Mean Value Theorem For Integrals Research Articles

A Generalization of Wayment’s Mean Value Theorem for Integrals

In this short note, we present a generalization of Wayment’s Mean Value Theorem for Integrals.

Robustness analysis of exponential stability of fuzzy inertial neural networks through the estimation of upper limits of perturbations

This paper characterizes the robustness of exponential stability of fuzzy inertial neural network which contains time delays or stochastic disturbance through the estimation of upper limits of perturbations. By utilizing Gronwall-Bellman lemma, stochastic analysis, Cauchy inequality, the mean value theorem of integrals, as well as the properties of integrations, the limits of both time delays and stochastic disturbances are derived in this paper which can make the disturbed system keep exponential stability. The constraints between the two types of disturbances are provided in this paper. Examples are offered to validate our results.

Adaptive Neural Consensus Tracking Control for Nonlinear Multiagent Systems Using Integral Barrier Lyapunov Functionals.

This article presents the adaptive tracking control scheme of nonlinear multiagent systems under a directed graph and state constraints. In this article, the integral barrier Lyapunov functionals (iBLFs) are introduced to overcome the conservative limitation of the barrier Lyapunov function with error variables, relax the feasibility conditions, and simultaneously solve state constrained and coupling terms of the communication errors between agents. An adaptive distributed controller was designed based on iBLF and backstepping method, and iBLF was differentiated by means of the integral mean value theorem. At the same time, the properties of neural network are used to approximate the unknown terms, and the stability of the systems is proven by the Lyapunov stability theory. This scheme can not only ensure that the output of all the followers meets the output trajectory of the leader but also make the state variables not violate the constraint bounds, and all the closed-loop signals are bounded. Finally, the efficiency of the proposed controller is revealed.

An Alternative Numerical Scheme to Approximate the Early Exercise Boundary of American Options

This paper deals with a new numerical method for the approximation of the early exercise boundary in the American option pricing problem. In more detail, using the mean-value theorem for integrals, we provide a flexible algorithm that allows for reaching a more accurate numerical solution with fewer calculations rather than other previously described methods.

Extracting the Oscillatory Component and Defining a Mean Amplitude of Thermokinetic Oscillations in the H/Pd System

Self-sustained oscillations may help in the research of catalytic reactions. But analysis of the resulting nonlinear time series may be complicated, not least due to the lack of a definite baseline. A mathematical method to transform such irregular time series into a more useful form that oscillates around the zero line has been proposed, which preserves both the frequency and the key topological aspects of the originals in the process. To this end, the mean value theorem for integrals has been applied in constructing a base curve for nonequilibrium, periodic thermokinetic oscillations, q(t), recorded microcalorimetrically in two experiments, respectively, with oscillatory sorptions of H2 and D2 in Pd. Once the mean values are so calculated for each period of q(t), they are subjected to cubic spline interpolation to form a new nonoscillatory curve, h(t). The latter can be used as an essential baseline for the oscillatory component of the original thermokinetic time series. Crucially, the areas under both q(t) and h(t) are supposed to be strictly identical. This being confirmed, the pointwise subtraction q(t) – h(t) yields another oscillatory time series g(t), considered to be the oscillatory component extracted from the original thermokinetic data. Subsequently, the method has been applied to a further group of nine microcalorimetric time series q(t) representing thermokinetic oscillations in the H/Pd system. Using the so-constructed g(t) curves (i.e., extracted oscillatory components), a new parameter, the mean amplitude, has been defined and used for correlating the observed gradation of intensities (i.e., amplitudes) of thermokinetic oscillations with a range of experimental conditions applied. The mean amplitude turns out to be a linear function of the first ionization potential of noble gases, admixed intentionally to H2 before its contacting Pd.

Dynamics on time scales of wave solutions for nonlinear neural networks

A time scale is an arbitrary non-empty closed subset of denoted by . In this work, the traveling wave solutions on time scales for a class of nonlinear delayed dynamical neural networks are investigated. The outcomes of this study are new and have been obtained by inequality technique, Ikehara's theorem, the weighted energy method, Taylor formula, comparison principle and the first integral mean value theorem. This is the first work focused for the wave solution in time space scales for a dynamical delayed neural networks model.

