Non-linear multidimensional singular integral equations in two-dimensional fluid mechanics

Ladopoulos, E. G.

doi:10.1016/s0020-7462(99)00052-9

Cited by 15 publications

(8 citation statements)

References 22 publications

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“…The multi-dimensional Volterra-Fredholm nonlinear integral equations arise in many physics, chemistry, biology and engineering applications, and they provide a vital tool for modeling many problems. Particular cases of such nonlinear integral equations arise in the mathematical design of the temporal-spatio development of an epidemic [1][2][3][4]. They also appear in the theory of porous filtering, antenna problems in electromagnetic theory, fracture mechanics, aerodynamics, in the quantum effects of electromagnetic fields in the black body whose interior is filled by Kerr nonlinear crystal.…”

Section: ≔ {(mentioning

confidence: 99%

On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra–Fredholm integral equations

Elkot¹,

Zaky²,

Doha³

et al. 2021

Commun. Theor. Phys.

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While the approximate solutions of one-dimensional nonlinear Volterra–Fredholm integral equations with smooth kernels are now well understood, no systematic studies of the numerical solutions of their multi-dimensional counterparts exist. In this paper, we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra–Fredholm integral equations based on the multi-variate Legendre-collocation approach. Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view. Consequently, rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors. The existence and uniqueness of the numerical solution are established. Numerical experiments are provided to support the theoretical convergence analysis. The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones.

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Section: ≔ {(mentioning

confidence: 99%

On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra–Fredholm integral equations

Elkot¹,

Zaky²,

Doha³

et al. 2021

Commun. Theor. Phys.

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“…The azimuthal angle θ indicates the position of the blade in the track circle, θ = θ 0 + ωt, in which ω is the angular velocity around the rotation center O and θ 0 is the initial azimuthal angle of the blade. At present, prediction methods for the hydrodynamic performance of vertical axis tidal turbines can broadly be divided into the following four types: Stream-tube model methods [4][5][6], vortex model methods [7][8][9], computational fluid dynamic (CFD) methods [2,10,11] and model test methods [12][13][14][15]. Stream-tube model methods are based on the momentum theorem and have difficulties predicting the flow field characteristics of turbines when the speed ratio increases to a point past which the momentum equation may diverge, leading to no solution.…”

Section: Methodsmentioning

confidence: 99%

An Unsteady Boundary Element Model for Hydrodynamic Performance of a Multi-Blade Vertical-Axis Tidal Turbine

Chen

2018

Water

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An unsteady boundary element model is developed to simulate the unsteady flow induced by the motion of a multi-blade vertical axis turbine. The distribution of the sources, bound vortices and wake vortices of the blades are given in detail. In addition, to make the numerical solution more robust, the Kutta condition is also introduced. The developed model is used to predict the hydrodynamic performance of a vertical axis tidal turbine and is validated by comparison with experimental data and other numerical solutions available in the literature. Good agreement is achieved and the calculation of the proposed model is simpler and more efficient than prior numerical solutions. The proposed model shows its capability for future profile design and optimization of vertical axis tidal turbines.

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“…According to the generalized Stokes theorem [15], for (∇, a) B − a (∇, B), we can write the following transformations:…”

Section: Definitionmentioning

confidence: 99%

Extension of Calculus Operations in Cartesian Tensor Analysis

Lu¹,

Peng²,

Wei³

2020

Mathematics

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In this paper, we derive and propose basic differential operations and generalized Green’s integral theorems applicable to multidimensional spaces based on Cartesian tensor analysis to solve some nonlinear problems in smooth spaces in the necessary dimensions. In practical applications, the theorem can be applied to numerical analysis in the conservation law, effectively reducing the dimensions of high-dimensional problems and reducing the computational difficulty, which can be effectively used in the solution of complex dimensional mechanical problems.

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Non-linear multidimensional singular integral equations in two-dimensional fluid mechanics

Cited by 15 publications

References 22 publications

On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra–Fredholm integral equations

On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra–Fredholm integral equations

An Unsteady Boundary Element Model for Hydrodynamic Performance of a Multi-Blade Vertical-Axis Tidal Turbine

Extension of Calculus Operations in Cartesian Tensor Analysis

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