Journal articles: 'Fractional noise'

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Wyss, Walter. "Fractional noise." Foundations of Physics Letters 4, no. 3 (June 1991): 235–46. http://dx.doi.org/10.1007/bf00665755.

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EL MELLALI, TARIK, and YOUSSEF OUKNINE. "WEAK CONVERGENCE FOR QUASILINEAR STOCHASTIC HEAT EQUATION DRIVEN BY A FRACTIONAL NOISE WITH HURST PARAMETER H ∈ (½, 1)." Stochastics and Dynamics 13, no. 03 (May 27, 2013): 1250024. http://dx.doi.org/10.1142/s0219493712500244.

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In this paper, we consider a quasi-linear stochastic heat equation in one dimension on [0, 1], with Dirichlet boundary conditions driven by an additive fractional white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter n ∈ ℕ which can approximate the fractional noise in some sense. Then, we provide sufficient conditions ensuring that the real-valued mild solution of the SPDE perturbed by this family of noises converges in law, in the space [Formula: see text] of continuous functions, to the solution of the fractional noise driven SPDE.

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Sun, Xichao, and Junfeng Liu. "Weak Convergence for a Class of Stochastic Fractional Equations Driven by Fractional Noise." Advances in Mathematical Physics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/479873.

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We consider a class of stochastic fractional equations driven by fractional noise ont,x∈0,T×0,1 ∂u/∂t=Dδαu+ft,x,u+∂2BHt,x/∂t ∂x, with Dirichlet boundary conditions. We formally replace the random perturbation by a family of sequences based on Kac-Stroock processes in the plane, which approximate the fractional noise in some sense. Under some conditions, we show that the real-valued mild solution of the stochastic fractional heat equation perturbed by this family of noises converges in law, in the space𝒞0,T×0,1of continuous functions, to the solution of the stochastic fractional heat equation driven by fractional noise.

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Jin, Bangti, Yubin Yan, and Zhi Zhou. "Numerical approximation of stochastic time-fractional diffusion." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 4 (July 2019): 1245–68. http://dx.doi.org/10.1051/m2an/2019025.

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We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order α ∈ (0,1), and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order γ ∈ [0,1] in the front). The numerical scheme approximates the model in space by the standard Galerkin method with continuous piecewise linear finite elements and in time by the classical Grünwald-Letnikov method (for both Caputo fractional derivative and Riemann-Liouville fractional integral), and the noise by the L2-projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the discrete solution operators for the deterministic inhomogeneous problem. One- and two-dimensional numerical results are presented to support the theoretical findings.

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Macias, Michal, Dominik Sierociuk, and Wiktor Malesza. "MEMS Accelerometer Noises Analysis Based on Triple Estimation Fractional Order Algorithm." Sensors 22, no. 2 (January 11, 2022): 527. http://dx.doi.org/10.3390/s22020527.

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This paper is devoted to identifying parameters of fractional order noises with application to noises obtained from MEMS accelerometer. The analysis and parameters estimation will be based on the Triple Estimation algorithm, which can simultaneously estimate state, fractional order, and parameter estimates. The capability of the Triple Estimation algorithm to fractional noises estimation will be confirmed by the sets of numerical analyses for fractional constant and variable order systems with Gaussian noise input signal. For experimental data analysis, the MEMS sensor SparkFun MPU9250 Inertial Measurement Unit (IMU) was used with data obtained from the accelerometer in x, y and z-axes. The experimental results clearly show the existence of fractional noise in this MEMS’ noise, which can be essential information in the design of filtering algorithms, for example, in inertial navigation.

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Safarinejadian, Behrouz, Nasrin Kianpour, and Mojtaba Asad. "State estimation in fractional-order systems with coloured measurement noise." Transactions of the Institute of Measurement and Control 40, no. 6 (March 15, 2017): 1819–35. http://dx.doi.org/10.1177/0142331217691219.

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This paper presents new estimation methods for discrete fractional-order state-space systems with coloured measurement noise. A novel approach is proposed to convert a fractional system with coloured measurement noise to a system with white measurement noise in which the process and measurement noises are correlated with each other. In this paper, two new Kalman filter algorithms for fractional-order linear state-space systems with coloured measurement noise, as well as a new extended Kalman filter algorithm for state estimation in nonlinear fractional-order state-space systems with coloured measurement noise, are proposed. The accuracy of the equations and relations is confirmed in several theorems. The validity and effectiveness of the proposed algorithms are verified by simulation results and compared with previous work. Results show that for linear and nonlinear fractional-order systems with coloured noise, the proposed methods are more accurate than conventional methods regarding estimation error and estimation error covariance. Simulation results demonstrate that the proposed algorithms can accurately perform estimation in fractional-order systems with coloured measurement noise.

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Lin, Lifeng, Huiqi Wang, Xipei Huang, and Yongxian Wen. "Generalized stochastic resonance for a fractional harmonic oscillator with bias-signal-modulated trichotomous noise." International Journal of Modern Physics B 32, no. 07 (March 5, 2018): 1850072. http://dx.doi.org/10.1142/s0217979218500728.

