Mean Value Theorem For Integrals Research Articles

A Class of Nonlinear Difference Inequality with Weakly Singularity in Engineering

In this paper, we discuss a class of new nonlinear weakly singular difference inequality, which is solved by change of variable, the mean-value theorem for integrals and amplification method, Gamma function, and explicit bounds for the unknown functions is given clearly.

Estimation of Unknown Function in a Class of Singular Difference Inequality in Engineering

In this paper, we discuss a class of new weakly singular difference inequality, which is solved using change of variable, discrete Jensen inequality, Beta function, the mean-value theorem for integrals and amplification method, and explicit bounds for the unknown functions is given clearly. The derived results can be applied in the study of fractional difference equations in Engineering.

Renewal and Renewal-Intensity Functions with Minimal Repair

The renewal and renewal-intensity functions with minimal repair are explored for the Normal, Gamma, Uniform, and Weibull underlying lifetime distributions. The Normal, Gamma, and Uniform renewal, and renewal-intensity functions are derived by the convolution method. Unlike these last three failure distributions, the Weibull except at shape β=1 does not have a closed-form function for the n-fold convolution. Since the Weibull is the most important failure distribution in reliability analyses, the approximate renewal and renewal-intensity functions of Weibull were obtained by the time-discretizing method using the Mean-Value Theorem for Integrals. A Matlab program outputs all reliability and renewal measures.

Analytical solution for combined heat and mass transfer in laminar falling film absorption using first type boundary conditions at the interface

Since the late seventies of the 20th century, several analytical models for combined heat and mass transfer in laminar falling film absorption have been proposed. Nevertheless the analytical solutions obtained with the Fourier method for the coupled process are complex and for short flow length a certain instability occurs which have been explained with the inconsistency of the initial and boundary conditions. Therefore boundary layer models have been justified in order to solve the transfer problem for short flow length. Moreover a linear approximation of the phase equilibrium is required.The analytical solutions for heat and mass transfer presented in this paper are obtained by using the Laplace transform to solve the partial differential equations for an isothermal as well as impermeable wall. An originally unknown constant temperature and mass fraction boundary condition at the interface are set. The temperature and mass fraction profile across the film are obtained formally independently. In order to determine the yet unknown interface temperature and mass fraction the phase equilibrium and the interface energy balance are applied, using averaged gradients with regard to the streamwise coordinate. The interface temperature and mass fraction obtained with this procedure are interpreted and treated as mean values. From the known evolution of the mean interface temperature and mass fraction, the local values are derived by inverting the first mean value theorem for integration. The results show very good agreement to the established analytical solutions. The solving procedure does not depend on the input parameters such as the Lewis number for instance, which is needed in order to determine the eigenvalues within the Fourier method. Moreover arbitrary correlations for the phase equilibrium are applicable. The present solution is mathematically stable and offer explicit expressions in order to calculate the mean heat and mass fluxes directly. Therefore this solution is favourable to implement entire absorption process simulation, yet describing the coupled heat and mass transfer process comprehensively.

Oscillation Criteria for a Class of Second Order Neutral Delay Differential Equations

Oscillation criteria for a class of second order neutral delay differential equations of the form [c(t)((x(t)+p(t)x(t-τ))^' )^α ]^'+q(t)f(x(t-σ) )=0,t≥t_0 is studied. By using first and second mean value theorem of integrals, the new sufficient condition is obtained and the corresponding result what was already obtained is generalized by the result in this paper.

A Class of Iterative Nonlinear Difference Inequality with Weakly Singularity

We discuss a class of new nonlinear weakly singular difference inequality, which is solved by change of variable, discrete Hölder inequality, discrete Jensen inequality, the mean-value theorem for integrals and amplification method, and Gamma function. Explicit bound for the unknown function is given clearly. Moreover, an example is presented to show the usefulness of our results.

Field-dependent critical state of high-Tc superconducting strip simultaneously exposed to transport current and perpendicular magnetic field

We present an exact analytical approach for arbitrary field-dependent critical state of high-Tc superconducting strip with transport current. The sheet current and flux-density profiles are derived by solving the integral equations, which agree with experiments quite well. For small transport current, the approximate explicit expressions of sheet current, flux-density and penetration depth for the Kim model are derived based on the mean value theorem for integration. We also extend the results to the field-dependent critical state of superconducting strip in the simultaneous presence of applied field and transport current. The sheet current distributions calculated by the Kim model agree with experiments better than that by the Bean model. Moreover, the lines in the Ia-Ba plane for the Kim model are not monotonic, which is quite different from that the Bean model. The results reveal that the maximum transport current in thin superconducting strip will decrease with increasing applied field which vanishes for the Bean model. The results of this paper are useful to calculate ac susceptibility and ac loss.

