A TABLE OF INTEGRALS INVOLVING POWERS, EXPONENTIALS, LOGARITHMS, AND THE EXPONENTIAL INTEGRAL
Table integrals involving powers, exponentials, logarithms and the exponential integral
EXPONENTIAL FUNCTION↗
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Theory, computation, and application of exponential splines
A generalization of the semiclassical cubic spline known in the literature as the exponential spline is discussed. In actuality, the exponential spline represents a continuum of interpolants ranging from the cubic spline to the linear spline. A particular member of this family is uniquely specified by the choice of certain tension parameters. The theoretical underpinnings of the exponential spline are outlined. This development roughly parallels the existing theory for cubic splines. The primary extension lies in the ability of the exponential spline to preserve convexity and monotonicity present in the data. Next, the numerical computation of the exponential spline is discussed. A variety of numerical devices are employed to produce a stable and robust algorithm. An algorithm for the selection of tension parameters that will produce a shape preserving approximant is developed. A sequence of selected curve-fitting examples are presented which clearly demonstrate the advantages of exponential splines over cubic splines.
Mccartin, B. J.↗
The matrix exponential in transient structural analysis
The primary usefulness of the presented theory is in the ability to represent the effects of high frequency linear response with accuracy, without requiring very small time steps in the analysis of dynamic response. The matrix exponential contains a series approximation to the dynamic model. However, unlike the usual analysis procedure which truncates the high frequency response, the approximation in the exponential matrix solution is in the time domain. By truncating the series solution to the matrix exponential short, the solution is made inaccurate after a certain time. Yet, up to that time the solution is extremely accurate, including all high frequency effects. By taking finite time increments, the exponential matrix solution can compute the response very accurately. Use of the exponential matrix in structural dynamics is demonstrated by simulating the free vibration response of multi degree of freedom models of cantilever beams.
Minnetyan, Levon↗
Simple quasi-exponential slope generator
Circuitry for digitally generating an exponentially decaying wave function permits discrete values to be sampled from the exponential waveform for comparison with a binary number of specified accuracy. This exponential-decay generator employs a simple binary counter to count in the sequence of exponential decay.
Anderson, T. O.↗
Numerical computation of exponential matrices using the Cayley-Hamilton theorem
A method for computing exponential matrices, which often arise naturally in the solution of systems of linear differential equations, is developed. An exponential matrix is generated as a linear combination of a finite number (equal to the matrix order) of matrices, the coefficients of which are scalar infinite sums. The method can be generalized to apply to any formal power series of matrices. Attention is focused upon the exponential function, and the matrix exponent is assumed tri-diagonal in form. In such cases, the terms in the coefficient infinite sums can be extracted, as recursion relations, from the characteristic polynomial of the matrix exponent. Two numerical examples are presented in some detail: (1) the three dimensional infinitesimal rotation rate matrix, which is skew symmetric, and (2) an N-dimensional tri-diagonal and symmetric finite difference matrix which arises in the numerical solution of the heat conduction partial differential equation. In the second example, the known eigenvalues and eigenvectors of the finite difference matrix permit an analytical solution for the exponential matrix, through the theory of diagonalization and similarity transformations, which is used for independent verification. The convergence properties of the scalar infinite summations are investigated for finite difference matrices of various orders up to ten, and it is found that the number of terms required for convergence increases slowly with the order of the matrix.
Walden, H.↗
Exponential-fitted methods for integrating stiff systems of ordinary differential equations: Applications to homogeneous gas-phase chemical kinetics
Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.
Pratt, D. T.↗
Future Exponential Economic Growth in Space
The human population and economy are now limited by the land area, materials and energy available on Earth. As explained by Malthus, human effort can increase this population limit only gradually and linearly at any fixed level of technology. The unexpected exponential increase in population since Malthus time is due to the even greater exponential increase in agricultural and industrial productivity due to modern technology. Current opinion is divided between the limiters who see fast approaching unavoidable limits to growth and growthers who expect continuing innovation and expansion. Given the minimum energy and food needed to support each human, the population of Earth is necessarily bounded. The past human exponential expansions were based on the invention of agriculture and industry. Continued expansion could occur with future colonization of the solar system. The initial occupation of near-Earth space would probably be for immediate military and economic gains rather than to escape the limits to growth on Earth. Space habitats could exploit solar system energy, materials, and living space, first in the Earth-moon system and later throughout the solar system. Space habitats would be supported by robotic power stations, mining, manufacturing, and transportation. These could travel to the stars with the addition of propulsion and nuclear power. Human expansion into the galaxy would allow continuing exponential growth and could establish a vast interconnected galactic civilization.