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For a fractional linear oscillator subjected to both parametric excitation of trichotomous noise and external excitation of bias-signal-modulated trichotomous noise, the generalized stochastic resonance (GSR) phenomena are investigated in this paper in case the noises are cross-correlative. First, the generalized Shapiro–Loginov formula and generalized fractional Shapiro–Loginov formula are derived. Then, by using the generalized (fractional) Shapiro–Loginov formula and the Laplace transformation technique, the exact expression of the first-order moment of the system’s steady response is obtained. The numerical results show that the evolution of the output amplitude amplification is nonmonotonic with the frequency of periodic signal, the noise parameters, and the fractional order. The GSR phenomena, including single-peak GSR, double-peak GSR and triple-peak GSR, are observed in this system. In addition, the interplay of the multiplicative trichotomous noise, bias-signal-modulated trichotomous noise and memory can induce and diversify the stochastic multi-resonance (SMR) phenomena, and the two kinds of trichotomous noises play opposite roles on the GSR.

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Li, Ming, Xichao Sun, and Xi Xiao. "Revisiting fractional Gaussian noise." Physica A: Statistical Mechanics and its Applications 514 (January 2019): 56–62. http://dx.doi.org/10.1016/j.physa.2018.09.008.

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Fa, Kwok Sau. "Fractional Ornstein–Uhlenbeck noise." Annals of Physics 393 (June 2018): 327–34. http://dx.doi.org/10.1016/j.aop.2018.04.019.

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Du, Wei, and Le Tong. "Introducing Robust Evolutionary Optimization in Noisy Fractional-Order Systems." International Journal of Bifurcation and Chaos 30, no. 08 (June 30, 2020): 2050119. http://dx.doi.org/10.1142/s0218127420501199.

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This paper investigates the problem of searching for the robust initial values of noisy fractional-order chaotic systems when the desired output is given. The problem is addressed under the framework of robust evolutionary optimization. Two different ways of adding noise are considered: (1) the noise is added to the initial point; (2) the noise is added to the orbit of the system. A series of experiments are conducted to validate the effectiveness of robust evolutionary optimization. The experimental results reveal that robust initial values of noisy fractional-order chaotic systems can be obtained in an efficient way by introducing robust evolutionary optimization.

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Li, Chujin, and Jinqiao Duan. "Impact of Correlated Noises on Additive Dynamical Systems." Mathematical Problems in Engineering 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/678976.

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Impact of correlated noises on dynamical systems is investigated by considering Fokker-Planck type equations under the fractional white noise measure, which correspond to stochastic differential equations driven by fractional Brownian motions with the Hurst parameterH>1/2. Firstly, by constructing the fractional white noise framework, one small noise limit theorem is proved, which provides an estimate for the deviation of random solution orbits from the corresponding deterministic orbits. Secondly, numerical experiments are conducted to examine the probability density evolutions of two special dynamical systems, as the Hurst parameterHvaries. Certain behaviors of the probability density functions are observed.

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Wang, Haibin, and Junbo Long. "Applications of Fractional Lower Order Synchrosqueezing Transform Time Frequency Technology to Machine Fault Diagnosis." Mathematical Problems in Engineering 2020 (August 3, 2020): 1–19. http://dx.doi.org/10.1155/2020/3983242.

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Synchrosqueezing transform (SST) is a high resolution time frequency representation technology for nonstationary signal analysis. The short time Fourier transform-based synchrosqueezing transform (FSST) and the S transform-based synchrosqueezing transform (SSST) time frequency methods are effective tools for bearing fault signal analysis. The fault signals belong to a non-Gaussian and nonstationary alpha (α) stable distribution with 1<α<2 and even the noises being also α stable distribution. The conventional FSST and SSST methods degenerate and even fail under α stable distribution noisy environment. Motivated by the fact that fractional low order STFT and fractional low order S-transform work better than the traditional STFT and S-transform methods under α stable distribution noise environment, we propose in this paper the fractional lower order FSST (FLOFSST) and the fractional lower order SSST (FLOSSST). In addition, we derive the corresponding inverse FLOSST and inverse FLOSSST. The simulation results show that both FLOFSST and FLOSSST perform better than the conventional FSSST and SSST under α stable distribution noise in instantaneous frequency estimation and signal reconstruction. Finally, FLOFSST and FLOSSST are applied to analyze the time frequency distribution of the outer race fault signal. Our results show that FLOFSST and FLOSSST extract the fault features well under symmetric stable (SαS) distribution noise.

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Verma, Atul Kumar, Barjinder Singh Saini, and Taranjit Kaur. "Image Denoising using Alexander Fractional Hybrid Filter." International Journal of Image and Graphics 18, no. 01 (January 2018): 1850003. http://dx.doi.org/10.1142/s0219467818500031.

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In this paper, a hybrid filter based on the concept of fractional calculus and Alexander polynomial is proposed. The hybrid filtering mask is constructed by convolving the designed Alexander fractional differential and integral masks. The hybrid mask shows high robustness for images corrupted with Gaussian, salt & pepper, and speckle noises. For the experimentation, the standard and real world noisy images are used. The qualitative comparison shows that the proposed hybrid filter has better denoising with high edge preserving capability as compared to the other existing filters. Quantitatively, the performance of the proposed hybrid filter is also evaluated by the measures such as peak signal to noise ratio (PSNR), normalized cross-correlation (NK), average difference (AD), structural content (SC), maximum difference (MD) and normalized absolute error (NAE) on standard set of images. The average values of these metrics for Gaussian noise with maximum standard deviation [Formula: see text] are PSNR [Formula: see text] 32.729, NK [Formula: see text] 0.8190, AD [Formula: see text] 0.01825, SC [Formula: see text] 0.8527, MD [Formula: see text] 87, NAE [Formula: see text] 0.0637. The experimentation reveals that the proposed hybrid filter gives better improvement as compared with other existing filters both qualitatively and quantitatively.

