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.

scholarly journals Hermite-Hadamard type inequalities for p-convex stochastic processes

2019 ◽  
Vol 9 (2) ◽  
pp. 148-153 ◽  
Author(s):  
Nurgul Okur ◽  
Imdat Işcan ◽  
Emine Yuksek Dizdar
Keyword(s):  
Stochastic Process ◽  
Mathematical Programming ◽  
Stochastic Processes ◽  
Mean Square ◽  
Important Place ◽  
Mathematical Programming Problems ◽  
Square Integrability ◽  
Convex Stochastic Processes ◽  
Convex Stochastic Process

In this study are investigated p-convex stochastic processes which are extensions of convex stochastic processes. A suitable example is also given for this process. In addition, in this case a p-convex stochastic process is increasing or decreasing, the relation with convexity is revealed. The concept of inequality as convexity has an important place in literature, since it provides a broader setting to study the optimization and mathematical programming problems. Therefore, Hermite-Hadamard type inequalities for p-convex stochastic processes and some boundaries for these inequalities are obtained in present study. It is used the concept of mean-square integrability for stochastic processes to obtain the above mentioned results.

scholarly journals Mean square calculus and random linear fractional differential equations: Theory and applications

2017 ◽  
Vol 2 (2) ◽  
pp. 317-328 ◽  
Author(s):  
C. Burgos ◽  
J.C Cortés ◽  
L. Villafuerte ◽  
R.J. Villanueva
Keyword(s):  
Stochastic Process ◽  
Second Order ◽  
Initial Condition ◽  
Mean Square ◽  
Initial Value ◽  
Mild Conditions ◽  
Fractional Differential ◽  
Linear Differential ◽  
Linear Fractional Differential Equations ◽  
Statistical Functions

AbstractThe aim of this paper is to study, in mean square sense, a class of random fractional linear differential equation where the initial condition and the forcing term are assumed to be second-order random variables. The solution stochastic process of its associated Cauchy problem is constructed combining the application of a mean square chain rule for differentiating second-order stochastic processes and the random Fröbenius method. To conduct our study, first the classical Caputo derivative is extended to the random framework, in mean square sense. Furthermore, a sufficient condition to guarantee the existence of this operator is provided. Afterwards, the solution of a random fractional initial value problem is built under mild conditions. The main statistical functions of the solution stochastic process are also computed. Finally, several examples illustrate our theoretical findings.

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.

An Application of Mean Square Calculus to Sliding Wear

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Cláudio R. Ávila da Silva ◽  
Giuseppe Pintaude ◽  
Hazim Ali Al-Qureshi ◽  
Marcelo Alves Krajnc
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Sliding Wear ◽  
Random Variables ◽  
Random Variable ◽  
Expected Value ◽  
Mean Square ◽  
Cauchy Problems ◽  
Numerical Examples ◽  
Correlation Models

In this paper the Archard model and classical results of mean square calculus are used to derive two Cauchy problems in terms of the expected value and covariance of the worn height stochastic process. The uncertainty is present in the wear and roughness coefficients. In order to model the uncertainty, random variables or stochastic processes are used. In the latter case, the expected value and covariance of the worn height stochastic process are obtained for three combinations of correlation models for the wear and roughness coefficients. Numerical examples for both models are solved. For the model based on a random variable, a larger dispersion in terms of worn height stochastic process was observed.

scholarly journals Analytic-Numerical Solution of Random Boundary Value Heat Problems in a Semi-Infinite Bar

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
M.-C. Casabán ◽  
J.-C. Cortés ◽  
B. García-Mora ◽  
L. Jódar
Keyword(s):  
Stochastic Process ◽  
Temperature Distribution ◽  
Numerical Solution ◽  
Closed Form ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mean Square ◽  
Random Boundary ◽  
Numerical Examples

This paper deals with the analytic-numerical solution of random heat problems for the temperature distribution in a semi-infinite bar with different boundary value conditions. We apply a random Fourier sine and cosine transform mean square approach. Random operational mean square calculus is developed for the introduced transforms. Using previous results about random ordinary differential equations, a closed form solution stochastic process is firstly obtained. Then, expectation and variance are computed. Illustrative numerical examples are included.

scholarly journals A fractional order epidemic model for the simulation of outbreaks of Ebola

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Weiqiu Pan ◽  
Tianzeng Li ◽  
Safdar Ali
Keyword(s):  
Numerical Solution ◽  
Root Mean Square ◽  
Fractional Order ◽  
Relative Error ◽  
Numerical Solutions ◽  
Real Data ◽  
Mean Square ◽  
Order Model ◽  
The Real ◽  
Fractional Orders