Jones, Harry W.↗
The effect of Be/+/ ion implanted exponential and uniform impurity profiles on the electrical characteristics of GaAs solar cells
The high surface recombination velocity is the major deterrent to obtaining efficient GaAs solar cells. If, however, an electric field is built in at the surface, the carriers will be swept away from the surface thus minimizing the surface recombination velocity problem. It has been previously shown theoretically that an exponential impurity distribution in the doped region of the cell results in a built-in electric field. Ion implantation was used to produce solar cells with an exponential impurity profile and cells with uniform profiles. It is shown that cells with an exponential impurity profile have higher open-circuit voltage, fill factors, and spectral response than those with a uniform impurity profile.
Vaidyanathan, K. V.↗
An Exponential Finite Difference Technique for Solving Partial Differential Equations
An exponential finite difference algorithm, as first presented by Bhattacharya for one-dimensianal steady-state, heat conduction in Cartesian coordinates, has been extended. The finite difference algorithm developed was used to solve the diffusion equation in one-dimensional cylindrical coordinates and applied to two- and three-dimensional problems in Cartesian coordinates. The method was also used to solve nonlinear partial differential equations in one (Burger's equation) and two (Boundary Layer equations) dimensional Cartesian coordinates. Predicted results were compared to exact solutions where available, or to results obtained by other numerical methods. It was found that the exponential finite difference method produced results that were more accurate than those obtained by other numerical methods, especially during the initial transient portion of the solution. Other applications made using the exponential finite difference technique included unsteady one-dimensional heat transfer with temperature varying thermal conductivity and the development of the temperature field in a laminar Couette flow.
Handschuh, Robert F.↗
Exponential ground impedance models and their interpretation
The authors compare the results of Donato's exponentially varying ground model, Attenborough's exponentially varying ground model and the rigid backed thin layer model. They show that these models produce similar results for slow variations. For rapid variations the results are quite different, but the basic theory used is only correct for the thin layer model. These results suggest that the exponentially varying models are not necessary for fitting ground impedance data.
Raspet, Richard↗
Quasi-random narrow-band model fits to near-infrared low-temperature laboratory methane spectra and derived exponential-sum absorption coefficients
Near-infrared 10/cm resolution spectra of methane obtained at various temperatures, pressures, and abundances are fit to a quasi-random narrow-band model. Exponential-sum absorption coefficients for three temperatures (112, 188, and 295 K), and 20 pressures from 0.0001 to 5.6 bars, applicable to the cold environments of the major planets, are then derived from the band model for the 230 wavelengths measured from 1.6 to 2.5 microns. RMS deviations between the laboratory and the exponential-sum synthetic transmissions are reported for the best fitting 50 wavelengths. Deviations relevant to broadband, 1-percent spectral resolution observations are also presented. The validity of exponential-sum coefficients derived from broadband (10/cm) transmission data is demonstrated via direct comparison with line-by-line calculations. The complete atlas of coefficients is available from the Planetary Data System-Planetary Atmospheres Discipline Node.
Baines, Kevin H.↗
Decoherence and Exponential Law: A Solvable Model
We analyze a modified version of the 'AgBr' Hamiltonian, solve exactly the equations of motion in terms of SU(2) coherent states, and study the weak-coupling, macroscopic limit of the model, obtaining an exponential behavior at all times. The asymptotic dominance of the exponential behavior is representative of a purely stochastic evolution and can be derived quantum mechanically in the so-called van Hove's limit (which is a weak-coupling, macroscopic limit). At the same time, a temporal behavior of the exponential type, yielding a 'probability dissipation' is closely related to dephasing ('decoherence') effects and one can expect a close connection with a dissipative and irreversible behavior. We stress the central relevance of the problem of dissipation to the quantum measurement theory and to the general topic of decoherence.