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Wang, Yuan Gan, Hong Lin Yu, and Xin Yu Liang. "Time Delay Model of Fractional Fourier Transform and the Application in Signal Filtering." Applied Mechanics and Materials 121-126 (October 2011): 3637–41. http://dx.doi.org/10.4028/www.scientific.net/amm.121-126.3637.

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Various frequency bands of noises are contained in the actual signal. And it's difficult to eliminate the noise portion, which has a time delay and same spectrum with the original signal with conventional filtering methods. Based on the time delay and the multiplication delay characteristics of the fractional Fourier transform (FRFT), we put forward a FRFT time delay model, which can increase distance between the signal and the noise component. Through corresponding Fractional Fourier Transform to noise-containing signal, the distance between the signal and common frequency noises can be constantly increased within the transform domain, thus easily separating the noise component. The algorithm of the model can be simply deduced, easy realized and converged fast. In the experiment, we simulated the separating characteristics of the transform, and used the method to de-noise the grating signal. Compared with other traditional methods, we find that the FRFT acquired a better result.

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Sun, Xichao, Zhi Wang, and Jing Cui. "On a Fractional SPDE Driven by Fractional Noise and a Pure Jump Lévy Noise inℝd." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/758270.

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We study a stochastic partial differential equation in the whole spacex∈ℝd, with arbitrary dimensiond≥1, driven by fractional noise and a pure jump Lévy space-time white noise. Our equation involves a fractional derivative operator. Under some suitable assumptions, we establish the existence and uniqueness of the global mild solution via fixed point principle.

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Lovejoy, Shaun. "Fractional relaxation noises, motions and the fractional energy balance equation." Nonlinear Processes in Geophysics 29, no. 1 (February 25, 2022): 93–121. http://dx.doi.org/10.5194/npg-29-93-2022.

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Abstract. We consider the statistical properties of solutions of the stochastic fractional relaxation equation and its fractionally integrated extensions that are models for the Earth's energy balance. In these equations, the highest-order derivative term is fractional, and it models the energy storage processes that are scaling over a wide range. When driven stochastically, the system is a fractional Langevin equation (FLE) that has been considered in the context of random walks where it yields highly nonstationary behaviour. An important difference with the usual applications is that we instead consider the stationary solutions of the Weyl fractional relaxation equations whose domain is −∞ to t rather than 0 to t. An additional key difference is that, unlike the (usual) FLEs – where the highest-order term is of integer order and the fractional term represents a scaling damping – in the fractional relaxation equation, the fractional term is of the highest order. When its order is less than 1/2 (this is the main empirically relevant range), the solutions are noises (generalized functions) whose high-frequency limits are fractional Gaussian noises (fGn). In order to yield physical processes, they must be smoothed, and this is conveniently done by considering their integrals. Whereas the basic processes are (stationary) fractional relaxation noises (fRn), their integrals are (nonstationary) fractional relaxation motions (fRm) that generalize both fractional Brownian motion (fBm) as well as Ornstein–Uhlenbeck processes. Since these processes are Gaussian, their properties are determined by their second-order statistics; using Fourier and Laplace techniques, we analytically develop corresponding power series expansions for fRn and fRm and their fractionally integrated extensions needed to model energy storage processes. We show extensive analytic and numerical results on the autocorrelation functions, Haar fluctuations and spectra. We display sample realizations. Finally, we discuss the predictability of these processes which – due to long memories – is a past value problem, not an initial value problem (that is used for example in highly skillful monthly and seasonal temperature forecasts). We develop an analytic formula for the fRn forecast skills and compare it to fGn skill. The large-scale white noise and fGn limits are attained in a slow power law manner so that when the temporal resolution of the series is small compared to the relaxation time (of the order of a few years on the Earth), fRn and its extensions can mimic a long memory process with a range of exponents wider than possible with fGn or fBm. We discuss the implications for monthly, seasonal, and annual forecasts of the Earth's temperature as well as for projecting the temperature to 2050 and 2100.

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Hou, Ming Liang, Yu Ran Liu, and Qi Wang. "An Image Information Extraction Algorithm for Salt and Pepper Noise on Fractional Differentials." Advanced Materials Research 179-180 (January 2011): 1011–15. http://dx.doi.org/10.4028/www.scientific.net/amr.179-180.1011.

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An image information extraction algorithm on fractional differentials is put forward in this paper that is based on the characteristics of fractional differential in signal processing. This paper has extracted the information of salt and pepper noise images with various coefficients, and analyzed and compared it with the information extraction results of classic integer-order operators as Prewitt, Roberts and Sobel. Experiments have shown that not only the high-frequency marginal information can be extracted by extracting information with fractional differentials, just as it is extracted with integer-order operators, but the texture information can also be extracted from the smooth region. Besides, this algorithm is featured with great noise immunity against salt and pepper noises.