An integral split-step fourier method for digital back propagation to compensate fiber nonlinearity

In optical fiber transmission systems, the split-step Fourier method (SSFM) has been widely used in digital back propagation (DBP) to compensate fiber nonlinearity. In this paper, by using the Lagrange's Integral Mean Value Theorem (LIMVT), we derive an analytical expression to calculate the optimal value of the nonlinearity calculation position (NLCP) for different systems and we propose an integral SSFM (I-SSFM) based on the expression. The I-SSFM can be performed more accurately and efficiently without parameter optimization. Simulations of various transmission links show that the I-SSFM outperforms the conventional asymmetric SSFM (A-SSFM) and the symmetric SSFM (S-SSFM) significantly, especially when we employ less amount of steps to ensure computation efficiency. The computation effort of the I-SSFM reaches as low as 50% of that of the S-SSFM.

Lyapunov-based design of locally collocated controllers for semi-linear parabolic PDE systems

The design problem of collocated feedback controllers is addressed in this paper for a class of semi-linear distributed parameter systems described by parabolic partial differential equation (PDE), where a finite number of local actuators and sensors are intermittently distributed in space. A Lyapunov direct method for the exponential stability analysis of the resulting closed-loop system is first presented for the system, in which the first mean value theorem for integration and the Wirtinger's inequality are employed. The corresponding stabilization condition is then derived through the analysis result. Finally, the proposed design method is implemented on the feedback control of a fisher equation and its effectiveness is evaluated through simulation results.

Note on the integral mean value method for Fredholm integral equations of the second kind

We study the paper of Avazzadeh et al. [Z. Avazzadeh, M. Heydari, G.B., Loghmani, Numerical solution of Fedholm integral equations of the second kind by using integral mean value theorem, Appl. Math. Model. 35 (2011) 2374–2383] with the integral mean value method for Fredholm integral equations of the second kind. The objective of the note is threefold. First, we point out a basic error in the paper. Second, we find that the given numerical examples are only related to the special cases of Fredholm integral equations of the second kind with the degenerate kernels, which can be solved simply. Third, due to the basic error, our observations reveal that generally the suggested method should not be considered for a Fredholm integral equation of the second kind.

Application of mean value theorem of integrals in the efficiency callibration technology of radioactive xeon sample

The radioxenon from environmental samples is usually measured by the HPGe detector. The relative measurement method is used in the analysis process. However, the half-life of radioactive isotopes of xenon is too short to be used as a standard, so the calibration of the HPGe detector is difficult. In order to resolve this problem, the mixed disk sources are used to simulate the efficiencies of four radioactive xenon isotopes, and the mean value theorem of integrals is adopted, thereby improving the accuracy and simplifying the process. The samples are obtained by the xenon system under environment of high activity of radon, then the γ spectrum with regions of interest for 133Xe are obtained by the sample measurement. The activity of 133Xe is given by using the detection efficiency of disk source at a certain source-detector distance. The detection efficiency simulated by disk source is consistent with the detection efficiency of gas cell. Therefore, the simulated technology of disk source and mean value theorem of integrals are identified to be correct and effective.

Integral mean value method for solving a general nonlinear Fredholm integro-differential equation under the mixed conditions

In this paper, the integral mean value method is employed to handle the general nonlinear Fredholm integro-differential equations under the mixed conditions. The application of the method is based on the integral mean value theorem for integrals. By using the integral mean value method, an integro-differential equation is transformed to an ordinary differential equation, then by solving it, the obtained solution is transformed to a system of nonlinear algebraic equations to calculate the unknown values. The efficiency of the approach will be shown by applying the procedure on some examples. In this respect, a comparison with series pattern solutions, obtained by some analytic methods, is given. For the approximate solution given by integral mean value method, the bounds of the absolute errors are given. The Mathematica program of the integral mean value method based on the procedure in this paper is designed.

