scholarly journals Stable time-domain differential equations which reproduce the warm plasma dielectric tensor

2014 ◽  
Author(s):  
W. Tierens ◽  
D. De Zutter
Keyword(s):  
Differential Equations ◽  
Time Domain ◽  
Dielectric Tensor ◽  
Warm Plasma

scholarly journals A Formal Time-domain Approach to Cold Magnetised Plasmas

1988 ◽  
Vol 41 (1) ◽  
pp. 55 ◽  
Author(s):  
Werner Weiglhofer
Keyword(s):  
Differential Equations ◽  
Time Domain ◽  
Scalar Potential ◽  
Coupled System ◽  
Dielectric Tensor ◽  
Tensor Operator ◽  
Transient Wave ◽  
The Time Domain ◽  
Limiting Case ◽  
Wave Phenomena

Representations of the electromagnetic and the average velocity field for a cold magnetised plasma are derived in terms of scalar potential functions. These Hertz potentials are solutions of a coupled system of integro-differential equations of second Qrder. Different from other approaches, the analysis is carried out in the time domain and is therefore especially suited for the investigation of transient wave phenomena. Furthermore, the dielectric tensor operator of the plasma is derived. Mter solving the system of integro-differential equations for a special limiting case, the applicability of the method presented is demonstrated and generalisations are discussed.

scholarly journals Photocatalytic degradation as Davidson–Cole relaxation in time domain

2019 ◽  
Vol 09 (01) ◽  
pp. 1950006 ◽  
Author(s):  
C. L. WANG
Keyword(s):  
Experimental Data ◽  
Differential Equations ◽  
Photocatalytic Degradation ◽  
Gamma Function ◽  
Time Domain ◽  
Degradation Kinetics ◽  
Incomplete Gamma Function ◽  
Response Functions ◽  
Degradation Processes

Photocatalytic degradation processes of different materials are fitted with Mittag-Leffler function and incomplete gamma function, which are response functions for Cole–Cole relaxation and Davidson–Cole relaxation. The fitting results show that both functions can fit experimental data fairly well. The order of derivative in the kinetic differential equations can be either less, or greater than one. In the case of the order of derivative is greater than one, only incomplete gamma function is reasonable for describing the photocatalytic degradation. This work further confirms the existence of the universality in photocatalytic degradation kinetics.

Using Pontryagin's Maximum Principle to Solve One-Dimensional Optimization Problems, with and without Constraints, on an Iterative Analog Computer

SIMULATION ◽  
1965 ◽  
Vol 4 (6) ◽  
pp. 382-389 ◽  
Author(s):  
Hans L. Steinmetz
Keyword(s):  
Maximum Principle ◽  
Differential Equations ◽  
Time Domain ◽  
Pontryagin’S Maximum Principle ◽  
Analog Computer ◽  
Solution Time ◽  
Pontryagin's Maximum Principle ◽  
Computer Technique ◽  
One Dimensional ◽  
Time Required

An analog computer technique is presented which enables application of Pontryagin's maximum prin ciple to the problem of optimizing control systems. The key problem in using Pontryagin's maximum principle is the extremization of the Hamiltonian function at every instant of time. Since the analog computer is an excellent differential equation solver, it is of advantage to convert this task into a dynamic problem. The technique used to do this is based upon the steepest ascent method. The method is applied to a one-dimensional control problem; higher-di mensional control problems can be treated using the same approach. The argument that an analog computer can solve differential equations with only one independent variable, corresponding to machine time, is true only in a technical sense. In practice it is feasible for cer tain types of problems to integrate one set of differ ential equations sufficiently fast enough so that, while integrating another set of differential equations at a much slower rate, the solution error associated with this approach remains within acceptable limits. When using the analog computer in this way, one time domain always corresponds to the solution time required for solving the differential equations de scribing the system; a second time domain corre sponds to the solution time required for solving an auxiliary set of differential equations which has no direct relationship with the system. Technological improvements and innovations made in the analog computer field during the recent past have contributed to the successful application of this approach.

Electromagnetic modeling of 3-D sources over 2-D inhomogeneities in the time domain

Geophysics ◽  
1992 ◽  
Vol 57 (8) ◽  
pp. 994-1003 ◽  
Author(s):  
Michael Leppin
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Time Domain ◽  
Magnetic Induction ◽  
Host Rock ◽  
Vertical Component ◽  
Wavenumber Domain ◽  
Transient Electromagnetic ◽  
Wide Range ◽  
The Time Domain

A numerical method is presented by which the transient electromagnetic response of a two‐dimensional (2-D) conductor, embedded in a conductive host rock and excited by a rectangular current loop, can be modeled. This 2.5-D modeling problem has been formulated in the time domain in terms of a vector diffusion equation for the scattered magnetic induction, which is Fourier transformed into the spatial wavenumber domain in the strike direction of the conductor. To confine the region of solution of the diffusion equation to the conductive earth, boundary values for the components of the magnetic induction on the ground surface have been calculated by means of an integral transform of the vertical component of the magnetic induction at the air‐earth interface. The system of parabolic differential equations for the three magnetic components has been integrated for 9 to 15 discrete spatial wavenumbers ranging from [Formula: see text] to [Formula: see text] using an implicit homogeneous finite‐difference scheme. The discretization of the differential equations on a grid representing a cross‐section of the conductive earth results in a large, sparse system of linear equations, which is solved by the successive overrelaxation method. The three‐dimensional (3-D) response has been computed by an inverse Fourier transformation of the cubic spline interpolated scattered magnetic induction in the wavenumber domain using a digital filtering technique. To test the algorithm, responses have been computed for a two‐layered half‐space and a vertical prism embedded in a conductive host rock. These examples were then compared with results obtained analytically or numerically using frequency‐domain finite‐element and time‐domain integral equation methods. The new numerical procedure gives satisfactory results for a wide range of 2-D conductivity distributions with conductivity ratios exceeding 1:100, provided the grid is sufficiently refined at the corners of the conductivity anomalies.