AbstractThe Ebola outbreak in 2014 caused many infections and deaths. Some literature works have proposed some models to study Ebola virus, such as SIR, SIS, SEIR, etc. It is proved that the fractional order model can describe epidemic dynamics better than the integer order model. In this paper, we propose a fractional order Ebola system and analyze the nonnegative solution, the basic reproduction number $R_{0}$ R 0 , and the stabilities of equilibrium points for the system firstly. In many studies, the numerical solutions of some models cannot fit very well with the real data. Thus, to show the dynamics of the Ebola epidemic, the Gorenflo–Mainardi–Moretti–Paradisi scheme (GMMP) is taken to get the numerical solution of the SEIR fractional order Ebola system and the modified grid approximation method (MGAM) is used to acquire the parameters of the SEIR fractional order Ebola system. We consider that the GMMP method may lead to absurd numerical solutions, so its stability and convergence are given. Then, the new fractional orders, parameters, and the root-mean-square relative error $g(U^{*})=0.4146$ g ( U ∗ ) = 0.4146 are obtained. With the new fractional orders and parameters, the numerical solution of the SEIR fractional order Ebola system is closer to the real data than those models in other literature works. Meanwhile, we find that most of the fractional order Ebola systems have the same order. Hence, the fractional order Ebola system with different orders using the Caputo derivatives is also studied. We also adopt the MGAM algorithm to obtain the new orders, parameters, and the root-mean-square relative error which is $g(U^{*})=0.2744$ g ( U ∗ ) = 0.2744 . With the new parameters and orders, the fractional order Ebola systems with different orders fit very well with the real data.

scholarly journals Stochastic Processes with a Particular Type of Variograms

2007 ◽  
Vol 2007 ◽  
pp. 1-5 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Series Expansion ◽  
Real Line ◽  
Random Fields ◽  
The Other ◽  
Simple Construction ◽  
The Real ◽  
Line Form

This paper is concerned with a class of stochastic processes or random fields with second-order increments, whose variograms have a particular form, among which stochastic processes having orthogonal increments on the real line form an important subclass. A natural issue, how big this subclass is, has not been explicitly addressed in the literature. As a solution, this paper characterizes a stochastic process having orthogonal increments on the real line in terms of its variogram or its construction. Our findings are a little bit surprising: this subclass is big in terms of the variogram, and on the other hand, it is relatively “small” according to a simple construction. In particular, every such process with Gaussian increments can be simply constructed from Brownian motion. Using the characterizations we obtain a series expansion of the stochastic process with orthogonal increments.

The Effect of Bandwidth of Semiactive Dampers on Vehicle Ride

1993 ◽  
Vol 115 (3) ◽  
pp. 571-575 ◽  
Author(s):  
J. Lieh
Keyword(s):  
Fourier Transform ◽  
Significant Influence ◽  
Switching Time ◽  
Control Valve ◽  
Mean Square ◽  
Active Suspensions ◽  
Car Model ◽  
Frequency Responses ◽  
And Control ◽  
Valve Switching

A passenger car model with a full car body and four wheel-axle assemblies is used to investigate the influence of semiactive suspensions on ride performance. Mean square values are evaluated for various damping levels and control valve switching times. Due to severe nonlinearities, frequency responses are not obtained directly. They are reconstructed from Fast Fourier Transform (FFT) using a Hanning window. The results are compared with those from LQR active suspensions and pure on-off dampers. The effect of control valve switching time (bandwidth) is studied and shows a significant influence on the vehicle ride, suspension travels, and tire deflections.

scholarly journals Study of Aerosol Influence on Nighttime Land Surface Temperature Retrieval Based on Two Methods

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
Caixia Gao ◽  
Enyu Zhao ◽  
Chuanrong Li ◽  
Yonggang Qian ◽  
Lingling Ma ◽  
...  
Keyword(s):  
Sensitivity Analysis ◽  
Water Vapor ◽  
Standard Deviation ◽  
Land Surface ◽  
Simulated Data ◽  
Mean Square ◽  
Surface Emissivity ◽  
Land Surface Emissivity ◽  
Instrument Noise ◽  
Temperature Retrieval

The aim of this study is to evaluate the aerosol influence on LST retrieval with two algorithms (split-window (SW) method and a four-channel based method) using simulated data under typical conditions. The results show that the root mean square error (RMSE) decreases to approximately 2.3 K for SW method and 1.5 K for four channel based method when VZA = 60° and visibility = 3 km; an RMSE would be increased by approximately 1.0 K when visibility varies from 3 km to 23 km. Moreover, a detailed sensitivity analysis under a visibility of 3 km and 23 km is performed in terms of uncertainties of land surface emissivity (LSE), water vapor content (WVC), and instrument noise, respectively. It is noted that the four-channel based method is more sensitive to LSE than SW method, especially for dry atmosphere; LST error caused by a WVC uncertainty of 20% is within 1.5 K for SW method and within 0.8 K for four-channel based method; the instrument noise would introduce LST error with a maximum standard deviation of 0.5 K and 0.04 K for the four-channel based method and SW method, respectively.

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