scholarly journals Approximating the Solution Stochastic Process of the Random Cauchy One-Dimensional Heat Model

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
A. Navarro-Quiles ◽  
J.-V. Romero ◽  
M.-D. Roselló ◽  
M. A. Sohaly
Keyword(s):  
Stochastic Process ◽  
Standard Deviation ◽  
Finite Difference ◽  
Numerical Solution ◽  
Numerical Scheme ◽  
Sufficient Conditions ◽  
Numerical Approximations ◽  
One Dimensional ◽  
The Mean ◽  
Theoretical Results

This paper deals with the numerical solution of the random Cauchy one-dimensional heat model. We propose a random finite difference numerical scheme to construct numerical approximations to the solution stochastic process. We establish sufficient conditions in order to guarantee the consistency and stability of the proposed random numerical scheme. The theoretical results are illustrated by means of an example where reliable approximations of the mean and standard deviation to the solution stochastic process are given.

Rigorous time-domain analysis of statistically oriented graphene sheet fluctuations

2017 ◽  
Vol 36 (5) ◽  
pp. 1351-1363 ◽  
Author(s):  
Athanasios N. Papadimopoulos ◽  
Stamatios A. Amanatiadis ◽  
Nikolaos V. Kantartzis ◽  
Theodoros T. Zygiridis ◽  
Theodoros D. Tsiboukis
Keyword(s):  
Monte Carlo ◽  
Standard Deviation ◽  
Finite Difference ◽  
Surface Waves ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Mean Value ◽  
Content Type ◽  
The Mean ◽  
Difference Time

Purpose Important statistical variations are likely to appear in the propagation of surface plasmon polariton waves atop the surface of graphene sheets, degrading the expected performance of real-life THz applications. This paper aims to introduce an efficient numerical algorithm that is able to accurately and rapidly predict the influence of material-based uncertainties for diverse graphene configurations. Design/methodology/approach Initially, the surface conductivity of graphene is described at the far infrared spectrum and the uncertainties of its main parameters, namely, the chemical potential and the relaxation time, on the propagation properties of the surface waves are investigated, unveiling a considerable impact. Furthermore, the demanding two-dimensional material is numerically modeled as a surface boundary through a frequency-dependent finite-difference time-domain scheme, while a robust stochastic realization is accordingly developed. Findings The mean value and standard deviation of the propagating surface waves are extracted through a single-pass simulation in contrast to the laborious Monte Carlo technique, proving the accomplished high efficiency. Moreover, numerical results, including graphene’s surface current density and electric field distribution, indicate the notable precision, stability and convergence of the new graphene-based stochastic time-domain method in terms of the mean value and the order of magnitude of the standard deviation. Originality/value The combined uncertainties of the main parameters in graphene layers are modeled through a high-performance stochastic numerical algorithm, based on the finite-difference time-domain method. The significant accuracy of the numerical results, compared to the cumbersome Monte Carlo analysis, renders the featured technique a flexible computational tool that is able to enhance the design of graphene THz devices due to the uncertainty prediction.

Phase shift of solitary wave in a one-dimensional granular chain

2019 ◽  
Vol 33 (10) ◽  
pp. 1950085
Author(s):  
Xian-Qing Yang ◽  
Yao Yang ◽  
Yang Jiao ◽  
Wei Zhang
Keyword(s):  
Phase Shift ◽  
Solitary Wave ◽  
Numerical Scheme ◽  
Binary Collision ◽  
One Dimensional ◽  
Binary Collision Approximation ◽  
Fifth Order ◽  
The Waves ◽  
Theoretical Results ◽  
Collision Approximation

In this paper, both the fifth-order Runge–Kutta numerical scheme and binary collision approximation are used to study the phase shift. Both numerical and theoretical results are shown that the solitary wave after head-on collision propagates along the chain behind the reference wave in both even and odd numbers of grain chains. It is the well-known feature of the appearance of the phase shift. Those results are in agreement with the experimental results. Furthermore, it is found that the phase shift is not only related to the collision position of the waves, but also to the position where the time is measured. The value of phase shift increases nonmonotonously with increasing the velocity of the opposite propagation of the wave. Binary collision approximation is applied to analyze the phase shift, and it is found that theoretical results agree well with numerical results, especially in the case of phase shift in odd chain.

scholarly journals Analytic-Numerical Solution of Random Parabolic Models: A Mean Square Fourier Transform Approach

2018 ◽  
Vol 23 (1) ◽  
pp. 79-100 ◽  
Author(s):  
Maria-Consuelo Casaban ◽  
Juan-Carlos Cortes ◽  
Lucas Jodar
Keyword(s):  
Stochastic Process ◽  
Fourier Transform ◽  
Standard Deviation ◽  
Stochastic Processes ◽  
Numerical Solution ◽  
Integral Form ◽  
Initial Condition ◽  
Mean Square ◽  
Improper Integrals ◽  
Gauss Hermite Quadrature

This paper deals with the construction of mean square analytic-numerical solution of parabolic partial differential problems where both initial condition and coefficients are stochastic processes. By using a random Fourier transform, an inf- nite integral form of the solution stochastic process is firstly obtained. Afterwards, explicit expressions for the expectation and standard deviation of the solution are obtained. Since these expressions depend upon random improper integrals, which are not computable in an exact manner, random Gauss-Hermite quadrature formulae are introduced throughout an illustrative example.