Output-feedback adaptive fuzzy control for a class of nonlinear systems with input delay and unknown control directions

In this paper, an adaptive fuzzy control scheme is proposed for a class of nonlinear systems with input delay, unknown time-varying delays and disturbances via dynamic output-feedback approach. During the controller design procedure, the input delay is dealt with by combining the smoothed dead-zone approach with the integral mean value theorem. Moreover, fuzzy logic systems (FLS) are employed to approximate the unknown continuous functions. The main advantages of this paper are that (1) the output-feedback adaptive fuzzy controller can dispose a class of nonlinear time-varying delay systems with input delay, in which the virtual control coefficients are all unknown and (2) only two parameters need to be adjusted online in controller design procedure, which reduces the online computation burden greatly. It is proven that the proposed control scheme renders the closed-loop system stable in the sense of semiglobal UUB. Finally, simulation results are provided to show the effectiveness of the proposed approach.

The HOL Light Theory of Euclidean Space

We describe the library of theorems about N-dimensional Euclidean space that has been formalized in the HOL Light prover. This formalization was started in 2005 and has been extensively developed since then, partly in direct support of the Flyspeck project, partly out of a general desire to develop a well-rounded and comprehensive theory of basic analytical, geometrical and topological machinery. The library includes various `big name' theorems (Brouwer's fixed point theorem, the Stone-Weierstrass theorem, the Tietze extension theorem), numerous non-trivial results that are useful in applications (second mean value theorem for integrals, power series for real and complex transcendental functions) and a host of supporting definitions and lemmas. It also includes some specialized automated proof tools. The library has as planned been applied to the Flyspeck project and has become the basis of a significant development of results in complex analysis, among others.

The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus

Abstract We generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz. the Riemann-Liouville operator, instead of a classical (firstorder) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.

Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem II. High dimensional problems

In this work, we generalize the numerical method discussed in [Z. Avazzadeh, M. Heydari, G.B. Loghmani, Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem, Appl. math. modelling, 35 (2011) 2374–2383] for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind. The presented method can be used for solving integral equations in high dimensions. In this work, we describe the integral mean value method (IMVM) as the technical algorithm for solving high dimensional integral equations. The main idea in this method is applying the integral mean value theorem. However the mean value theorem is valid for multiple integrals, we apply one dimensional integral mean value theorem directly to fulfill required linearly independent equations. We solve some examples to investigate the applicability and simplicity of the method. The numerical results confirm that the method is efficient and simple.

Rolling With Simplified Stream Function Velocity and Strain Rate Vector Inner Product

The dual-stream function velocity field is reduced in order to analyze two-dimensional plate broadside rolling in roughing. The strain rate vector inner product and integral mean value theorem, as well as co-line vector inner product are used respectively in plastic deformation power, friction losses and shear power. A theoretical solution of roll torque and separating force for the rolling is obtained and the calculated results by the solution are compared with those measured in broadside rolling on-line. It shows that both the force and torque calculated are higher than those of measured, but the maximum relative error between them is no more than 11%

A stronger version of the second mean value theorem for integrals

We prove a stronger version of the classic second mean value theorem for integrals.

Solution of rectangular bar forging with bulging of sides by strain rate vector inner-product

A new linear integration solution was constructed for determining the total pressure developed during forging of a rectangular bar with flat tools. The effective strain rate for rectangular bar forging with bulging was expressed in terms of four-dimensional strain rate vector. The inner-product of the vector was termwise integrated and summed. The integral mean value theorem was applied to determining the ratio of the strain rate components and the values of direction cosine of the vector and then an analytical solution of stress effective factor was obtained. The compression experiments of pure lead bar were performed to test the accuracy of the solution. The optimized results of total pressure by golden section search were compared with those of the indicator readings of the testing machine. It indicates that the optimized total pressures are 2.60%-10.14% higher than those measured. The solution is available and still an upper-bound solution.

Analysis of Creep and Shrinkage Mechanism of Bridge Considering the Effect of Shrinkage on Creep Stress Reduction

The effect of concrete shrinkage and creep on the deformation of the bridge structure is posing a serious issue with increasing numbers of large-span bridges are being built. At the moment, only the creep coefficient is usually considered for calculating bridge creep, without taking other factors into consideration. This paper, based on aging theory, considers the influence of shrinkage on creep stress reduction, by simplifying the formula of creep strain under discrete load by using the integral mean-value theorem and deriving the formula of concrete creep strain. Taking a continuous rigid frame bridge for example, the paper calculated the creep considering creep stress reduction under shrinkage, which showed that the method can well predict the final shrinkage and creep values of large-span concrete bridge.