Regression-based Monte Carlo integration

Monte Carlo integration is typically interpreted as an estimator of the expected value using stochastic samples. There exists an alternative interpretation in calculus where Monte Carlo integration can be seen as estimating a constant function---from the stochastic evaluations of the integrand---that integrates to the original integral. The integral mean value theorem states that this constant function should be the mean (or expectation) of the integrand. Since both interpretations result in the same estimator, little attention has been devoted to the calculus-oriented interpretation. We show that the calculus-oriented interpretation actually implies the possibility of using a more complex function than a constant one to construct a more efficient estimator for Monte Carlo integration. We build a new estimator based on this interpretation and relate our estimator to control variates with least-squares regression on the stochastic samples of the integrand. Unlike prior work, our resulting estimator is provably better than or equal to the conventional Monte Carlo estimator. To demonstrate the strength of our approach, we introduce a practical estimator that can act as a simple drop-in replacement for conventional Monte Carlo integration. We experimentally validate our framework on various light transport integrals. The code is available at https://github.com/iribis/regressionmc.

Lyapunov‐based iterative learning feedback control design for parabolic MIMO PDEs with time‐varying delays

This paper presents an iterative learning control approach to achieve precise tracking control for a class of repeatable parabolic multiple-input–multiple-output (MIMO) partial differential equations (PDEs) with time-varying delays over a finite time interval. Feedback control is utilised in the iterative learning control system to improve the convergence speed of the repeatable system. A Lyapunov–Krasovskii functional is constructed to deal with the time-varying delay problem caused by the model uncertainty of MIMO PDEs. Collocated piecewise control and piecewise measurement is considered based on the distributions of actuators and sensors to generate multiple inputs and multiple outputs. By using the mean value theorem for integrals, variants of the Poincaré–Wirtinger inequality in the 1D space, Cauchy–Schwartz inequality and Gronwall–Bellman lemma, sufficient conditions that guarantee the convergence of the iterative learning feedback control system are presented. Numerical simulation experiments and comparisons verify the effectiveness and advantages of the proposed method.

Iterative learning feedback control for linear parabolic distributed parameter systems with multiple collocated piecewise observation

This paper presents a novel iterative learning feedback control method for linear parabolic distributed parameter systems with multiple collocated piecewise observation. Multiple actuators and sensors distributed at the same position of the spatial domain are utilized to perform collocated piecewise control and measurement operations. The advantage of the proposed method is that it combines the iterative learning algorithm and feedback technique. Not only can it use the iterative learning algorithm to track the desired output trajectory, but also the feedback control approach can be utilized to achieve real-time online update. By utilizing integration by parts, triangle inequality, mean value theorem for integrals and Gronwall lemma, two sufficient conditions based on the inequality constraints for the convergence analysis of the tracking error system are presented. Some simulation experiments are provided to prove the effectiveness of the proposed method.

Numerical treatment of the generalized time-fractional Huxley-Burgers’ equation and its stability examination

Abstract In this paper, we show how to approximate the solution to the generalized time-fractional Huxley-Burgers’ equation by a numerical method based on the cubic B-spline collocation method and the mean value theorem for integrals. We use the mean value theorem for integrals to replace the time-fractional derivative with a suitable approximation. The approximate solution is constructed by the cubic B-spline. The stability of the proposed method is discussed by applying the von Neumann technique. The proposed method is shown to be conditionally stable. Several numerical examples are introduced to show the efficiency and accuracy of the method.

Novel traveling waves solutions for nonlinear delayed dynamical neural networks with leakage term

In this work, the exponential stability of monotone traveling wave solutions for a class of nonlinear delayed dynamical neural networks with leakage term and distributed delays is investigated. The outcomes of this study are new and has obtained by applying of Ikeharas theorem, the weighted energy method, Taylor formula, comparison principle and the first integral mean value theorem. In addition, the leakage term is considered for the first time in this work to the dynamical model of cellular neural networks.

