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Liu, Chein‐Shan

doi:10.1016/j.cam.2012.05.006

Cited by 13 publications

(3 citation statements)

References 17 publications

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“…As a practical application of the proposed iterative schemes, we developed a Lie symmetry method based on the Lie group SL(2, R) to solve the second-order nonlinear boundary-value problem. This Lie symmetry method was first developed in [34] for computing the eigenvalues of the generalized Sturm-Liouville problem.…”

Section: A Lie Symmetry Methodsmentioning

confidence: 99%

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

Liu,

Chang,

Kuo

2024

Symmetry

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In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values f′(r), f″(r), and f‴(r) of a nonlinear equation f(x)=0 with r being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound 22/3=1.587 with three function evaluations and over the bound 21/2=1.414 with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the Dzˇunic´ method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed.

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Section: A Lie Symmetry Methodsmentioning

confidence: 99%

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

Liu,

Chang,

Kuo

2024

Symmetry

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“…First example of a double eigenvalue. The following example from [12] was also discussed in [16][17][18]. Consider the spectral problem…”

Section: Examplesmentioning

confidence: 99%

Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter

Aliyev

2023

Preprint

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In the paper we study Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter. The necessary and sufficient conditions for minimality and completeness of the chosen system of root functions of the corresponding operator were given in two forms, one of which can be done by direct computations. This computationally simpler way was known for the affine case and even found some applications in the literature. But, until now, it was not developed for the quadratic case. The aim of the present paper is to fill this gap for the quadratic case and to consider the affine and quadratic cases together in a unified way. 2020 MSC: 34B24, 34L10.

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“…The Lie-group S L(2, R) has been used to develop the shooting method to solve the generalized Sturm-Liouville problem by Liu (2012), to solve a singular ϕ-Laplacian nonlinear ODE by Liu (2013a), and to solve a nonlinear heat transfer equation by Hashemi (2015).…”

Section: Introductionmentioning

confidence: 99%

The Integrating Factors for Riccati and Abel Differential Equations

Liu¹

2015

JMR

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We can recast the Riccati and Abel differential equations into new forms in terms of introduced integrating factors. Therefore, the Lie-type systems endowing with transformation Lie-groups S L(2, R) can be obtained. The solution of second-order linear homogeneous differential equation is an integrating factor of the corresponding Riccati differential equation. The numerical schemes which are developed to fulfil the Lie-group property have better accuracy and stability than other schemes. We demonstrate that upon applying the group-preserving scheme (GPS) to the logistic differential equation, it is not only qualitatively correct for all values of time stepsize h, and is also the most accurate one among all numerical schemes compared in this paper. The group-preserving schemes derived for the Riccati differential equation, second-order linear homogeneous and non-homogeneous differential equations, the Abel differential equation and higher-order nonlinear differential equations all have accuracy better than O(h 2 ).

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Computing the eigenvalues of the generalized Sturm–Liouville problems based on the Lie-group SL(2,R)

Cited by 13 publications

References 17 publications

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter

The Integrating Factors for Riccati and Abel Differential Equations

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