Mean Value Theorem For Integrals Research Articles

On the uniform convergence of double sine integrals over –ℝ 2+

We investigate the convergence behavior of the family of double sine integrals of the form ∫ ∞ 0 ∫ ∞ 0 f ( x,y ) sin ux sin vy dx dy , where ( u,v ) ∈ ℝ 2 + := ℝ + × ℝ + , ℝ + := (0, ∞), and f is a monotonically nonincreasing function. We give necessary and sufficient conditions for the uniform convergence of the ‘remainder’ integrals ∫ b 1 a 1 ∫ b 2 a 2 to zero in ( u,v ) ∈ ℝ 2 + as max{ a 1 , a 2 } → ∞ , where b j > a j ≥ 0 , j = 1,2 (called uniform convergence in the regular sense). This implies the uniform existence of the finite limits of the partial integrals ∫ b 1 0 ∫ b 2 0 in ( u,v ) ∈ ℝ 2 + as min{ b 1 , b 2 } → ∞ (called uniform convergence in Pringsheim´s sense). Our basic tool is the second mean value theorem for certain double integrals over a rectangle.

Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem

In this paper, we present a new semi-analytical method for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind and the systems including them. The main idea in this method is applying the mean value theorem for integrals. Some examples are presented to show the ability of the model. The results confirm that the method is very effective and simple.

An extension of the mean value theorem for integrals

In this note we present an extension of the mean value theorem for integrals. The extension we consider is motivated by an older result (here referred as Corollary 2), which is quite classical for the literature of Mathematical Analysis or Calculus. We also show an interesting application for computing the sum of a harmonic series.

Sahoo- and Wayment-type integral mean value theorems

In this article, by using Rolle's theorem, we establish some results related to the mean value theorem for integrals. Our results are different from the set of integral mean value theorems which are given by Wayment [An integral mean value theorem, Math. Gazette 54 (1970), pp. 300–301] and Sahoo [Some results related to the integral mean value theorem, Int. J. Math. Ed. Sci. Tech. 38(6) (2007), pp. 818–822]. The importance of our results are illustrated by interesting examples.

Improving explicit bounds for the solutions of second order linear differential equations

This paper improves the algorithm for the construction of explicit bounds for the solutions of second order linear differential equations of the type [ p ( x ) y ′ ( x ) ] ′ + q ( x ) y ( x ) = 0 , p ( x ) , q ( x ) > 0 , x > x 0 described in a recent paper of the authors. The main improvements come from the introduction of new functionals devised by Trevisan and from the application of an enhanced version of the mean value theorem for integrals when some functions of the integrand are monotonic.

Measuring Curvature in Concrete Slabs and Connecting the Data to Slab Modeling Theory

Slab curling, warping, and load deflection basins are analyzed by quantifying slab curvature. Warping and curling curvature can cause the edges or midslab regions of slabs to lift off the foundation, with increases in slab-bending stress and deflections caused by loading. This paper provides an overview and comparison of methods for measuring slab curvature from curling and warping. Methods described include the end-slope, string-line, and straight-edge methods, best-fit polynomials, and shape functions. In addition, for multiple slabs, a rapid traveling profiler method is described: the profiler uses thousands of closely spaced data points and a generalized analysis of variation of curvature known as the road profile curvature index. To show key differences, the methods are compared by means of a test slab. The three primary ways of matching measured slab shape data to theoretical models are matching on the basis of the mean value theorem for integrals for continuous functions, matching on the basis of a best-fit or least squares error approach for the full slab length, or matching the difference in magnitude of the elevation between the ends and middle portions of the slabs. It is shown that fitting methods based on the mean value theorem for integrals are the most accurate and theoretically correct and will closely match the end of slab slope conditions near joints and overall average slab curvature simultaneously. On the basis of the mean value theorem for integrals, how to fit theoretical equations by Westergaard or any finite element method solution to the site average slab curvature measurement is described.

METHOD FOR CALCULATING THE OPACITY OF THE ATOMIC LINES IN STELLAR ATMOSPHERES

A new approximate method for calculating the opacity of the atomic lines in the computation of stellar model atmospheres is presented. Transforming the sums in the equation of the opacity of the lines into integrals and considering the Kramers equation for the oscillator strengths for level transitions in the hydrogenic atoms as a continuous function of frequency, we can apply the mean value theorem for integrals in order to describe the variation of the global opacity of the lines of all the chemical elements. The high-lying levels in nonhydrogenic atoms as well as their hydrogen-like transitions can also benefit from this method. For the case of stellar atmospheres not in local thermodynamic equilibrium, we can apply our method directly. The compact analytic expressions obtained are easy to use. The application of the method to the hydrogen lines shows the classical line-blanketing effects.

