Mean Value Theorem Research Articles

An Error Compensation Method for Improving the Properties of a Digital Henon Map Based on the Generalized Mean Value Theorem of Differentiation.

Continuous chaos may collapse in the digital world. This study proposes a method of error compensation for a two-dimensional digital system based on the generalized mean value theorem of differentiation that can restore the fundamental performance of chaotic systems. Different from other methods, the compensation sequence of our method comes from the chaotic system itself and can be applied to higher-dimensional digital chaotic systems. The experimental results show that the improved system is highly consistent with the real chaotic system, and it has excellent chaotic characteristics such as high complexity, randomness, and ergodicity.

Numerical approach of riemann-liouville fractional derivative operator

<p>This article introduces some new straightforward and yet powerful formulas in the form of series solutions together with their residual errors for approximating the Riemann-Liouville fractional derivative operator. These formulas are derived by utilizing some of forthright computations, and by utilizing the so-called weighted mean value theorem (WMVT). Undoubtedly, such formulas will be extremely useful in establishing new approaches for several solutions of both linear and nonlinear fractionalorder differential equations. This assertion is confirmed by addressing several linear and nonlinear problems that illustrate the effectiveness and the practicability of the gained findings.</p>

A New Conformable Fractional Derivative and Applications

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

Semi-quasidifferentiability in nonsmooth nonconvex multiobjective optimization

In this paper, we extend the concept of quasidifferential to a new notion called semi-quasidifferential. This generalization is motivated by the convexificator notion. Some important properties of semi-quasidifferentials are established. The relationship between semi-quasidifferentials and the Clarke subdifferential is studied, and a mean value theorem in terms of semi-quasidifferentials is proved. It is shown that this notion is helpful to investigate nonsmooth optimization problems even when the objective and/or constraint functions are discontinuous. Considering a multiobjective optimization problem, a characterization of some cones related to the feasible set is provided. They are used for deriving necessary and sufficient optimality conditions. We close the paper by obtaining optimality conditions in multiobjective optimization in terms of semi-quasidifferentials. Some outcomes of the current work generalize the related results existing in the literature.

High-Accuracy Time Discretization of Stochastic Fractional Diffusion Equation

A high-accuracy time discretization is discussed to numerically solve the nonlinear fractional diffusion equation forced by a space-time white noise. The main purpose of this paper is to improve the temporal convergence rate by modifying the semi-implicit Euler scheme. The solution of the equation is only Hölder continuous in time, which is disadvantageous to improve the temporal convergence rate. Firstly, the system is transformed into an equivalent form having better regularity than the original one in time. But the regularity of nonlinear term remains unchanged. Then, combining Lagrange mean value theorem and independent increments of Brownian motion leads to a higher accuracy discretization of nonlinear term which ensures the implementation of the proposed time discretization scheme without loss of convergence rate. Our scheme can improve the convergence rate from $${\min \{\frac{\gamma }{2\alpha },\frac{1}{2}\}}$$ to $${\min \{\frac{\gamma }{\alpha },1\}}$$ in the sense of mean-squared $$L^2$$ -norm. The theoretical error estimates are confirmed by extensive numerical experiments.

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.

Multiplicative derivative and its basic properties on time scales

We define the multiplicative derivative and its properties on time scales. Then, we restate many concepts for multiplicative analysis such as derivative, Rolle's theorem, mean value theorem, and increasing‐decreasing property on time scales. We aim to create important fields of study by carrying this most important issue of multiplicative analysis, which has applications in economics, finance and many other fields, to time scale calculus.

Linear first order Riemann-Liouville fractional differential and perturbed Abel's integral equations

Linear first order Riemann-Liouville fractional differential equations are studied. These new equations unify and generalize the Riemann-Liouville, modified Caputo and Caputo fractional differential equations. The equivalences between the fractional differential equations and the corresponding perturbed Abel's integral equations are obtained. These results are useful not only to study the initial or boundary value problems for nonlinear first order Riemann-Liouville fractional differential equations but also to study the solutions of the perturbed Abel's integral equations arising in a problem of mechanics and many other physical problems. The well-known Tonelli's result on solvability of the Abel's integral equation is generalized. We exhibit that there are nonconstant equilibria for some first order Caputo fractional equations. This is different from nonlinear first order ordinary differential equations which have only constant equilibria. The equivalence results are applied to generalize the classical Mean Value Theorem to the first order Riemann-Liouville fractional derivatives.

