Exponential integral equation

Exponential Integral, equation (11) nonlinear function in Astarita s kinetics Gamma distribution, equation (5)... 

M. Hochbruck, Ch. Lubich, and H. Selhofer Exponential integrators for large systems of differential equations. SIAM J. Sci. Comp. (1998) (to appear)... 

Equation 9-15 gives the conversion expression for the second order reaction of a macrofluid in a mixed flow. An exponential integral, ei(a), which is a function of a, and its value can be found from tables of integrals. However, the conversion from Equation 9-15 is different from that of a perfectly mixed reactor without reference to RTD. An earlier analysis in Chapter 5 gives... 

From equation 14.3-19, with MA defined by equation 4.3-4. From equation 13.52, with 3.4-10 or 14.3-20 and 13.4-2. dE is an exponential integral defined by E x) = y le y dy, where y is a dummy variable the integral must be evaluated numerically (e.g., using E-Z Solve) tabulated values also exist. 

Closely related to the gamma function are the exponential-integral ei(a) defined by the equation... 

Equation (41) involves the exponential integral which can be evaluated numerically or else is available as a standard tabulated function (see, for instance, ref. 33). Therefore... 

Following the approach taken by Currie [82], this problem can be solved analytically in terms of exponential integrals. Noting that 2 fdf = df2, Eq. 5.10 can easily be integrated once to produce a second-order equation as... 

The inequalities for the exponential integral stated above, equation (12), then give... 

To test whether the reaction is first order, we simply fit the data to the exponential integrated first-order rate equation (Table 3.1) using a non-linear optimisation procedure and the result is shown in Fig. 3.3. The excellent fit shows that the reaction follows the mathematical model and, therefore, that the process is first order with respect to [N2O5], i.e. the rate law is r = A bsI Os]. The rate constant is also obtained in the fitting procedure, k0bs = (6.10 0.06) x 10 4 s 1. We see that, even with such a low number of experimental points, the statistical error is lower than 1%, which shows that many data points are not needed if... 

The integrals in Eqs. (B) and (C) must be evaluated through the exponential integral, E(x), a special function whose values are tabulated in handbooks and are also found from such software packages as MAPLE . The necessary equations, as found from MAPLE , are ... 

The velocity dependence of the distribution function, which is not of primary interest here, may be eliminated from the spray equation by integrating equation (2) over all velocity space. Sincej. 0 very rapidly (at least exponentially) as [ v oo for all physically reasonable flows, the divergence theorem shows that the integral of the last term in equation (2) is zero, whence... 

Radioactive decay can be described as a first-order process. Thus, for any first-order decay process, the amount of material present declines in an exponential fashion with time. This is easy to see by integrating Equation (1.5.3) to give ... 

The exponential accounts for the stored nutrient, which washes out of the vessel - along with the cells containing it - during the storage period. Balancing input and output with consumption and washout yields integral equations of the form... 

The second term in the series expansion of the exponential in Equation (1.29), iK-rj, leads to an integral that may be separated into two parts. The square of the first part... 

For an exponential representation of the local retention the integral equation (16) assumes the following form [142] ... 

The exponential in Equation (19.24) is expanded about an average and integrated term. 

In general, the NMR spectrum of the object shows more than one resonance line, or in solids the lineshape is non-Lorentzian. Thus, the first exponential in Equation (5.6) needs to be replaced by the FID f t) in a homogeneous magnetic field. If the spectrum changes within the sample, the FID has to be placed inside the integral. 

Rate laws derived for reaction schemes consisting of any number and combination of consecutive, parallel and independent first-order reaction steps can be integrated in closed form to give a sum of exponential terms, so that the total absorbance A of a solution in which these reactions proceed can also be expressed as a sum of exponential terms (Equation 3.9), but see last paragraph of this Section. 

The integral inside the parenthesis in Equation 16.28 is known as the exponential-integral function Ei(x) ... 

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