A hybrid asymptotic and augmented compact finite volume method for nonlinear singular two point boundary value problems

Zhao, Tengjin; Zhang, Zhiyue; Wang, Tongke

doi:10.1016/j.amc.2020.125745

Cited by 8 publications

(4 citation statements)

References 46 publications

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“…Its application extends to various geological phenomena, including block motion, earthquake faulting, and the sliding behavior of rocks in geological and geotechnical contexts. The model expresses the relationship between frictional resistance and sliding velocity as a power-law function, defined by the equation (37),…”

Section: 1: Power-law Friction Modelmentioning

confidence: 99%

“…In an endeavor to overcome this impediment, the concept of local fractional derivatives was introduced [30][31][32], leading to the enhancement of the fractional Taylor series expansion formula [33][34][35][36]. This refined formula has been successfully applied by scholars to solve univariate non-differentiable problems with explicit variable expressions [37,38]. Nevertheless, the challenge of effectively resolving differential equations that lack explicit variable expressions remains an ongoing area of research.…”

Section: Introductionmentioning

confidence: 99%

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Advanced Numerical Algorithm for Non-smoothness Differential Equations: Integrating Fractional Interpolation with Predictive- Corrective Techniques

Ye,

Chen,

Liu

et al. 2024

Preprint

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In this study, we investigate numerical methods for non-smooth differential equations (NSDEs), which are pivotal in simulating abrupt phenomena in natural and engineering systems. We introduce the fractional interpolation method (FIM), a novel technique that utilizes fractional power functions to approximate solutions at points where derivatives are infinite. This method’s principal innovation is its adept handling of NSDEs' inherent discontinuities, offering a stable and convergent solution framework. Our findings confirm that FIM is both theoretically sound and practically reliable. Through rigorous numerical experiments, we have demonstrated its superior performance compared to conventional high-order numerical methods and MATLAB’s built-in functions. To further affirm FIM’s practicality, we applied it to two distinct non-smooth system types: systems with dry friction and binary wing systems with clearances. These applications substantiate the effectiveness of FIM and highlight its potential to tackle real-world challenges. Furthermore, this research equips scientists and engineers with a robust new tool for addressing NSDEs, setting the stage for further exploration and practical uses, especially in scenarios requiring accurate simulation of abrupt system behaviors. We anticipate the broader application of FIM in analyzing and designing non-smooth systems and are enthusiastic about its role in enhancing our understanding and prediction of complex dynamics across various natural and technical systems.

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Section: 1: Power-law Friction Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Advanced Numerical Algorithm for Non-smoothness Differential Equations: Integrating Fractional Interpolation with Predictive- Corrective Techniques

Ye,

Chen,

Liu

et al. 2024

Preprint

View full text Add to dashboard Cite

show abstract

“…High-order FEMs such as P 2 or P 3 have been developed with increased complexity and denser coefficient matrices. In the literature, there is a rich collection of different numerical methods such as spline collocation methods, discontinuous Galerkin (DG) methods, techniques for singular two-point BVPs, and nonlinear two-point BVPs, see [5][6][7][8][9][10][11][12][13][14][15][16][17] for an incomplete list.…”

Section: Introductionmentioning

confidence: 99%

Higher Order Finite Element Methods for Some One-dimensional Boundary Value Problems

Dong

Ruiz-Álvarez

2023

Research Reports on Computer Science

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In this paper, third-order compact and fourth-order finite element methods (FEMs) based on simple modifications of traditional FEMs are proposed for solving one-dimensional Sturm-Liouville boundary value problems (BVPs). The key idea is based on interpolation error estimates. A simple posterior error analysis of the original piecewise linear finite element space leads to a third-order accurate solution in the L2 norm, second-order in the H1, and the energy norm. The novel idea is also applied to obtain a fourth-order FEM based on the quadratic finite element space. The basis functions in the new fourth-order FEM are more compact compared with that of the classic cubic basis functions. Numerical examples presented in this paper have confirmed the convergence order and analysis. A generalization to a class of nonlinear two-point BVPs is also discussed and tested.

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“…Problems ( 1)-( 3) frequently occur in different fields of applied science and engineering. Some of these include chemical reactor, control and optimization, boundary layer theory, astrophysics theory, isothermal gas sphere, spherical cloud thermal behavior, nuclear physics, atomic structure, electrohydrodynamics and tumor growth problem (Verma et al 2020b;Ramos 2020, 2021b, a;Ramos and Vigo-Aguiar 2008;Moaaz et al 2021;Tomar 2021a;Verma et al 2020a;Van Gorder and Vajravelu 2008;Verma et al 2021b;Tomar 2021b;Pandey and Tomar 2021;Verma et al 2021a;Zhao et al 2021). Therefore, in physical models, these problems occur naturally, often due to an impulsive sink or source term and having singularity associated with the independent variable.…”

Section: Introductionmentioning

confidence: 99%

An effective method for solving singular boundary value problems with some relevant physical applications

2021

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In this paper, an effective iterative technique for analytical solutions of a class of nonlinear singular boundary value problems (SBVPs) occurring in different physical situations is presented. In constructing the recursive approach for the iterative solution components, a technique that relies on establishing a corresponding integral representation is used. This approach provides a highly accurate approximate solution with a few iterations. We also discussed the convergence of the methodology. Furthermore, we consider some numerical examples from various physical situations, including real-life problems, to demonstrate the efficacy of the technique. As seen in the presented numerical tests, our new proposal outperforms traditionally existing iterative methods in the literature. To demonstrate the comparable efficiency and robustness of the proposed iterative procedure, the numerical results are compared to existing results in the literature. It is apparently, an advanced approach to deal with various forms of highly nonlinear problems. This approach works well for SBVPs with nonlinear boundary conditions also as depicted by a numerical example.

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A hybrid asymptotic and augmented compact finite volume method for nonlinear singular two point boundary value problems

Cited by 8 publications

References 46 publications

Advanced Numerical Algorithm for Non-smoothness Differential Equations: Integrating Fractional Interpolation with Predictive- Corrective Techniques

Advanced Numerical Algorithm for Non-smoothness Differential Equations: Integrating Fractional Interpolation with Predictive- Corrective Techniques

Higher Order Finite Element Methods for Some One-dimensional Boundary Value Problems

An effective method for solving singular boundary value problems with some relevant physical applications

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