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Zarei, Jafar, and Mahmood Tabatabaei. "Fractional order unknown input filter design for fault detection of discrete fractional order linear systems." Transactions of the Institute of Measurement and Control 40, no. 16 (February 1, 2018): 4321–29. http://dx.doi.org/10.1177/0142331217746493.

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In this study, a new method is introduced to design an estimator for discrete-time linear fractional order systems, which are affected by unknown disturbances. The main goal of this study is decoupling disturbance and uncertainties from the true states for discrete fractional order systems in noisy environment. The fractional Kalman filter framework is exploited to develop a robust estimator against unknown inputs (UIs) in noisy environment. The proposed filter is exploited to detect faults in fractional order systems. Simulation results illustrate the advantages of this robust filter for state estimation and fault detection of fractional order model of ultra-capacitor (UC). The robustness of the designed filter is shown in the sense of disturbance decoupling in the presence of noise.

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Chen, Ming. "Fractional-Order Adaptive P -Laplace Equation-Based Art Image Edge Detection." Advances in Mathematical Physics 2021 (August 31, 2021): 1–10. http://dx.doi.org/10.1155/2021/2337712.

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In recent years, with the rapid development of image processing research, the study of nonstandard images has gradually become a research hotspot, for example, fabric images, remote sensing images, and gear images. Some of the remote sensing images have a complex background and low illumination compared with standard images and are easy to be mixed with noise during acquisition; some of the fabric images have rich texture information, which adds difficulty to the related processing, and are also easy to be mixed with noise during acquisition. In this paper, we propose a fractional-order adaptive P -Laplace equation image edge detection algorithm for the problem of image edge detection in which the edge and texture information of the image is lost. The algorithm can apply for the order adaptively to filter the noise according to the noise distribution of the image, and the adaptive diffusion factor is determined by both the fractional-order curvature and fractional-order gradient of the iso-illumination line and combined with the iterative approach to realize the fine-tuning of the noisy image. The experimental results demonstrate that the algorithm can remove the noise while preserving the texture and details of the image. A fractional-order partial differential equation image edge detection model with a fractional-order fidelity term is proposed for Gaussian noise. The model incorporates a fractional-order fidelity term because this fidelity term smoothes out the rougher parts of the image while preserving the texture in the original image in greater detail and eliminating the step effect produced by other models such as the Perona-Malik (PM) and Rudin-Osher-Fatemi (ROF) models. By comparing with other algorithms, the image edge detection effect is measured with the help of evaluation metrics such as peak signal-to-noise ratio and structural similarity, and the optimal value is selected iteratively so that the image with the best edge detection result is retained. A convolutional mask image edge detection model based on adaptive fractional-order calculus is proposed for the scattered noise in medical images. The adaption is mainly reflected in the model algorithm by constructing an exponential parameter relation that is closely related to the image, which can dynamically adjust the parameter values, thus making the model algorithm more practical. The model achieves the scattering noise removal in four steps.

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Zunino, L., D. G. Pérez, M. T. Martín, M. Garavaglia, A. Plastino, and O. A. Rosso. "Permutation entropy of fractional Brownian motion and fractional Gaussian noise." Physics Letters A 372, no. 27-28 (June 2008): 4768–74. http://dx.doi.org/10.1016/j.physleta.2008.05.026.

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Zunino, L., D. G. Pérez, A. Kowalski, M. T. Martín, M. Garavaglia, A. Plastino, and O. A. Rosso. "Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy." Physica A: Statistical Mechanics and its Applications 387, no. 24 (October 2008): 6057–68. http://dx.doi.org/10.1016/j.physa.2008.07.004.

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Sandler, Nancy P., Claudio de C. Chamon, and Eduardo Fradkin. "Noise measurements and fractional charge in fractional quantum Hall liquids." Physical Review B 59, no. 19 (May 15, 1999): 12521–36. http://dx.doi.org/10.1103/physrevb.59.12521.

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Levajkovic, Tijana, and Dora Selesi. "Chaos expansion methods for stochastic differential equations involving the Malliavin derivative, Part II." Publications de l'Institut Math?matique (Belgrade) 90, no. 104 (2011): 85–98. http://dx.doi.org/10.2298/pim1104085l.

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We solve stochastic differential equations involving the Malliavin derivative and the fractional Malliavin derivative by means of a chaos expansion on a general white noise space (Gaussian, Poissonian, fractional Gaussian and fractional Poissonian white noise space). There exist unitary mappings between the Gaussian and Poissonian white noise spaces, which can be applied in solving SDEs.

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Barton, R. J., and H. V. Poor. "Signal detection in fractional Gaussian noise." IEEE Transactions on Information Theory 34, no. 5 (September 1988): 943–59. http://dx.doi.org/10.1109/18.21218.

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Fa, Kwok Sau. "Fractional oscillator noise and its applications." International Journal of Modern Physics B 34, no. 26 (September 18, 2020): 2050234. http://dx.doi.org/10.1142/s0217979220502343.

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It is shown that a fractional oscillator (FO) noise, which is a generalization of the ordinary overdamped linear oscillator driven by the white noise may be ‘applied to diverse systems; its stationary correlation function presents power-law-like function, exponential-like function, exponential function, and oscillatory decays. The model may be employed to describe the fluctuation of the distance between a fluorescein–tyrosine pair within a single protein complex and the internal dynamics of a lysozyme molecule in solution. It also has the possibility of describing a Brownian particle in an oscillatory viscoelastic shear flow.

