On Solvability of the Integro-Differential Equations

In this paper, we study the existence of solution to integro-differential equations in the space of Lebesgue-integrable on un-bounded interval after transformed to nonlinear integral functional equation, the used tool is the fixed point theorem due to Schauder with weak measure of non compactness, due to De-Blasi. In addition, we give an example which satisfies the conditions of our existence theorem.

Lupaş post quantum Bernstein operators over arbitrary compact intervals

This paper deals with Lupaş post quantum Bernstein operators over arbitrary closed and bounded interval constructed with the help of Lupaş post quantum Bernstein bases. Due to the property that these bases are scale invariant and translation invariant, the derived results on arbitrary intervals are important from computational point of view. Approximation properties of Lupaş post quantum Bernstein operators on arbitrary compact intervals based on Korovkin type theorem are studied. More general situation along all possible cases have been discussed favouring convergence of sequence of Lupaş post quantum Bernstein operators to any continuous function defined on compact interval. Rate of convergence by modulus of continuity and functions of Lipschitz class are computed. Graphical analysis has been presented with the help of MATLAB to demonstrate approximation of continuous functions by Lupaş post quantum Bernstein operators on different compact intervals.

Neural Network-Based Electric Vehicle Range Prediction for Smart Charging Optimization

Range prediction is a standard feature in most modern road vehicles, allowing drivers to make informed decisions about when to refuel. Most vehicles make range predictions through data- or model-driven means, monitoring the average fuel consumption rate or using a tuned vehicle model to predict fuel consumption. The uncertainty of future driving conditions makes the range prediction problem challenging, particularly for less pervasive battery electric vehicles (BEV). Most contemporary machine learning-based methods attempt to forecast the battery SOC discharge profile to predict vehicle range. In this work, we propose a novel approach using two recurrent neural networks (RNNs) to predict the remaining range of BEVs and the minimum charge required to safely complete a trip. Each RNN has two outputs which can be used for statistical analysis to account for uncertainties; the first loss function leads to mean and variance estimation (MVE), while the second results in bounded interval estimation (BIE). These outputs of the proposed RNNs are then used to predict the probability of a vehicle completing a given trip without charging, or if charging is needed, the remaining range and minimum charging required to finish the trip with high probability. Training data was generated using a low-order physics model to estimate vehicle energy consumption from historical drive cycle data collected from medium-duty last-mile delivery vehicles. The proposed method demonstrated high accuracy in the presence of day-to-day route variability, with the root-mean-square error (RMSE) below 6% for both RNN models.

Logit Truncated-Exponential Skew-Logistic Distribution with Properties and Applications

In recent years, bounded distributions have attracted extensive attention. At the same time, various areas involve bounded interval data, such as proportion and ratio. In this paper, we propose a new bounded model, named logistic Truncated exponential skew logistic distribution. Some basic statistical properties of the proposed distribution are studied, including moments, mean residual life function, Renyi entropy, mean deviation, order statistics, exponential family, and quantile function. The maximum likelihood method is used to estimate the unknown parameters of the proposed distribution. More importantly, the applications to three real data sets mainly from the field of engineering science prove that the logistic Truncated exponential skew logistic distribution fits better than other bounded distributions.

Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations

In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: $$ \textstyle\begin{cases} u_{t}=\alpha u_{xx}+\beta [\varphi (u) ]_{xx}+f(u) &\text{in} \ Q:=\Omega \times (0,T), \\ u=0 &\text{on} \ \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) &\text{in} \ \Omega , \end{cases} $$ { u t = α u x x + β [ φ ( u ) ] x x + f ( u ) in Q : = Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , where $T>0$ T > 0 , $\Omega \subset \mathbb{R}$ Ω ⊂ R is a bounded interval, $u_{0}$ u 0 is nonnegative bounded Radon measure on Ω, and $\alpha , \beta \geq 0$ α , β ≥ 0 , under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.

Periodic Points of Asymmetric Bernoulli Shifts

It is well-known that Sharkovskii’s theorem gives a complete structure of periodic order for a continuous self-map on a closed bounded interval. As a further study, a natural problem is how to determine the location and number of periodic points for a specific map. This paper considers the periodic points of asymmetric Bernoulli shift, which is a piecewise linear chaotic map.

Pseudo Almost Automorphic Solutions for Stochastic Differential Equations Driven by Lévy Noise and Its Optimal Control

As we know, the periodic functions are symmetric within a cycle time, and it is meaningful to generalize the periodicity into more general cases, such as almost periodicity or almost automorphy. In this work, we introduce the concept of Poisson Sγ2-pseudo almost automorphy (or Poisson generalized Stepanov-like pseudo almost automorphy) for stochastic processes, which are almost-symmetric within a suitable period, and establish some useful properties of such stochastic processes, including the composition theorems. In addition, we apply a Krasnoselskii–Schaefer type fixed point theorem to obtain the existence of pseudo almost automorphic solutions in distribution for some semilinear stochastic differential equations driven by Lévy noise under Sγ2-pseudo almost automorphic coefficients. In addition, then we establish optimal control results on the bounded interval. Finally, an example is provided to illustrate the theoretical results obtained in this paper.

Limit measures and ergodicity of fractional stochastic reaction–diffusion equations on unbounded domains

This paper deals with invariant measures of fractional stochastic reaction–diffusion equations on unbounded domains with locally Lipschitz continuous drift and diffusion terms. We first prove the existence and regularity of invariant measures, and then show the tightness of the set of all invariant measures of the equation when the noise intensity varies in a bounded interval. We also prove that every limit of invariant measures of the perturbed systems is an invariant measure of the corresponding limiting system. Under further conditions, we establish the ergodicity and the exponentially mixing property of invariant measures.

Banach-Steinhaus Theorem for the Space P of All Primitives of Henstock-Kurzweil Integrable Functions

In this paper, it is shown how the Banach-Steinhaus theorem for the space P of all primitives of Henstock-Kurzweil integrable functions on a closed bounded interval, equipped with the uniform norm, can follow from the Banach-Steinhaus theorem for the Denjoy space by applying the classical Hahn-Banach theorem and Riesz representation theorem.

The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach

The model reduction of a mesoscopic kinetic dynamics to a macroscopic continuum dynamics has been one of the fundamental questions in mathematical physics since Hilbert's time. In this paper, we consider a diagram of the diffusion limit from the Vlasov-Poisson-Fokker-Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson-Nernst–Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system by computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP) scheme. Also, we provide several theoretical evidence that the Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solutions of each system if the total loss function vanishes.