On a nonlocal problem for the Laplace equation in the unit ball with fractional boundary conditions

2015 ◽  
Vol 39 (5) ◽  
pp. 1121-1128 ◽  
Author(s):  
Mokhtar Kirane ◽  
Berikbol T. Torebek
Keyword(s):  
Boundary Conditions ◽  
Unit Ball ◽  
Laplace Equation ◽  
Nonlocal Problem ◽  
Fractional Boundary Conditions

Hydrodynamic Coupling of Viscous and Non-Viscous Numerical Wave Solutions Within the Open-Source Hydrodynamics Framework REEF3D

2021 ◽  
Author(s):  
Weizhi Wang ◽  
Csaba Pákozdi ◽  
Arun Kamath ◽  
Tobias Martin ◽  
Hans Bihs
Keyword(s):  
High Resolution ◽  
Boundary Conditions ◽  
Open Source ◽  
Laplace Equation ◽  
Scale Model ◽  
Surface Boundary ◽  
Cfd Model ◽  
Hydrodynamic Coupling ◽  
Coupling Strategy ◽  
Numerical Wave

Abstract A comprehensive understanding of the marine environment in the offshore area requires phase-resolved wave information. For the far-field wave propagation, computational efficiency is crucial, as large spatial and temporal scales are involved. For the near-field extreme wave events and wave impacts, high resolution is required to resolve the flow details and turbulence. The combined use of a computationally efficient large-scale model and a high-resolution local-scale solver provides a solution the combines accuracy and efficiency. This article introduces a coupling strategy between the efficient fully nonlinear potential flow (FNPF) solver REEF3D::FNPF and the high-fidelity computational fluid dynamics (CFD) model REEF3D::CFD within in the open-source hydrodynamics framework REEF3D. REEF3D::FNPF solves the Laplace equation together with the boundary conditions on a sigma-coordinate. The free surface boundary conditions are discretised using high-order finite difference methods. The Laplace equation for the velocity potential is solved with a conjugated gradient solver preconditioned with geometric multi-grid provided by the open-source library hypre. The model is fully parallelised following the domain decomposition strategy and the MPI protocol. The waves calculated with the FNPF solver are used as wave generation boundary condition for the CFD based numerical wave tank REEF3D::CFD. The CFD model employs an interface capturing two-phase flow approach that can resolve complex wave structure interaction, including breaking wave kinematics and turbulent effects. The presented hydrodynamic coupling strategy is tested for various wave conditions and the accuracy is fully assessed.

Two point fractional boundary value problems with a fractional boundary condition

2018 ◽  
Vol 21 (2) ◽  
pp. 442-461 ◽  
Author(s):  
Jeffrey W. Lyons ◽  
Jeffrey T. Neugebauer
Keyword(s):  
Boundary Condition ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence Of Positive Solutions ◽  
Fractional Boundary Conditions ◽  
Fractional Boundary Value Problems

Abstract In this paper, we employ Krasnoseľskii’s fixed point theorem to show the existence of positive solutions to three different two point fractional boundary value problems with fractional boundary conditions. Also, nonexistence results are given.

scholarly journals A computing method for bending problem of thin plate on Pasternak foundation

2020 ◽  
Vol 12 (7) ◽  
pp. 168781402093933
Author(s):  
Jiarong Gan ◽  
Hong Yuan ◽  
Shanqing Li ◽  
Qifeng Peng ◽  
Huanliang Zhang
Keyword(s):  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Integral Equations ◽  
Thin Plate ◽  
Laplace Equation ◽  
Two Dimensional ◽  
Pasternak Foundation ◽  
Helmholtz Operator ◽  
Numerical Computing ◽  
Matlab Programming

The governing equation of the bending problem of simply supported thin plate on Pasternak foundation is degraded into two coupled lower order differential equations using the intermediate variable, which are a Helmholtz equation and a Laplace equation. A new solution of two-dimensional Helmholtz operator is proposed as shown in Appendix 1. The R-function and basic solutions of two-dimensional Helmholtz operator and Laplace operator are used to construct the corresponding quasi-Green function. The quasi-Green’s functions satisfy the homogeneous boundary conditions of the problem. The Helmholtz equation and Laplace equation are transformed into integral equations applying corresponding Green’s formula, the fundamental solution of the operator, and the boundary condition. A new boundary normalization equation is constructed to ensure the continuity of the integral kernels. The integral equations are discretized into the nonhomogeneous linear algebraic equations to proceed with numerical computing. Some numerical examples are given to verify the validity of the proposed method in calculating the problem with simple boundary conditions and polygonal boundary conditions. The required results are obtained through MATLAB programming. The convergence of the method is discussed. The comparison with the analytic solution shows a good agreement, and it demonstrates the feasibility and efficiency of the method in this article.

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