Approximate analytic temperature distribution and efficiency for annular fins of uniform thickness

The salient feature in the quasi one-dimensional differential equation for annular fins of uniform thickness is without question the presence of the variable coefficient 1 / r multiplying the first order derivative, d T / d r . A good-natured manipulation of the variable coefficient 1 / r is the principal objective of the present work. Specifically, the manipulation applies the mean value theorem for integration to 1 / r in the proper fin domain extending from the inner radius r 1 to the outer radius r 2 . It is demonstrated that approximate analytic temperature profiles and heat transfer rates of good quality are easily obtainable without resorting to the exact analytic temperature distribution and heat transfer rate embodying modified Bessel functions. For enhanced visualization, the computed temperature profiles, tip temperatures and fin efficiencies of approximate nature are graphed and tabulated for realistic combinations of the normalized radii ratio c and the thermo-geometric fin parameter ξ of interest in thermal engineering applications.

Temperature/Heat Analysis of Annular Fins of Hyperbolic Profile Relying on the Simple Theory for Straight Fins of Uniform Profile

This technical brief addresses an elementary analytic procedure for solving approximately the quasi-1D heat conduction equation (a generalized Airy equation) governing the annular fin of hyperbolic profile. The importance of this fin configuration stems from the fact that its geometrical shape and heat transfer performance are reminiscent of the annular fin of convex parabolic profile, the so-called optimal annular fin. To avoid the disturbing variable coefficient in the quasi-1D heat conduction equation, usage of the mean value theorem for integration is made. Thereafter, invoking a coordinate transformation, the product is a differential equation, which is equivalent to the quasi-1D heat conduction equation for the simple straight fin of uniform profile. The nearly exact analytic temperature distribution is conveniently written in terms of the two controlling parameters: the normalized radii ratio c and the dimensionless thermogeometric parameter M2, also called the enlarged Biot number. For engineering analysis and design, the estimates of temperatures and heat transfer rates for annular fins of hyperbolic profile owing realistic combinations of c and M2 give evidence of good quality.

Two new analytic approximations of the Chandrasekhar's H function for isotropic scattering

The Chandrasekhar's H functions play a central role in theoretical descriptions of radiative transfer in planetary and stellar atmospheres. In the present work our aim is to obtain in a simple way new analytic approximations of the Chandrasekhar's function for isotropic scattering, which would be sufficiently simple but more accurate than the existing approximations. We apply the mean value theorem for definite integrals in the nonlinear integral equation for the Chandrasekhar's H function. In this way the integral equation is formally solved but the solution depends on a new unknown parameter. We determine this parameter approximately, from the condition that the obtained H function matches the zero-order moment of the H function, which is known exactly, as accurately as possible in the whole range of the single particle albedo. The result gives our first analytic approximation for the H function. Using it as a starting approximation in the corresponding integral equation, after only one iteration which may be performed analytically, we obtain our second analytic approximation. The maximum relative error of our first analytic approximation, which is very simple in structure, is below 2.5%. The accuracy of our second approximation is within 0.07%, so that it highly surpasses the accuracy of the other analytic approximations available in the literature.

Some results related to the integral mean value theorem

In this paper, some new results related to the integral mean value theorem are proven.

A new way of obtaining analytic approximations of Chandrasekhar’s H function

Applying the mean value theorem for definite integrals in the non-linear integral equation for Chandrasekhar?s H function describing conservative isotropic scattering, we have derived a new, simple analytic approximation for it, with a maximal relative error below 2.5%. With this new function as a starting-point, after a single iteration in the corresponding integral equation, we have obtained a new, highly accurate analytic approximation for the H function. As its maximal relative error is below 0.07%, it significantly surpasses the accuracy of other analytic approximations.

90.20 Mean value theorem for integrals generalised to involve two functions

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Parameterization of the Autoconversion Process.Part I: Analytical Formulation of the Kessler-Type Parameterizations

Various commonly used Kessler-type parameterizations of the autoconversion of cloud droplets to embryonic raindrops are theoretically derived from the same formalism by applying the generalized mean value theorem for integrals to the general collection equation. The new formalism clearly reveals the approximations and assumptions that are implicitly embedded in these different parameterizations. A new Kessler-type parameterization is further derived by eliminating the incorrect and/or unnecessary assumptions inherent in the existing Kessler-type parameterizations. The new parameterization exhibits a different dependence on liquid water content and droplet concentration, and provides theoretical explanations for the multitude of values assigned to the tunable coefficients associated with the commonly used parameterizations. Relative dispersion of the cloud droplet size distribution (defined as the ratio of the standard deviation to the mean radius of the cloud droplet size distribution) is explicitly included in the new parameterization, allowing for investigation of the influences of the relative dispersion on the autoconversion rate and, hence, on the second indirect aerosol effect. The new analytical parameterization compares favorably with those parameterizations empirically obtained by curve-fitting results from simulations of detailed microphysical models.