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Jiang, Yuan Yuan, You Ren Wang, and Hui Luo. "Denoising Method for Unknown Image Noise Based on FWT Optimal Order Selection." Advanced Materials Research 765-767 (September 2013): 2776–80. http://dx.doi.org/10.4028/www.scientific.net/amr.765-767.2776.

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The optimal fractional order is got for image denoising by 2-D fractional wavelet transform (FWT). But, the actual application environment is complex, and the input image has already been polluted by unknown noise frequently in the process of capture and transmission. And it's impossible to get the optimal fractional order on the basis of the objective evaluation standard in existence. Therefore, in view of the unknown image noise, a method to get the estimated value of optimal fractional order is put forward. Firstly, new objective evaluation standards for image denoising in fractional wavelet domain are defined, and its optimal value is obtained based on noise estimation. Then the optimal estimated fractional order is got. The experiment results show that, the optimal order of 2-D FWT can be selected reasonably by the proposed method and the unknown image noise can be filtered effectively in the estimated optimal fractional wavelet domain.

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Ji, Un Cig, Mi Ra Lee, and Peng Cheng Ma. "Fractional Langevin type equations for white noise distributions." Fractional Calculus and Applied Analysis 24, no. 4 (August 1, 2021): 1160–92. http://dx.doi.org/10.1515/fca-2021-0050.

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Abstract In this paper, by applying the intertwining properties, we introduce the fractional powers of the number operator perturbed by generalized Gross Laplacians (infinite dimensional Laplacians), which are special types of the infinitesimal generators of generalized Mehler semigroups. By applying the intertwining properties and semigroup approach, we study the Langevin type equations associated with the infinite dimensional Laplacians and with white noise distributions as forcing terms. Then we investigate the unique solution of the fractional Langevin type equations associated with the Riemann-Liouville and Caputo time fractional derivatives, and the fractional power of the infinite dimensional Laplacians, for which we apply the intertwining properties again. For our purpose, we discuss the fractional integrals and fractional derivatives of white noise distribution valued functions.

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Sabzikar, Farzad, Jinu Kabala, and Krzysztof Burnecki. "Tempered fractionally integrated process with stable noise as a transient anomalous diffusion model." Journal of Physics A: Mathematical and Theoretical 55, no. 17 (March 29, 2022): 174002. http://dx.doi.org/10.1088/1751-8121/ac5b92.

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Abstract We present here the autoregressive tempered fractionally integrated moving average (ARTFIMA) process obtained by taking the tempered fractional difference operator of the non-Gaussian stable noise. The tempering parameter makes the ARTFIMA process stationary for a wider range of the memory parameter values than for the classical autoregressive fractionally integrated moving average, and leads to semi-long range dependence and transient anomalous behavior. We investigate ARTFIMA dependence structure with stable noise and construct Whittle estimators. We also introduce the stable Yaglom noise as a continuous version of the ARTFIMA model with stable noise. Finally, we illustrate the usefulness of the ARTFIMA process on a trajectory from the Golding and Cox experiment.

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BO, LIJUN, YIMING JIANG, and YONGJIN WANG. "STOCHASTIC CAHN–HILLIARD EQUATION WITH FRACTIONAL NOISE." Stochastics and Dynamics 08, no. 04 (December 2008): 643–65. http://dx.doi.org/10.1142/s0219493708002500.

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We study the existence and uniqueness of global mild solutions to a class of stochastic Cahn–Hilliard equations driven by fractional noises (fractional in time and white in space), through a weak convergence argument.

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Xia, Dengfeng, Litan Yan, and Weiyin Fei. "Mixed Fractional Heat Equation Driven by Fractional Brownian Sheet and Lévy Process." Mathematical Problems in Engineering 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/8059796.

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We consider the stochastic heat equation of the form∂u/∂t=(Δ+Δα)u+(∂f/∂x)(t,x,u)+σ(t,x,u)L˙+W˙H,whereW˙His the fractional noise,L˙is a (pure jump) Lévy space-time white noise,Δis Laplacian, andΔα=-(-Δ)α/2is the fractional Laplacian generator onR, andf,σ:[0,T]×R×R→Rare measurable functions. We introduce the existence and uniqueness of the solution by the fixed point principle under some suitable assumptions.

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Röösli, Marc P., Michael Hug, Giorgio Nicolí, Peter Märki, Christian Reichl, Bernd Rosenow, Werner Wegscheider, Klaus Ensslin, and Thomas Ihn. "Fractional Coulomb blockade for quasi-particle tunneling between edge channels." Science Advances 7, no. 19 (May 2021): eabf5547. http://dx.doi.org/10.1126/sciadv.abf5547.

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In the fractional quantum Hall effect, the elementary excitations are quasi-particles with fractional charges as predicted by theory and demonstrated by noise and interference experiments. We observe Coulomb blockade of fractional charges in the measured magneto-conductance of a 1.4-micron-wide quantum dot. Interaction-driven edge reconstruction separates the dot into concentric compressible regions with fractionally charged excitations and incompressible regions acting as tunnel barriers for quasi-particles. Our data show the formation of incompressible regions of filling factors 2/3 and 1/3. Comparing data at fractional filling factors to filling factor 2, we extract the fractional quasi-particle charge e*/e = 0.32 ± 0.03 and 0.35 ± 0.05. Our investigations extend and complement quantum Hall Fabry-Pérot interference experiments investigating the nature of anyonic fractional quasi-particles.