Pascazio, Saverio↗
Modeling the Role of Dislocation Substructure During Class M and Exponential Creep
The different substructures that form in the power-law and exponential creep regimes for single phase crystalline materials under various conditions of stress, temperature and strain are reviewed. The microstructure is correlated both qualitatively and quantitatively with power-law and exponential creep as well as with steady state and non-steady state deformation behavior. These observations suggest that creep is influenced by a complex interaction between several elements of the microstructure, such as dislocations, cells and subgrains. The stability of the creep substructure is examined in both of these creep regimes during stress and temperature change experiments. These observations are rationalized on the basis of a phenomenological model, where normal primary creep is interpreted as a series of constant structure exponential creep rate-stress relationships. The implications of this viewpoint on the magnitude of the stress exponent and steady state behavior are discussed. A theory is developed to predict the macroscopic creep behavior of a single phase material using quantitative microstructural data. In this technique the thermally activated deformation mechanisms proposed by dislocation physics are interlinked with a previously developed multiphase, three-dimensional. dislocation substructure creep model. This procedure leads to several coupled differential equations interrelating macroscopic creep plasticity with microstructural evolution.
Raj, S. V.↗
Slow Crack Growth of Brittle Materials With Exponential Crack-Velocity Formulation: Analysis - Part 1
Extensive slow-crack-growth (SCG) analysis was made using a primary exponential crack-velocity formulation under three widely used load configurations: constant stress rate, constant stress, and cyclic stress. Although the use of the exponential formulation in determining SCG parameters of a material requires somewhat inconvenient numerical procedures, the resulting solutions presented gave almost the same degree of simplicity in both data analysis and experiments as did the power-law formulation. However, the fact that the inert strength of a material should be known in advance to determine the corresponding SCG parameters was a major drawback of the exponential formulation as compared with the power-law formulation.
Choi, Sung R.↗
Slow Crack Growth of Brittle Materials With Exponential Crack-Velocity Formulation: Constant Stress Rate Experiments - Part 2
The previously determined life prediction analysis based on an exponential crack-velocity formulation was examined using a variety of experimental data on glass and advanced structural ceramics in constant stress rate and preload testing at ambient and elevated temperatures. The data fit to the relation of strength versus the log of the stress rate was very reasonable for most of the materials. Also, the preloading technique was determined equally applicable to the case of slow-crack-growth (SCG) parameter n greater than 30 for both the power-law and exponential formulations. The major limitation in the exponential crack-velocity formulation, however, was that the inert strength of a material must be known a priori to evaluate the important SCG parameter n, a significant drawback as compared with the conventional power-law crack-velocity formulation.
Choi, Sung R.↗
Slow Crack Growth of Brittle Materials With Exponential Crack-Velocity Formulation: Constant Stress and Cyclic Stress Experiments - Part 3
The previously determined life prediction analysis based on an exponential crack-velocity formulation was examined using a variety of experimental data on advanced structural ceramics tested under constant stress and cyclic stress loading at ambient and elevated temperatures. The data fit to the relation between the time to failure and applied stress (or maximum applied stress in cyclic loading) was very reasonable for most of the materials studied. It was also found that life prediction for cyclic stress loading from data of constant stress loading in the exponential formulation was in good agreement with the experimental data, resulting in a similar degree of accuracy as compared with the power-law formulation. The major limitation in the exponential crack-velocity formulation, however, was that the inert strength of a material must be known a priori to evaluate the important slow-crack-growth (SCG) parameter n, a significant drawback as compared with the conventional power-law crack-velocity formulation.
Choi, Sung R.↗
Evidence of the Exponential Decay Emission in the Swift Gamma-ray Bursts
We present a systematic study of the steep decay emission of gamma-ray bursts (GRBs) observed by the Swift X-Ray Telescope (XRT). In contrast to the analysis in recent literature, instead of extrapolating the data of Burst Alert Telescope (BAT) down into the XRT energy range, we extrapolated the XRT data up to the BAT energy range, 15-25 keV, to produce the BAT and XRT composite light curve. Based on our composite light curve fitting, we have confirmed the existence of an exponential decay component which smoothly connects the BAT prompt data to the XRT steep decay for several GRBs. We also find that the XRT steep decay for some of the bursts can be well fitted by a combination of a power-law with an exponential decay model. We discuss that this exponential component may be the emission from an external shock and a sign of the deceleration of the outflow during the prompt phase.
Sakamoto, T.↗
A property of the order statistics of an exponential variate
Order statistics of exponential variate used in estimating scale of exponential distribution of know location
DISTRIBUTION FUNCTION↗