Adaptive event‐triggered control of pure‐feedback systems with quantized input and unknown input delay

AbstractThis article presents an adaptive event‐triggered dynamic surface control for pure‐feedback systems with unknown input time‐delay and quantized input. Event‐triggered and quantized input are both of discontinuity, which will make their integral item nondifferentiable and the integral mean value theorem cannot be applied. In order to solve the design problem of input delay system with discontinuous input, an auxiliary tracking error, an auxiliary system and Lyapunov–Krasovskii functionals are well designed, which can effectively deal with the input delay and allows the time delay to be unknown. Moreover, the improved event‐triggered quantized control can greatly reduce the amount of calculation of traditional quantized control and avoid unnecessary network access. The stability analysis shows that all signals in the closed‐loop system are semi‐globally uniformly ultimately bounded, and the output constraint is not violated. A simulation example of practical control system is conducted to demonstrate the effectiveness of proposed control protocol.

SEMI-ANALYTICAL METHOD FOR UNSTABLE PRESSURE OF 3D FRACTURE NETWORK

Due to the influence of in-situ stress and fracturing technology, the distribution of hydraulic fracture network in highly deviated wells is complex with different inclined directions, different distribution forms and different penetration degrees. In this paper, the fracture surface is discretized into several rectangular micro elements to realize the effective characterization of fracture morphology. The seepage process is divided into two stages: matrix flow to fracture and fracture flow to wellbore. The numerical solution of unsteady seepage in discrete fracture surface is constructed by using finite difference method, and the analytical solution of unsteady seepage in matrix is constructed by combining closed boundary source function and superposition principle. The solution of the unstable pressure of the 3D fracture network is obtained by coupling the numerical solution of flow in fracture and the analytical solution of flow in matrix. Based on the integral mean value theorem, a solution method of point source and special line source instead of surface source is proposed to solve the seepage in matrix. The feasibility and applicable conditions of this method are analyzed, which can ensure the accuracy of the model and improve the calculation efficiency. The research shows that the point source function area fraction method can accurately solve the bottom hole pressure dynamic of 3D pressure fracture network in the matrix, but the calculation is quite inefficient. The point source and special line source approximate surface source method based on the integral mean value theorem can greatly improve the calculation efficiency, and can achieve higher accuracy when the fracture micro element division is more precise (the dimensionless side length of micro element is less than 0.15). Based on this model, the flow regimes and related sensitivity analysis are analyzed. The results show that the fracture conductivity, fracture dip angle, fracture height and fracture interval have great influence on the typical well test curves of highly deviated wells. Fracture dip angle and fracture interval mainly affect the pressure and pressure derivative curve from linear flow to early radial flow, especially when the fracture height is small, the pressure derivative has backflow phenomenon.

On reachable set estimation of nonlinear singular systems with distributed delay

SummaryThe estimation of the reachable set for singular systems with distributed delay and nonlinear term under zero initial condition is studied in this article. Based on the Lyapunov functional with triple integrals, a sufficient condition for bounding the reachable set is established by combining the free‐weighting matrix method, the integral mean value theorem, and the new inequality scaling method. Then the reachable set of nonlinear singular system is bounded by the prescribed ellipsoids. Finally, two numerical examples and a practical example verify the validity of the theoretical analysis.

An Improved Nonhomogeneous Grey Model with Fractional-Order Accumulation and Its Application

The nonhomogeneous grey model has been seen as an effective method for forecasting time series with approximate nonhomogeneous index law, which has been widely used in diverse disciplines on account of its high prediction precision. However, there remains room for improvements. For this, this study presents an improved nonhomogeneous grey model by incorporating the dynamic integral mean value theorem and fractional accumulation simultaneously. In order to promote the efficacy of the optimised model, we apply the whale optimization algorithm (WOA) to ascertain its optimal parameter. In particular, two examples are conducted to validate the superiority of the proposed model in contrast with other benchmarks, and the experimental results show that the mean absolute percentage error of the proposed approach is 808692% and 6.0706%, respectively, indicating the proposed approach performs better than other competing models.