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Ismail, Samar M., Lobna A. Said, Ahmed H. Madian, and Ahmed G. Radwan. "Fractional-Order Edge Detection Masks for Diabetic Retinopathy Diagnosis as a Case Study." Computers 10, no. 3 (March 5, 2021): 30. http://dx.doi.org/10.3390/computers10030030.

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Edge detection is one of the main steps in the image processing field, especially in biomedical imaging, to diagnose a disease or trace its progress. The transfer of medical images makes them more susceptible to quality degradation due to any imposed noise. Hence, the protection of this data against noise is a persistent need. The efficiency of fractional-order filters to detect fine details and their high noise robustness, unlike the integer-order filters, it renders them an attractive solution for biomedical edge detection. In this work, two novel central fractional-order masks are proposed with their detailed mathematical proofs. The fractional-order parameter gives an extra degree of freedom in designing different masks. The noise performance of the proposed masks is evaluated upon applying Salt and Pepper noise and Gaussian noise. Numerical results proved that the proposed masks outperform the integer-order masks regarding both types of noise, achieving higher Peak Signal to Noise Ratio. As a practical application, the proposed fractional-order edge detection masks are employed to enhance the Diabetic Retinopathy disease diagnosis.

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Zhu, Jianguang, Juan Wei, Haijun Lv, and Binbin Hao. "Truncated Fractional-Order Total Variation for Image Denoising under Cauchy Noise." Axioms 11, no. 3 (February 25, 2022): 101. http://dx.doi.org/10.3390/axioms11030101.

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In recent years, the fractional-order derivative has achieved great success in removing Gaussian noise, impulsive noise, multiplicative noise and so on, but few works have been conducted to remove Cauchy noise. In this paper, we propose a novel nonconvex variational model for removing Cauchy noise based on the truncated fractional-order total variation. The new model can effectively reduce the staircase effect and keep small details or textures while removing Cauchy noise. In order to solve the nonconvex truncated fractional-order total variation regularization model, we propose an efficient alternating minimization method under the framework of the alternating direction multiplier method. Experimental results illustrate the effectiveness of the proposed model, compared to some previous models.

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Chen, Yong, Yaozhong Hu, and Zhi Wang. "Parameter Estimation of Complex Fractional Ornstein-Uhlenbeck Processes with Fractional Noise." Latin American Journal of Probability and Mathematical Statistics 14, no. 1 (2017): 613. http://dx.doi.org/10.30757/alea.v14-30.

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Shi, Kehua, and Yongjin Wang. "On a stochastic fractional partial differential equation with a fractional noise." Stochastics 84, no. 1 (October 3, 2011): 21–36. http://dx.doi.org/10.1080/17442508.2011.566336.

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Delignières, Didier. "Correlation Properties of (Discrete) Fractional Gaussian Noise and Fractional Brownian Motion." Mathematical Problems in Engineering 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/485623.

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The fractional Gaussian noise/fractional Brownian motion framework (fGn/fBm) has been widely used for modeling and interpreting physiological and behavioral data. The concept of 1/fnoise, reflecting a kind of optimal complexity in the underlying systems, is of central interest in this approach. It is generally considered that fGn and fBm represent a continuum, punctuated by the boundary of “ideal” 1/fnoise. In the present paper, we focus on the correlation properties of discrete-time versions of these processes (dfGn and dfBm). We especially derive a new analytical expression of the autocorrelation function (ACF) of dfBm. We analyze the limit behavior of dfGn and dfBm when they approach their upper and lower limits, respectively. We show that, asHapproaches 1, the ACF of dfGn tends towards 1 at all lags, suggesting that dfGn series tend towards straight line. Conversely, asHapproaches 0, the ACF of dfBm tends towards 0 at all lags, suggesting that dfBm series tend towards white noise. These results reveal a severe breakdown of correlation properties around the 1/fboundary and challenge the idea of a smooth transition between dfGn and dfBm processes. We discuss the implications of these findings for the application of the dfGn/dfBm model to experimental series, in terms of theoretical interpretation and modeling.

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Sanyal, Arindam, Xueyi Yu, Yanlong Zhang, and Nan Sun. "Fractional‐ N PLL with multi‐element fractional divider for noise reduction." Electronics Letters 52, no. 10 (May 2016): 809–10. http://dx.doi.org/10.1049/el.2016.0680.

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Sun, Xichao, Ming Li, and Wei Zhao. "Moderate Deviations for Stochastic Fractional Heat Equation Driven by Fractional Noise." Complexity 2018 (July 10, 2018): 1–17. http://dx.doi.org/10.1155/2018/7402764.

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Liu, Junfeng, and Litan Yan. "Solving a Nonlinear Fractional Stochastic Partial Differential Equation with Fractional Noise." Journal of Theoretical Probability 29, no. 1 (October 5, 2014): 307–47. http://dx.doi.org/10.1007/s10959-014-0578-4.

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Levajkovic, Tijana, and Dora Selesi. "Chaos expansion methods for stochastic differential equations involving the Malliavin derivative, Part I." Publications de l'Institut Math?matique (Belgrade) 90, no. 104 (2011): 65–84. http://dx.doi.org/10.2298/pim1104065l.

