, biology, boundary layer theory, deflection in cables, diffusion process, electromagnetism, heat transfer, and other topics. It is well known that closed form analytical solution of such problems cannot be obtained in many cases; therefore, numerical techniques such as collocation method [1,2], Bspline interpolation [3], Hermite cubic collocation [4][5][6], finite difference method [7][8][9], nonlinear shooting method [10], geometric Hermite interpolation [11], quintic B-spline collocation method [12], polynomial and nonpolynomial spline approaches [13][14][15], quartic spline solution [16], cubic spline collocation method [17], and finite volume element method [18] are frequently used.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the decoupling technique [4] used to solve elliptic problems with Neumann and Dirichlet conditions is a particular case of present technique. In this paper, linear and nonlinear boundary value problems reported in recent papers [2,3,7,9,13,14] are solved using CHCM. It is worth mentioning that CHCM is giving better results than finite difference method, finite element method, finite volume method, Bspline method, and polynomial and nonpolynomial spline approach with fourth order of convergence.…”

Section: Introductionmentioning

confidence: 99%

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Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet, Neumann, and Robin Conditions

Ganaie

Arora

Kukreja

2014

International Journal of Engineering Mathematics

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Cubic Hermite collocation method is proposed to solve two point linear and nonlinear boundary value problems subject to Dirichlet, Neumann, and Robin conditions. Using several examples, it is shown that the scheme achieves the order of convergence as four, which is superior to various well known methods like finite difference method, finite volume method, orthogonal collocation method, and polynomial and nonpolynomial splines and B-spline method. Numerical results for both linear and nonlinear cases are presented to demonstrate the effectiveness of the scheme. Proposed TechniqueIn the present method, the domain is divided into finite elements and then orthogonal collocation method with cubic Hermite as basis function is applied within each element.

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Section: Numerical Examples and Discussionmentioning

confidence: 99%

“…A system of (2 + 2) equations is obtained from (9) to (13). It includes all parameters of the system and the dependent variables at the boundaries.…”

Section: Symbolic Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet, Neumann, and Robin Conditions

Ganaie

Arora

Kukreja

2014

International Journal of Engineering Mathematics

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“…A smart nonstandard finite difference scheme has been proposed by Erdogan and Ozis [11] for solving second-order nonlinear BVPs. Besides that, Liu et al [12] solve BVPs with Neumann type by using polynomial spline approach. The aim of this research is to propose a three-step block method to solve the BVPs directly using multiple shooting techniques with variable step size strategy.…”

Section: Introductionmentioning

confidence: 99%

Three-Step Block Method for Solving Nonlinear Boundary Value Problems

See

Majid

Suleiman

2014

Abstract and Applied Analysis

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We propose a three-step block method of Adam’s type to solve nonlinear second-order two-point boundary value problems of Dirichlet type and Neumann type directly. We also extend this method to solve the system of second-order boundary value problems which have the same or different two boundary conditions. The method will be implemented in predictor corrector mode and obtain the approximate solutions at three points simultaneously using variable step size strategy. The proposed block method will be adapted with multiple shooting techniques via the three-step iterative method. The boundary value problem will be solved without reducing to first-order equations. The numerical results are presented to demonstrate the effectiveness of the proposed method.

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Quadratic spline collocation method and efficient preconditioner for the Helmholtz equation with the Sommerfeld boundary conditions

Luo

Huang

et al. 2016

Japan J. Indust. Appl. Math.

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Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions

Cited by 18 publications

References 17 publications

Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet, Neumann, and Robin Conditions

Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet, Neumann, and Robin Conditions

Three-Step Block Method for Solving Nonlinear Boundary Value Problems

Quadratic spline collocation method and efficient preconditioner for the Helmholtz equation with the Sommerfeld boundary conditions

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