Integral mean value theorem for discontinuous function

This study examines the modification of the integral mean value theorem for discontinuous functions. Modification is studied by proving that a function that is not continuous at a certain and bounded interval can be integrated (finite integral) both in Rieman’s integral and Newton’s integral (integral as antiderivative). The discontinuity of the intended function, namely; f is defined on [a,b] but the value of a function and its limit are not equal at some points or infinite points and countable on (a,b), f is undefined on [a,b] at some points or infinite points and countable on (a,b) but its limit exists there. The results of this study provide a modification of the integral mean value theorem by replacing the value of f in the implication of the theorem with its limit value so that the integral mean value theorem is obtained for the non-continuous function.

Wall Thinning Assessment for Ferromagnetic Plate with Pulsed Eddy Current Testing Using Analytical Solution Decoupling Method

The wall-thinning measurement of ferromagnetic plates covered with insulations and claddings is a main challenge in petrochemical and power generation industries. Pulsed eddy current testing (PECT) is considered as a promising method. However, the accuracy is limited due to the interference factors such as lift-off and cladding. In this study, by decoupling analytic solution, a feature only sensitive to plate thickness is proposed. Based on the electromagnetic waves reflection and transmission theory, cladding-induced interference is firstly decoupled from the analytical model. Moreover, by using the first integral mean value theorem, interferences of insulation and the lift-off are decoupled, too. Hence, the method is proposed by calculating Euclidean distances between the normalized detection signal and normalized reference signal as the feature to assess wall thinning. Its effectiveness under various conditions is examined and results show that the proposed feature is only sensitive to the ferromagnetic plate thickness. Finally, the experiment is carried on to verify this method practicable.

Time-varying IBLFs-based adaptive control of uncertain nonlinear systems with full state constraints

This paper presents an adaptive control design for nonlinear systems with time-varying full state constraints. It is the first time to introduce the novel time-varying Integral Barrier Lyapunov functions (TVIBLFs) into the adaptive control design, which not only overcomes the limitation of conservatism existing in the traditional BLFs, but also guarantees that the full state time-varying constraint bounds are not violated. The TVIBLFs are combined with the backstepping design procedure to construct the controllers and adaptation laws, and the integral mean value theorem is used to differentiate TVIBLFs. It can be proven that all the states are forced in the time-varying regions and the stability of the closed-loop system is achieved. The effectiveness of the proposed adaptive control strategy can be illustrated through a simulation example.

An accurate and novel numerical simulation with convergence analysis for nonlinear partial differential equations of Burgers–Fisher type arising in applied sciences

Abstract In this study, the second-order nonlinear partial differential equations of Burgers–Fisher type are considered under a unique formulation by introducing a novel highly accurate numerical method based on the Nörlund rational polynomial and matrix-collocation computational system. The method aims to obtain a sustainable approach since it contains the rational structure of the Nörlund polynomial. A unique computer program module, which involves very few routines, is constructed to discuss the precision and efficiency of the method and these few steps are described via an algorithm. A residual function is employed in both the error and convergence analyses with mean value theorem for double integrals. The considered equations in the numerical tests stand for model phenomena arising widely in applied sciences. Graphical and numerical comparisons provide a clear observation about the consistency of the method. All results prove that the method is highly accurate, eligible, and provides the ultimate operation for aforementioned problems.

Research on Transmission Capacity Characteristics of the Cluster Flight Spacecraft Network

In order to analyze the transmission capacity performance of the cluster flight spacecraft network, there are two different types of outage performance theory which are derived in this paper. First of all, by applying the mean value theorem of integrals, the expression of the outage probability of decode-and-forward relaying is derived. Subsequently, according to the Macdonald random variable form, the expression of the outage probability of amplify-and-forward is derived. By simulating the transmission capacity of decode-and-forward, the transmission capacity characteristics of a single hop and dual hops are analyzed. The simulation results showed that transmission capacity performance changes with the change of the time slot in the orbital hyperperiod, and the transmission capacity of a dual-hop relay has better performance than a single-hop transmission in the cluster flight spacecraft network.