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We consider Gaussian, Poissonian, fractional Gaussian and fractional Poissonian white noise spaces, all represented through the corresponding orthogonal basis of the Hilbert space of random variables with finite second moments, given by the Hermite and the Charlier polynomials. There exist unitary mappings between the Gaussian and Poissonian white noise spaces. We investigate the relationship of the Malliavin derivative, the Skorokhod integral, the Ornstein-Uhlenbeck operator and their fractional counterparts on a general white noise space.

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Afterman, Danielle, Pavel Chigansky, Marina Kleptsyna, and Dmytro Marushkevych. "Linear Filtering with Fractional Noises: Large Time and Small Noise Asymptotics." SIAM Journal on Control and Optimization 60, no. 3 (May 26, 2022): 1463–87. http://dx.doi.org/10.1137/20m1360359.

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Teodoru, Emil. "Reducing the Phase-Noise In ΔΣ Fractional-N Synthezis – A Simulink Model." International conference KNOWLEDGE-BASED ORGANIZATION 23, no. 3 (June 27, 2017): 131–34. http://dx.doi.org/10.1515/kbo-2017-0166.

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AbstractThe resolution in fractional-N synthesis results as a fractional part of the reference frequency. This category of synthesizers permits a greater frefand a smaller N, a larger loop bandwidth, faster lock times and reduced output phase-noise. In ΔΣ fractional-N PLL’s the main problem is the specific quantization noise. To reduce them many techniques are used. The paper presents a Simulink model of the influence of the requantisation in the phase-noise cancellation process.

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Chen, Xi, Maokang Luo, and Lu Zhang. "Consensus of Fractional-Order Double-Integral Multi-Agent System in a Bounded Fluctuating Potential." Fractal and Fractional 6, no. 3 (March 7, 2022): 147. http://dx.doi.org/10.3390/fractalfract6030147.

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At present, the consensus problem of fractional complex systems has received more attention. However, there is little literature on the consensus problem of fractional-order complex systems under noise disturbance. In this paper, we present a fractional-order double-integral multi-agent system affected by a common bounded fluctuating potential, where the protocol term consists of both the relative position and velocity information of neighboring agents. The consensus conditions of the presented system in the absence of noise are analytically given and verified by a numerical simulation algorithm. Then, the influences of the system order and other system parameters on the consensus of the presented system in the presence of bounded noise are also analyzed. It is found that when compared with the classical integer-order system, the presented fractional-order system has a larger range of consensus parameters and has more rich dynamic characteristics under the action of random noise. Especially, the bounded noise has a promoting effect on the consensus of the presented fractional-order system, while there is no similar phenomenon in the corresponding integer-order system.

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Yan, Zhi, Juan L. G. Guirao, Tareq Saeed, Huatao Chen, and Xianbin Liu. "Different Stochastic Resonances Induced by Multiplicative Polynomial Trichotomous Noise in a Fractional Order Oscillator with Time Delay and Fractional Gaussian Noise." Fractal and Fractional 6, no. 4 (March 30, 2022): 191. http://dx.doi.org/10.3390/fractalfract6040191.

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A general investigation on the mechanism of stochastic resonance is reported in a time-delay fractional Langevin system, which endues a nonlinear form multiplicative colored noise and fractional Gaussian noise. In terms of theoretical analysis, both the expressions of output steady-state amplitude and that of the first moment of system response are obtained by utilizing stochastic averaging method, fractional Shapiro and Laplace methods. Due to the presence of trichotomous colored noise, the excitation frequency can induce fresh multimodal Bona fide stochastic resonance, exhibiting much more novel dynamical behaviors than the non-disturbance case. It is verified that multimodal pattern only appears with small noise switching rate and memory damping order. The explicit expressions of critical noise intensity corresponding to the generalized stochastic resonance are given for the first time, by which it is determined that nonlinear form colored noise induces much more of a comprehensive resonant phenomena than the linear form. In the case of slow transfer rate noise, a newfangled phenomenon of double hypersensitive response induced by a variation in noise intensity is discovered and verified for the first time, with the necessary range of parameters for this phenomenon given. In terms of numerical scheme, an efficient and feasible algorithm for generating trichotomous noise is proposed, by which an algorithm based on the Caputo fractional derivative are applied. The numerical results match well with the analytical ones.

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Saggaf, M. M., and Enders A. Robinson. "A unified framework for the deconvolution of traces of nonwhite reflectivity." GEOPHYSICS 65, no. 5 (September 2000): 1660–76. http://dx.doi.org/10.1190/1.1444854.

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One of the fundamental assumptions of conventional deconvolution methods is that reflection coefficients follow the white‐noise model. However, analysis of well logs in various regions of the world confirms that in the majority of cases, reflectivity tends to depart from the white‐noise behavior. The assumption of white noise leads to a conventional deconvolution operator that can recover only the white component of reflectivity, thus yielding a distorted representation of the desired output. Various alternative processes have been suggested to model reflection coefficients. In this paper, we will examine some of these processes, apply them, contrast their stochastic properties, and critique their use for modeling reflectivity. These processes include autoregressive moving average (ARMA), scaling Gaussian noise, fractional Brownian motion, fractional Gaussian noise, and fractionally integrated noise. We then present a consistent framework to generalize the conventional deconvolution procedure to handle reflection coefficients that do not follow the white‐noise model. This framework represents a unified approach to the problem of deconvolving signals of nonwhite reflectivity and describes how higher‐order solutions to the deconvolution problem can be realized. We test generalized filters based on the various stochastic models and analyze their output. Because these models approximate the stochastic properties of reflection coefficients to a much better degree than white noise, they yield generalized deconvolution filters that deliver a significant improvement on the accuracy of seismic deconvolution over the conventional operator.