A non-parametric procedure for evaluating treatment effect in the meta-analysis of survival data.

This paper addresses the problem of combining information from independent clinical trials which compare survival distributions of two treatment groups. Current meta-analytic methods which take censoring into account are often not feasible for meta-analyses which synthesize summarized results in published (or unpublished) references, as these methods require information usually not reported. The paper presents methodology which uses the log(-log) survival function difference, (i.e. log(-logS2(t))-log(-logS1(t)), as the contrast index to represent the multiplicative treatment effect on survival in independent trials. This article shows by the second mean value theorem for integrals that this contrast index, denoted as theta, is interpretable as a weighted average on a natural logarithmic scale of hazard ratios within the interval [0,t] in a trial. When the within-trial proportional hazards assumption is true, theta is the logarithm of the proportionality constant for the common hazard ratio for the interval considered within the trial. In this situation, an important advantage of using theta as a contrast index in the proposed methodology is that the estimation of theta is not affected by length of follow-up time. Other commonly used indices such as the odds ratio, risk ratio and risk differences do not have this invariance property under the proportional hazard model, since their estimation may be affected by length of follow-up time as a technical artefact. Thus, the proposed methodology obviates problems which often occur in survival meta-analysis because trials do not report survival at the same length of follow-up time. Even when the within-trial proportional hazards assumption is not realistic, the proposed methodology has the capability of testing a global null hypothesis of no multiplicative treatment effect on the survival distributions of two groups for all studies. A discussion of weighting schemes for meta-analysis is provided, in particular, a weighting scheme based on effective sample sizes is suggested for the meta-analysis of time-to-event data which involves censoring. A medical example illustrating the methodology is given. A simulation investigation suggested that the methodology performs well in the presence of moderate censoring.

A Generalization of the Mean Value Theorem for Integrals

Let f(x) be a continuous function on [a, b]. The Mean Value Theorem for Integrals asserts that there is a point c in (a, b) such that fab f(x) dx = f(c)(b a). Unlike the proof of the Mean Value Theorem for derivatives, the proof of the Mean Value Theorem for Integrals typically does not use Rolle's Theorem. In this note, we use Rolle's Theorem to introduce a generalization of the Mean Value Theorem for Integrals. Our generalization involves two functions instead of one, and has a very clear geometric explanation.

A converse of the mean value theorem for integrals

This note considers continuous functions f on an interval [a, b] and develops sufficient conditions to answer the question, for what points c in the interval is there a rectangle with height ƒ(c) and some width β- α that has a signed area equal to ∫α βƒ(x)dx?

Existence of Solutions to u″ + u + g(t, u, u′) = p(t), u(0) = u(π) = 0

Existence and multiplicity results for the boundary value problem[formula]are presented. The proofs are based on the alternative method, a connectedness result, the contraction mapping principle, and a detailed analysis of the bifurcation equation utilizing, e.g., a generalization of the mean value theorem for integrals. We shall obtain results with g bounded or unbounded, having finite limits at ±∞ or without limits, thus extending some recent results in the literature. The proofs offer a constructive way to find the bounds for p̄ and to find numerically the number of solutions and the approximative solutions.

An improved energy criterion for dynamic buckling of imperfection sensitive nonconservative systems

Abstract Imperfection sensitive, multi-degree-of-freedom, autonomous, structural systems under partial follower compressive loading, which lose their stability via divergence are investigated both qualitatively and quantitatively. Attention is focused on the global instability of that equilibrium state on the locally stable primary path, which at a certain level of the loading becomes globally unstable. Previous work valid for potential systems under step loading is extended here to nonpotential, imperfection sensitive systems. The serious difficulty of the lack of potential of the follower type of loading is overcome by formulating an appropriate energy balance equation, including loss of energy. Then, similar considerations to those for potential systems can be established, and geometric criteria can be formulated for an “equivalent energy” surface. Using the mean-value theorem for integrals one can obtain approximate dynamic buckling loads that are very good for structural design purposes. The efficiency and reliability of the proposed method is comprehensively demonstrated through numerous examples.

A comparison of the multigroup and collocation methods for solving the low-energy neutron Boltzmann equation

A low-energy neutron transport algorithm for use in space-radiation protection is developed. The algorithm is based upon a multiple energy group analysis of the straight ahead Boltzmann equation utilizing a mean value theorem for integrals. The algorithm developed is then verified by using a collocation method solution on the same straight ahead Boltzmann equation. This algorithm was then coupled to the existing NASA Langley HZETRN (high charge and energy transport) code through the evaporation source term. Evaluation of the neutron fluence generated by the February 23, 1956 solar particle event for an aluminum-water shield-target configuration is then compared with the LAHET Monte Carlo calculation for the same shield-target configuration. The algorithm developed showed a great improvement in results over the unmodified HZETRN solution. A bidirectional modification of the evaporation source produced further improvement of the fluence.