Numerical solutions of the first boundary-value problem for the telegraph equation on open contours

1994 ◽  
Vol 71 (4) ◽  
pp. 2581-2585
Author(s):  
R. S. Khapko
Keyword(s):  
Boundary Value Problem ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Telegraph Equation

scholarly journals An Integral Equation Formalism for Integrating a Nonlinear Initial-Boundary Value Problem for a Boussinesq Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
T. S. Jang
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Boussinesq Equation ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Coupled System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Successive Approximations ◽  
Initial Boundary Value

In this paper, a new nonlinear initial-boundary value problem for a Boussinesq equation is formulated. And a coupled system of nonlinear integral equations, equivalent to the new initial-boundary value problem, is constructed for integrating the initial-boundary value problem, but which is inherently different from other conventional formulations for integral equations. For the numerical solutions, successive approximations are applied, which leads to a functional iterative formula. A propagating solitary wave is simulated via iterating the formula, which is in good agreement with the known exact solution.

Boundary-Layer Similarity Near the Edge of a Rotating Disk

1975 ◽  
Vol 42 (3) ◽  
pp. 584-590 ◽  
Author(s):  
R. J. Bodonyi ◽  
K. Stewartson
Keyword(s):  
Boundary Layer ◽  
Approximate Solution ◽  
Flow Field ◽  
Boundary Value Problem ◽  
Asymptotic Expansions ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Rotating Disk ◽  
Angular Speed ◽  
Rotating Fluid

Numerical solutions of the similarity equations governing the flow near the edge of a finite rotating disk are found to be possible only for −2.06626 ≤ α ≤ 1, where α is the ratio of the disk’s angular speed to that of the rigidly rotating fluid far from the disk. Furthermore, for α ≤ −1 the solutions of the boundary-value problem are not unique, and along one of the solution branches a singular structure of the flow field is approached as α → −1. Using the method of matched asymptotic expansions an approximate solution is found along the singular branch which explains some of the problems encountered in finding numerical solutions.

scholarly journals Hydrodynamic interactions among multiple circular cylinders in an inviscid flow

2012 ◽  
Vol 712 ◽  
pp. 505-530 ◽  
Author(s):  
R. Sun ◽  
C. O. Ng
Keyword(s):  
Boundary Value Problem ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Cyclic Permutation ◽  
Neumann Boundary ◽  
Hydrodynamic Interactions ◽  
Circular Cylinders ◽  
Constructive Method ◽  
Nonlinear Phenomena ◽  
Inviscid Fluid

AbstractHydrodynamic interactions among multiple circular cylinders translating in an otherwise undisturbed inviscid fluid are theoretically investigated. A constructive method for solving a Neumann boundary-value problem in a domain outside $N$ circles (one kind of Hilbert boundary-value problem in the complex plane) is presented in the study to derive the velocity potential of the liquid. The method employs successive offset functions combined with a ‘generalized cyclic permutation’ in turn to satisfy the impenetrable boundary condition on each circle. The complex potential is therefore expressed as $N$ isolated singularities in power series form and used to get instantaneous added masses of $N$ submerged circular cylinders. Then, based on the Hamilton variational principle, a dynamical equation of motion in vector form is derived to predict nonlinear translations of the submerged bodies under fully hydrodynamic interactions. Also, the equivalence of the energy-based Lagrangian framework and a momentum-type one in the two-dimensional body–liquid system is proved. It implies that the pressure integration around a submerged body is holographic, which provides information about velocities and accelerations of all bodies. The numerical solutions indicate some typical dynamical behaviours of more than two circular cylinders which reveal that interesting nonlinear phenomena would appear in such a system with simple physical assumptions.

