Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates

Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.

On the Analytical Inversion of Solver Matrices Used in Numerical Approximations for the Diffusion Equation

The goal of this paper is to introduce an analytical approach for the inversion of nxn solver matrices, which are typically used in Finite Difference Method approximations. In the present case, they are used to solve the Diffusion Equation numerically, since in many physics and engineering fields, partial differential equations cannot be solved analytically. The method presented in this work is primarily formulated for cylindrical coordinates, which are often used in Gas Release Experiments as those described in [8]. However, it is possible to introduce a generalized method, which also allows solutions for Cartesian solvers. The advantage of having the explicit inverse is considerable, since the computational effort is reduced. In this paper we also carry out an investigation on the eigenvalues of the backward and forward solver matrix in order to determine an optimal range for the discretization parameters.

Numerical Solution of Heat Equation in Polar Cylindrical Coordinates by the Meshless Method of Lines

We propose a numerical solution to the heat equation in polar cylindrical coordinates by using the meshless method of lines approach. The space variables are discretized by multiquadric radial basis function, and time integration is performed by using the Runge-Kutta method of order 4. In radial basis functions (RBFs), much of the research are devoted to the partial differential equations in rectangular coordinates. This work is an attempt to explore the versatility of RBFs in nonrectangular coordinates as well. The results show that application of RBFs is equally good in polar cylindrical coordinates. Comparison with other cited works confirms that the present approach is accurate as well as easy to implement to problems in higher dimensions.

An analysis of axisymmetric Sezawa waves in elastic solids

The wave propagation in elastic solids covered by a thin layer has received significant attention due to the existence of Sezawa waves in many applications such as medical imaging. With a Helmholtz decomposition in cylindrical coordinates and subsequent solutions with Bessel functions, it is found that the velocity of such Sezawa waves is the same as the one in Cartesian coordinates, but the displacement will be decaying along the radius with eventual conversion to plane waves. The decaying with radius exhibits a strong contrast to the uniform displacement in the Cartesian formulation, and the asymptotic approximation is accurate in the range about one wavelength away from the origin. The displacement components in the vicinity of origin are naturally given in Bessel functions which can be singular, making it more suitable to analyze waves excited by a point source with solutions from cylindrical coordinates. This is particularly important in extracting vital wave properties and reconstructing the waveform in the vicinity of source of excitation with measurement data from the outer region.

The Emerging Field Trends Erosion-Free Electric Hall Thrusters Systems

The Hall-type accelerator with closed Hall current and open (that is unbounded by metal or dielectric) walls was proposed and considered both theoretically and experimentally. The novelty of this accelerator is the use of a virtual parallel surface of the anode and the cathode due to the principle of equipotentialization of magnetic field lines, which allows to avoid sputtering of the cathode surface and preserve the dynamics of accelerated ions. The formation of the actual traction beam should be due to the acceleration of ions with the accumulated positive bulk charge. A two-dimensional hybrid model in cylindrical coordinates is created in the framework of which the possibility of creation a positive space charge at the system axes is shown. It is shown that the ions flow from the hump of electrical potential can lead to the creation of a powerful ion flow, which moves along the symmetry axis in both sides from the center.