Numerical vs. Experimental Predictions of Yaw Motion of Single Point Moored Vessel

2005 ◽  
Author(s):  
Mathieu Brotons ◽  
Philippe Jean
Keyword(s):  
Numerical Simulation ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Time Domain ◽  
Model Tests ◽  
Linear Differential Equations ◽  
Integration Scheme ◽  
Engineering Model ◽  
Non Linear ◽  
Linear Differential

The accurate prediction of SPM vessel yaw motion is important to its mooring system design. Inconsistencies have been observed between the numerical and model test predictions of offloading responses. In some cases, the numerical simulation predicted unstable yaw behavior of the vessel (fishtailing) while the model tests did not show such instability. This discrepancy between experiment and theory casts doubt as to whether the numerical simulation predicts correctly the vessel yaw motion. The work presented in this paper investigates the following two hypotheses to possibly explain the non-expected fishtailing in the numerical simulations: The mooring software may not accurately integrate non-linear differential equations that describe the yaw motion of the SPM vessel. Some damping terms may be under-estimated in the software (user input issue). To validate the integration scheme of the system of non-linear differential equations as implemented in the mooring software, a stability analysis has been conducted on a shuttle tanker moored to a West Africa deep water buoy. Variations of parameters like the hawser length, its axial stiffness and the vessel’s drag coefficients have been studied to explore their impacts on the vessel yaw stability. The approach is to identify without performing any time domain simulations, the domains of stability by linearizing the differential equations of SPM vessel’s yaw motion around its equilibrium point. The validity of the developed approach is then confirmed by performing time domain simulations of the same case. The second conjecture which may explain the non-expected fishtailing in numerical simulations was that some damping terms may be under-estimated. A semi empirical formula for the drag moment can be derived from rotation tests and comparisons were performed with the engineering model implemented in the mooring analysis software. The results show that by calibrating this damping term with the one derived from the experiments, the numerical simulations would match the stable yaw motion behavior as predicted during model tests. Following the above findings, a tool has been developed to fit the yaw drag moment engineering model based on experimental measurements, for any case of mooring analysis.

A Comparative Assessment of Simplified Models for Simulating Parametric Roll

2017 ◽  
Vol 139 (2) ◽  
Author(s):  
Abhilash S. Somayajula ◽  
Jeffrey Falzarano
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Time Domain ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Phenomenon ◽  
Concept Design ◽  
Parametric Roll ◽  
Statistical Confidence ◽  
Restoring Forces

The motion of a ship/offshore platform at sea is governed by a coupled set of nonlinear differential equations. In general, analytical solutions for such systems do not exist and recourse is taken to time-domain simulations to obtain numerical solutions. Each simulation is not only time consuming but also captures only a single realization of the many possible responses. In a design spiral when the concept design of a ship/platform is being iteratively changed, simulating multiple realizations for each interim design is impractical. An analytical approach is preferable as it provides the answer almost instantaneously and does not suffer from the drawback of requiring multiple realizations for statistical confidence. Analytical solutions only exist for simple systems, and hence, there is a need to simplify the nonlinear coupled differential equations into a simplified one degree-of-freedom (DOF) system. While simplified methods make the problem tenable, it is important to check that the system still reflects the dynamics of the complicated system. This paper systematically describes two of the popular simplified parametric roll models in the literature: Volterra GM and improved Grim effective wave (IGEW) roll models. A correction to the existing Volterra GM model described in current literature is proposed to more accurately capture the restoring forces. The simulated roll motion from each model is compared against a corresponding simulation from a nonlinear coupled time-domain simulation tool to check its veracity. Finally, the extent to which each of the models captures the nonlinear phenomenon accurately is discussed in detail.

scholarly journals Stability Analysis for Linear Uncertain Switched Systems in Infinite-time Domain

2021 ◽  
Author(s):  
Yadong Shu ◽  
Bo Li
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Time Domain ◽  
Switched Systems ◽  
Uncertainty Theory ◽  
Switched System ◽  
Almost Sure Stability ◽  
Infinite Time ◽  
Internal Property ◽  
Stability Issues

Abstract In this work, an uncertain switched system expressed as a series of uncertain differential equations is considered in depth. Stability issues have been widely investigated on switched systems while few results related to stability analysis for uncertain switched systems can be found. Due to such fact, three different stabilities, including stability in measure, almost sure stability and stability in mean, are comprehensively studied for linear uncertain switched systems in infinite-time domain. Internal property of the systems is able to be illustrated from different perspectives with the help of above stability analysis. By employing uncertainty theory and the feature of switched systems, corresponding judgement theorems of these stabilities are proposed and verified. An example with respect to stability in measure is provided to display the validness of the results derived.

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