On attached masses of panels oscillating in incompressible medium

2021 ◽  
pp. 56-64
Author(s):  
Valery A. BUZHINSKIY
Keyword(s):  
Boundary Value Problem ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Oscillation Mode ◽  
Boundary Elements ◽  
Boundary Integral ◽  
Solar Array ◽  
Attached Mass ◽  
Linear Superposition ◽  
Incompressible Medium

The paper discusses small oscillations of a panel in an incompressible medium. Air can be considered an incompressible medium during modal tests of solar array panels for spacecraft deployed on the ground in a lab environment. A panel is represented as a two-sided boundary surface. Conditions are determined for applicability of the potential motion of the medium. Calculation of the attached mass is reduced to the solution of the Neumann boundary value problem. To solve the boundary value problem, the method of boundary elements is used in the piecewise constant approximation variant, which provides a solution of the hypersingular boundary integral equation. Numerical solutions are obtained for the three fundamental modes of rectangular panels. The obtained numerical values are refined using non-linear Shanks transformation. Dependence of attached mass on panel elongation and the amount of the gap between its fragments is studied. For any in-plane oscillation mode of a panel fragment, the attached mass is determined using the principle of linear superposition. An estimate is given of the effect of the distance from the panel to the wall on the attached mass value. Key words: oscillations, incompressible medium, air, attached mass, rectangular panels, boundary elements method.

THE METHOD OF FUNDAMENTAL SOLUTIONS FOR AN INVERSE INTERNAL BOUNDARY VALUE PROBLEM FOR THE BIHARMONIC EQUATION

2009 ◽  
Vol 06 (04) ◽  
pp. 557-567 ◽  
Author(s):  
D. LESNIC ◽  
A. ZEB
Keyword(s):  
Boundary Value Problem ◽  
Biharmonic Equation ◽  
Numerical Solutions ◽  
Least Squares Method ◽  
Regularization Parameter ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Internal Boundary ◽  
Unknown Boundary

In this paper, an inverse internal boundary value problem associated to the biharmonic equation is considered. The problem consists of determining unknown boundary conditions from extra interior measurements. The method of fundamental solutions (MFS) is used to discretize the problem and the resulting ill-conditioned system of linear equations is solved using the Tikhonov regularization technique. It is shown that, unlike the least-squares method, the MFS-regularization numerical technique produces stable and accurate numerical solutions for an appropriate choice of the regularization parameter given by the L-curve criterion.

scholarly journals On Conditioning of a Schwarz Method for Singularly Perturbed Convection–diffusion Equations in the Case of Disturbances in the Data of the Boundary-value Problem

2003 ◽  
Vol 3 (3) ◽  
pp. 459-487 ◽  
Author(s):  
Gregorii I. Shishkin
Keyword(s):  
Boundary Value Problem ◽  
Numerical Solutions ◽  
Difference Operator ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Maximum Norm ◽  
Schwarz Method ◽  
Base Scheme ◽  
Decomposition Scheme ◽  
Number Of Iterations

AbstractIn this paper we discuss conditioning of a discrete Schwarz method on piecewise–uniform meshes with an example of a one-dimensional singularly perturbed boundary-value problem. We consider a Dirichlet problem for singularly perturbed ordinary differential equations with convection terms and a small perturbation parameter ε. To solve the problem numerically we use an ε-uniformly convergent (in the maximum norm) difference scheme on special piecewise–uniform meshes. For this base scheme we construct a decomposition scheme based on a Schwarz technique with overlapping subdomains, which converges ε-uniformly with respect to both the number of mesh points and the number of iterations. The step-size of such special meshes is extremely small in the neighborhood of the layer and changes sharply on its boundary, that (as was shown by A.A. Samarskii) can generally lead to a loss of well-conditioning of the above schemes. For the decomposition scheme we study the conditioning of the system (difference scheme) and the conditioning of the system matrix (difference operator), and also the influence of perturbations in the data of the boundary-value problem on disturbances of its numerical solutions. We derive estimates for the disturbances of the numerical solutions (in the maximum norm) depending on the subdomain in which the disturbance of the data appears. It is shown that the condition number of the difference operator associated with the Schwarz method, just as for the base scheme, is not ε-uniformly bounded. However, these difference schemes are well-conditioned ε-uniformly (with the ε-uniform estimate for the condition number being the same as for the schemes on uniform meshes for regular problems) when the right-hand side of the discrete equations is considered in a “natural” norm, i.e., in the maximum norm with a special weight multiplier. In the case of the boundary-value problem with perturbed data we give conditions under which the solution of the iterative scheme based on the overlapping Schwarz method is convergent ε-uniformly to the solution of this Dirichlet problem as the number of mesh points and the number of iterations increase.

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