A process-based flood frequency analysis within a trivariate statistical framework. Application to a semi-arid Mediterranean case study

This paper proposes a trivariate methodology for flood frequency estimation. It combines the flood peak, storm magnitude, and initial soil moisture condition (ISMC) as the main flood-related statistical variables to be considered. The semi-arid Mediterranean “Rambla del Poyo” catchment has been used as a representative case study where the influence of the spatio-temporal variability of the storms and the ISMC on floods can lead to differences of up to two orders of magnitude in quantiles when the most commonly used methods are applied. In order to incorporate the main flood-generating mechanisms, the integrated use of a multidimensional storm generator with distributed hydrological modelling is proposed. Flood quantiles are then estimated by combining the maximum flows with the storm magnitude and ISMC in a trivariate probability distribution function through the application of Bayes’ theorem and Lagrange’s Mean Value theorem. Although the methodology proposed in this paper has been applied and tested in only one case study, it can be extended to other case studies due to its process-based orientation.

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.

Model-Free Tracking Control with Prescribed Performance for a Shape Memory Alloy-Based Robotic Hand

The shape memory alloy (SMA)-based robotic hand has been a new emerging technology with potential applications ranging from life service to surgical treatment, because of the characteristics of SMA, such as high power-to-weight ratio, small volume and low driving voltage. However, due to the complex dynamic model and nonlinear aspects of SMA, it is complicated to control an SMA-based robotic hand. This paper presents a novel model free adaptive control for the SMA-based robotic hand system. By applying the Taylor series expansion method and the differential mean value theorem, the SMA based robotic hand system can be transformed into an equivalent linearization model, which merely depends on measurement data without any information on the system. Combined with prescribed performance control, the novel control method can constrain the tracking error in a preassigned domain. Experiments are conducted on the SMA-based robotic hand system to verify the performance of the presented control method.

Emerging approaches for nonlinear parameter varying systems

Emerging approaches for nonlinear parameter varying systems

A simplified adaptive tracking control for nonlinear pure‐feedback systems with input delay and full‐state constraints

SummaryThis article investigated the adaptive backstepping tracking control for a class of pure‐feedback systems with input delay and full‐state constraints. With the help of mean value theorem, the system is transformed into strict‐feedback one. By introducing the Pade approximation method, the effect of input delay was compensated. Radial basis function neural networks are utilized to approximate the unknown nonlinear functions. Furthermore, in order to reduce the computational burden by introducing backstepping design technique, dynamic surface control technique was employed. In addition, the number of the adaptive parameters that should be updated online was also reduced. By utilizing the barrier Lyapunov function, the closed‐loop nonlinear system is guaranteed to be semi‐globally ultimately uniformly bounded. Finally, a numerical simulation example is given to show the effectiveness of the proposed control strategy.

Finite-time impulsive observers for nonlinear systems represented by Takagi–Sugeno models: Application to a chaotic system

In this paper, observer design for nonlinear systems represented by Takagi Sugeno models (T-S) is investigated. The first main contribution concerns the finite time convergence of the estimations, ensured by an impulsive observer with state updates. The second contribution, lies with taking into account unmeasurable parameters, using the Differential Mean Value Theorem (DMVT) to express the disturbed error dynamics into a Linear Parameter Varying system. The stability conditions are formulated in terms of Linear Matrix Inequalities (LMI). To prove the efficiency of the proposed procedure, applications are performed on a chaotic system. The obtained results are pretty satisfying.

On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis

The discrete delta Caputo-Fabrizio fractional differences and sums are proposed to distinguish their monotonicity analysis from the sense of Riemann and Caputo operators on the time scale Z. Moreover, the action of Q− operator and discrete delta Laplace transform method are also reported. Furthermore, a relationship between the discrete delta Caputo-Fabrizio-Caputo and Caputo-Fabrizio-Riemann fractional differences is also studied in detail. To better understand the dynamic behavior of the obtained monotonicity results, the fractional difference mean value theorem is derived. The idea used in this article is readily applicable to obtain monotonicity analysis of other discrete fractional operators in discrete fractional calculus.

Mean-Value Theorem for Multiple Trigonometric Sums on the Sequence of Bell Polynomials

Mean-Value Theorem for Multiple Trigonometric Sums on the Sequence of Bell Polynomials

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.

On a basic mean value theorem with explicit exponents

We follow a paper by Sedunova regarding Vaughan’s basic mean value Theorem to improve and complete a more general demonstration for a suitable class of arithmetic functions as started by Cojocaru and Murty. As an application we derive a basic mean value theorem for the von Mangoldt generalized functions.

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.

Further analysis for consensus of hybrid multiagent systems: A unified framework

AbstractHybrid multiagent systems exist widely in real world and have been applied in many engineering fields. In this article, we consider the consensus problem for the hybrid multiagent system consisting of continuous‐time and discrete‐time dynamic agents. Different from the previous works, in this article, the interactions between different agents no longer only occur at the sampling time, but continuous‐time dynamic agents can interact with their continuous‐time dynamic neighbors in real time. By using graph theory and differential mean value theorem of matrix function, a consensus criterion is obtained for the hybrid multiagent system. A unified framework is also established for the consensus of continuous‐time and discrete‐time multiagent systems. Finally, a simulation example is given to illustrate the validity of our protocol.