scholarly journals Cubic Spline Iterative Method for Poisson’s Equation in Cylindrical Polar Coordinates

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
R. K. Mohanty ◽  
Rajive Kumar ◽  
Vijay Dahiya
Keyword(s):  
Iterative Method ◽  
Spline Approximation ◽  
Cubic Spline ◽  
Poisson’S Equation ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Poisson's Equation ◽  
Polar Coordinate ◽  
Cylindrical Polar Coordinates ◽  
Diffusion Convection

Using nonpolynomial cubic spline approximation in x- and finite difference in y-direction, we discuss a numerical approximation of O(k2+h4) for the solutions of diffusion-convection equation, where k>0 and h>0 are grid sizes in y- and x-coordinates, respectively. We also extend our technique to polar coordinate system and obtain high-order numerical scheme for Poisson’s equation in cylindrical polar coordinates. Iterative method of the proposed method is discussed, and numerical examples are given in support of the theoretical results.

Additional Separated-Variable Solutions of the Biharmonic Equation in Polar Coordinates

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
I. H. Stampouloglou ◽  
E. E. Theotokoglou
Keyword(s):  
Fourth Order ◽  
Biharmonic Equation ◽  
Direct Determination ◽  
Additional Term ◽  
Elastic Solid ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Radial Coordinate ◽  
Displacement Fields ◽  
Polar Coordinate

From the biharmonic equation of the plane problem in the polar coordinate system and taking into account the variable-separable form of the partial solutions, a homogeneous ordinary differential equation (ODE) of the fourth order is deduced. Our study is based on the investigation of the behavior of the coefficients of the above fourth order ODE, which are functions of the radial coordinate r. According to the proposed investigation additional terms, φ¯−m(r,θ)(1≤m≤n) other than the usually tabulated in the Michell solution (1899, “On the Direct Determination of Stress in an Elastic Solid, With Application to the Theory of Plates,” Proc. Lond. Math. Soc., 31, pp. 100–124) are found. Finally the stress and the displacement fields due to each one additional term of φ¯−m(r,θ) are determined.

scholarly journals Determination of Ship Approach Parameters in the Polar Coordinates System

2014 ◽  
Vol 96 (1) ◽  
pp. 1-8
Author(s):  
Andrzej Banachowicz ◽  
Adam Wolski
Keyword(s):  
Coordinate System ◽  
Target Position ◽  
Geometric Interpretation ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Polar Coordinate ◽  
Radar Measurements ◽  
Cartesian Coordinate ◽  
Radar Antenna

Abstract An essential aspect of the safety of navigation is avoiding collisions with other vessels and natural or man made navigational obstructions. To solve this kind of problem the navigator relies on automatic anti-collision ARPA systems, or uses a geometric method and makes radar plots. In both cases radar measurements are made: bearing (or relative bearing) on the target position and distance, both naturally expressed in the polar coordinates system originating at the radar antenna. We first convert original measurements to an ortho-Cartesian coordinate system. Then we solve collision avoiding problems in rectangular planar coordinates, and the results are transformed to the polar coordinate system. This article presents a method for an analysis of a collision situation at sea performed directly in the polar coordinate system. This approach enables a simpler geometric interpretation of a collision situation

scholarly journals Cubic Spline Method for 1D Wave Equation in Polar Coordinates

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
R. K. Mohanty ◽  
Rajive Kumar ◽  
Vijay Dahiya
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Spline Approximation ◽  
Cubic Spline ◽  
Polar Coordinates ◽  
Spline Method ◽  
Implicit Difference Scheme ◽  
Time Direction ◽  
Numerical Examples ◽  
Theoretical Results

Using nonpolynomial cubic spline approximation in space and finite difference in time direction, we discuss three-level implicit difference scheme of O(k2+h4) for the numerical solution of 1D wave equations in polar coordinates, where k>0 and h>0 are grid sizes in time and space coordinates, respectively. The proposed method is applicable to problems with singularity. Stability theory of the proposed method is discussed, and numerical examples are given in support of the theoretical results.

scholarly journals The Symplectic Method in Polar Coordinates for Linear Viscoelastic Materials

2015 ◽  
Vol 9 (1) ◽  
pp. 130-134
Author(s):  
W.X. Zhang ◽  
Y. Bai
Keyword(s):  
Analytical Form ◽  
Fundamental Solutions ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Symplectic Method ◽  
Polar Coordinate ◽  
Linear Viscoelastic ◽  
Annular Sector ◽  
Particular Solution ◽  
Variable Substitution

In this study, the symplectic method is applied to a two-dimensional annular-sector viscoelastic domain under the polar coordinate system. By applying variable separation approach, all fundamental solutions are derived in analytical form. Furthermore, using the method of variable substitution, lateral conditions are transformed into finding a particular solution for the governing equations, and this particular solution is derived with the use of eigensolution expansion. In the numerical example, the boundary condition problem is discussed in detail to analyze the stress response of viscoelastic solids.

Sign in / Sign up

Export Citation Format

Share Document