A Numerical Model for the Radial Flow Through Porous and Deformable Shells

2004 ◽  
Vol 4 (1) ◽  
pp. 34-47 ◽  
Author(s):  
Francisco J. Gaspar ◽  
Francisco J. Lisbona ◽  
Petr N. Vabishchevich
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Energy Estimates ◽  
One Dimensional ◽  
Consolidation Model ◽  
Finite Difference Discretization ◽  
Flow Through ◽  
The One ◽  
Theoretical Results ◽  
Difference Discretization

AbstractEnergy estimates and convergence analysis of finite difference methods for Biot's consolidation model are presented for several types of radial ow. The model is written by a system of partial differential equations which depend on an integer parameter (n = 0; 1; 2) corresponding to the one-dimensional ow through a deformable slab and the radial ow through an elastic cylindrical or spherical shell respectively. The finite difference discretization is performed on staggered grids using separated points for the approximation of pressure and displacements. Numerical results are given to illustrate the obtained theoretical results.

scholarly journals On the numerical solution of the one-dimensional convection-diffusion equation

2005 ◽  
Vol 2005 (1) ◽  
pp. 61-74 ◽  
Author(s):  
Mehdi Dehghan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Diffusion Equation ◽  
Numerical Solution ◽  
Partial Differential ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
The One

The numerical solution of convection-diffusion transport problems arises in many important applications in science and engineering. These problems occur in many applications such as in the transport of air and ground water pollutants, oil reservoir flow, in the modeling of semiconductors, and so forth. This paper describes several finite difference schemes for solving the one-dimensional convection-diffusion equation with constant coefficients. In this research the use of modified equivalent partial differential equation (MEPDE) as a means of estimating the order of accuracy of a given finite difference technique is emphasized. This approach can unify the deduction of arbitrary techniques for the numerical solution of convection-diffusion equation. It is also used to develop new methods of high accuracy. This approach allows simple comparison of the errors associated with the partial differential equation. Various difference approximations are derived for the one-dimensional constant coefficient convection-diffusion equation. The results of a numerical experiment are provided, to verify the efficiency of the designed new algorithms. The paper ends with a concluding remark.

scholarly journals A Finite Difference Method for the Variational p-Laplacian

2021 ◽  
Vol 90 (1) ◽  
Author(s):  
Félix del Teso ◽  
Erik Lindgren
Keyword(s):  
Nonlinear System ◽  
Finite Difference ◽  
Numerical Scheme ◽  
Explicit Method ◽  
Dirichlet Problems ◽  
Finite Difference Discretization ◽  
Newton Raphson ◽  
First Time ◽  
Theoretical Results ◽  
Difference Discretization

AbstractWe propose a new monotone finite difference discretization for the variational p-Laplace operator, $$\Delta _pu=\text{ div }(|\nabla u|^{p-2}\nabla u),$$ Δ p u = div ( | ∇ u | p - 2 ∇ u ) , and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

Excursions above a fixed level by n-dimensional random fields

1976 ◽  
Vol 13 (2) ◽  
pp. 276-289 ◽  
Author(s):  
Robert J. Adler
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
Random Fields ◽  
Mean Value ◽  
Regularity Conditions ◽  
Gaussian Field ◽  
Differential Topology ◽  
One Dimensional ◽  
Excursion Set ◽  
The Mean

For an n-dimensional random field X(t) we define the excursion set A of X(t) by A = [t ∊ S: X(t) ≧ u] for real u and compact S ⊂ Rn. We obtain a generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields via a characteristic of the set A related to the Euler characteristic of differential topology. When X(t) is a homogeneous Gaussian field satisfying certain regularity conditions we obtain an explicit formula for the mean value of this characteristic.

Modeling and Test of an Electropneumatic Servovalve Controlled Long Rodless Actuator

1996 ◽  
Vol 118 (3) ◽  
pp. 457-462 ◽  
Author(s):  
X. Lin ◽  
F. Spettel ◽  
S. Scavarda
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Transmission Line ◽  
Dynamic Behavior ◽  
Numerical Scheme ◽  
Distributed Parameter ◽  
One Dimensional ◽  
Flow Parameters ◽  
Characteristics Method ◽  
Cylinder Model

The usual way of modeling an electropneumatic servodrive system consists of supposing the homogeneity of the flow parameters in the cylinder. In order to study the influence of the transmission line effects on the dynamic behavior of a long pneumatic rodless cylinder, a generalized one-dimensional distributed parameter cylinder model is presented. The proposed numerical scheme combining the finite difference and the “characteristics” method enables the simulate of a long rodless cylinder by taking into account the servovalve boundary conditions and the piston movement. The simulated results are compared with experimental results.

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