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The distribution models of wind speed data are essential to assess the potential wind speed energy because they decrease the uncertainty in estimating wind energy output. Therefore, before performing a detailed potential energy analysis, the precise distribution model for data relating to wind speed must be found. This research contains material from numerous goodness-of-fit tests, such as Kolmogorov–Simonov, Anderson–Darling, chi-square, root mean square error, Akaike information criterion, and Bayesian information criterion, which were combined finally to determine the wind speed of the best-fitted distribution. The suggested method collectively makes each criterion. This method was useful in statistically fitting 14 distribution models to wind speed data collected at four sites in Pakistan. The consequences show that this method provides the best source for selecting the most suitable wind speed statistical distribution. Also, the graphical representation is consistent with the analytical consequences. This research presents three estimation methods that can be used to calculate the different distributions used to estimate the wind. In the suggested maximum likelihood method, method of moments, and maximum likelihood estimation, the third-order moment used in the wind energy formula is a crucial function because it contributes to the precise estimate of wind energy. In order to prove the presence of the suggested method of moments, it was compared with well-known estimation methods, such as the method of linear moments and maximum likelihood estimation. In the relative analysis, given several goodness-of-fit tests, the presentation of the considered techniques is estimated based on the actual wind speed evaluated in different periods. The results show that the method of moments provides a more precise estimation than other commonly used methods for estimating wind energy based on the 14 distributions. Therefore, the method of moments can be a better technique for assessing wind energy. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a version of the Feynman-Kac formula for the tempered fractional general diffusion equation$ \begin{align} \partial^{\beta, \eta}_{t} u(t,x) = \mathfrak{L}u(t,x) +b(t)u(t,x),\; \; x\in\mathcal{X},\; t\geq0, \end{align} $with initial value $ f $ belonging to a Banach space $ (\mathbb{B}, \|\cdot\|) $, where $ \partial^{\beta, \eta}_{t} $ denotes the Caputo tempered fractional derivative with order $ \beta\in(0,1) $ and tempered parameter $ \eta>0 $, $ b(t) $ is a bounded and continuous external potential on $ [0, \infty) $, $ \mathfrak{L} $ is the infinitesimal generator of a general time-homogeneous strong Markov process $ \{X_{t}\}_{t\geq0} $, and $ \mathcal{X} $ denotes a Lusin space that is a topological space being homeomorphic to a Borel subset of a compact metric space. By using the properties of the tempered $ \beta $-stable subordinator $ S_{\beta,\eta}(t) $ and the inverse tempered $ \beta $-stable subordinator $ D_{\beta,\eta}(t) $, and the stochastic calculus for the stochastic integral driven by $ D_{\beta,\eta}(t) $, we show that the Feynman-Kac representation $ u(t,x) $ defined by$ \begin{align} u(t,x) = {\mathbb{E}}^{x}\bigg[f(X_{D_{\beta,\eta}(t)}) e^{\int_{0}^{t}b(r)dD_{\beta,\eta}(r)}\bigg] \end{align} $is the unique mild and weak solutions to the tempered fractional general diffusion equation. From the Feynman-Kac formula, we further show the continuity of the solution with respect to time based on the integral properties of the Mittag-Leffler function and differential formula of covariance for $ D_{\beta,\eta}(t) $. By exploring the scaling property of $ D_{\beta,\eta}(t) $, the explicit order is also presented for the continuity of the solution with respect to tempered parameter $ \eta $. [ABSTRACT FROM AUTHOR]

In this paper, we consider stochastic evolution equations with finite delay driven by fractional Brownian motion (fBm) with Hurst index $ H\in(0,1) $. First, the global existence and uniqueness of mild solutions are established by using a temporally weighted norm, the semigroup theory, the Banach fixed point theorem, and a new estimate of the stochastic integral with respect to fBm. Then after extending the integral inequality to the general case, we obtain the mean-square exponential stability of mild solutions. In particular, the convolution method and the Young convolution inequality are utilized to deal with the difficulty of the noise term caused by fBm with Hurst index $ H\in(0,1) $. Finally, the stochastic evolution equation with multiple time-varying delays and the reaction diffusion neural networks system with time-varying delay are considered, and the global existence, uniqueness and exponential stability of mild solutions are proved by using the abstract results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and uniqueness of mild solutions for stochastic nonlocal evolution equations with Lévy diffusion operator and nonlocal initial conditions. Based on the continuity of the Lévy semigroup and the technique of the measure of noncompactness, we establish the local existence of mild solutions in C 0 , T ; L q (Ω ; L q (ℝ N)) under some weaker growth conditions. Moreover, we obtain the existence of mild solutions on any finite interval by using the general growth conditions on the nonlinear. Finally, the global existence and uniqueness of mild solutions follow from the additional Lipschitz conditions on nonlinear terms. [ABSTRACT FROM AUTHOR]

The objective of this article is to introduce and study Itô type stochastic integrals with respect to tempered fractional Brownian motion (TFBM) of Hurst index H ∈ (1 2 , 1) and tempering parameter λ > 0 , by using the Wick product. The main tools are fractional calculus and Malliavin calculus. The Itô formula for this stochastic integral is established for the Itô type processes driven by TFBM. Based on this new Itô formula, we analyze the stability of stochastic differential equations driven by TFBM in the sense of p -th moment. A numerical example is given to illustrate our stability results. [ABSTRACT FROM AUTHOR]

This paper is concerned with stochastic delay evolution equations driven by tempered fractional Brownian motion (tfBm) $ B_Q^{\sigma, \lambda}(t) $ with time fractional operator of order $ \alpha\in (1/2+\sigma, 1) $, where $ \sigma\in (-1/2, 0) $ and $ \lambda>0 $. First, we establish the global existence and uniqueness of mild solutions by using the new established estimation of stochastic integrals with respect to tfBm. Moreover, based on the relations between the stochastic integrals with respect to tfBm and fBm, we show the continuity of mild solutions for stochastic delay evolution equations when tempered fractional noise is reduced to fractional noise. Finally, we analyze the stability with general decay rate (including exponential, polynomial and logarithmic stability) of mild solutions for stochastic delay evolution equations with tfBm and time tempered fractional operator. [ABSTRACT FROM AUTHOR]

To model wave propagation in inhomogeneous media with frequency-dependent power-law attenuation, it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time. The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative Gaussian noise and fractional Gaussian noise, because of the potential fluctuations of the external sources. We first give a representation of the mild solution and some stability estimates for the homogeneous problem, and then prove the existence and uniqueness of the mild solution by using fixed point theorem. Finally, the decay and regularity theory of the solution are provided. [ABSTRACT FROM AUTHOR]