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Zhang, Jintian, Zhongkui Sun, Xiaoli Yang, and Wei Xu. "Controlling Bifurcations in Fractional-Delay Systems with Colored Noise." International Journal of Bifurcation and Chaos 28, no. 11 (October 2018): 1850137. http://dx.doi.org/10.1142/s0218127418501377.

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Comparing with the traditional integer-order model, fractional-order systems have shown enormous advantages in the analysis of new materials and anomalous diffusion dynamics mechanism in the past decades, but the research has been confined to fractional-order systems without delay. In this paper, we study the fractional-delay system in the presence of both the colored noise and delayed feedback. The stationary density functions (PDFs) are derived analytically by means of the stochastic averaging method combined with the principle of minimum mean-square error, by which the stochastic bifurcation behaviors have been well identified and studied. It can be found that the fractional-orders have influences on the bifurcation behaviors of the fractional-order system, but the bifurcation point of stationary PDF for amplitude differs from the bifurcation point of joint PDF. By merely changing the colored noise intensity or correlation time the shape of the PDFs can switch between unimodal distribution and bimodal one, thus announcing the occurrence of stochastic bifurcation. Further, we have demonstrated that modulating the time delay or delayed feedback may control bifurcation behaviors. The perfect agreement between the theoretical solution and the numerical solution obtained by the predictor–corrector algorithm confirms the correctness of the conclusion. In addition, fractional-order dominates the bifurcation control in the fractional-delay system, which causes the sensitive dependence of other bifurcation parameters on fractional-order.

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Luo, Qi, and Hongxia Wang. "The Matrix Completion Method for Phase Retrieval from Fractional Fourier Transform Magnitudes." Mathematical Problems in Engineering 2016 (2016): 1–6. http://dx.doi.org/10.1155/2016/4617327.

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Inspired by the implementation of the fractional Fourier transform (FRFT) and its applications in optics, we address the problem of reconstructing a signal from its several FRFT magnitudes (or intensities). The matrix completion method is adopted here. Through numerical tests, the matrix completion method is proven effective in both noisy and noise-free situations. We also compare our method with the Gerchberg-Saxton (GS) algorithm based on FRFT. Numerical tests show that the matrix completion method gains a certain advantage in recovering uniqueness and convergence over the GS algorithm in the noise-free case. Furthermore, in terms of noisy signals, the matrix completion method performs robustly and adding more measurements can generally increase accuracy of recovered signals.

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Mohammed, Wael W., Meshari Alesemi, Sahar Albosaily, Naveed Iqbal, and M. El-Morshedy. "The Exact Solutions of Stochastic Fractional-Space Kuramoto-Sivashinsky Equation by Using (G′G)-Expansion Method." Mathematics 9, no. 21 (October 26, 2021): 2712. http://dx.doi.org/10.3390/math9212712.

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In this paper, we consider the stochastic fractional-space Kuramoto–Sivashinsky equation forced by multiplicative noise. To obtain the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation, we apply the G′G-expansion method. Furthermore, we generalize some previous results that did not use this equation with multiplicative noise and fractional space. Additionally, we show the influence of the stochastic term on the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation.

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HU, YAOZHONG, and BERNT ØKSENDAL. "FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE." Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, no. 01 (March 2003): 1–32. http://dx.doi.org/10.1142/s0219025703001110.

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The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Itô type of stochastic differential equations driven by fractional Brownian motion BH(t); 1/2 < H < 1. We show that if we use an Itô type of stochastic integration with respect to BH(t) (as developed in Ref. 8), then the corresponding Itô fractional Black–Scholes market has no arbitrage, contrary to the situation when the pathwise integration is used. Moreover, we prove that our Itô fractional Black–Scholes market is complete and we compute explicitly the price and replicating portfolio of a European option in this market. The results are compared to the classical results based on standard Brownian motion B(t).

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Wang, Hai Bin, Jun Bo Long, and Dai Feng Zha. "Pseudo Cohen Time-Frequency Distributions in Infinite Variance Noise Environment." Applied Mechanics and Materials 475-476 (December 2013): 253–58. http://dx.doi.org/10.4028/www.scientific.net/amm.475-476.253.

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stable distribution has been suggested as a more appropriate model in impulsive noise environment.The performance of conventional time-frequency distributions (TFDs) degenerate in stable distribution noise environment. Hence, three improved methods are proposed based on Fractional Low Order statistics, Fractional Low Order Wigner-Ville Distribution (FLO-WVD), Fractional Low Order Statistic pseudo Wigner-Ville Distribution (FLO-PWVD), Fractional Low Order Statistic Cohen class distribution (FLO-Cohen). In order for real-time, on-line operation and fairly long signals processing, a new smoothed pseudo Fractional Low Order Cohen class distribution (PFLO-Cohen) is proposed.Simulations show that the methods demonstrate the advantages in this paper, are robust.

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Journal articles on the topic 'Fractional noise'

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