Error function, erf

The integral in equation 10.31 cannot be evaluated analytically but it can be written in terms of the error function erf(rj) defined as... 

The error function erf x and its complementary function erfc x appear in the solution of dilferential equations describing dilfusive processes (see, for instance, section 11.6.3). They are defined by the integrals... 

The solutions of a diffusion equation under the transient case (non-steady state) are often some special functions. The values of these functions, much like the exponential function or the trigonometric functions, cannot be calculated simply with a piece of paper and a pencil, not even with a calculator, but have to be calculated with a simple computer program (such as a spreadsheet program, but see later comments for practical help). Nevertheless, the values of these functions have been tabulated, and are now easily available with a spreadsheet program. The properties of these functions have been studied in great detail, again much like the exponential function and the trigonometric functions. One such function encountered often in one-dimensional diffusion problems is the error function, erf(z). The error function erf(z) is defined by... 

A similar procedure is considered for the upscaled, ID problems, obtained either by our approach or by taking the simple mean. It is refined in the situations when we have explicit formulas for the solution, using the direct numerical evaluation of the error function erf. 

The error function erf(x) is defined in Section A.2. Table A. 1 gives specific numerical values of the normal distribution and its integral 7,(x). 

Obtain a general formula for the most probable three-dimensional translational quantum number j = jmax for a gas (assume a Boltzmann distribution). Evaluate this expression for NO2 at 1000 K (assume a cubic container 0.1 m on each side). Determine the translational energy that this corresonds to (J/mole). Find the fraction of molecules having a translational energy level greater than jmax. Hint Solution to this problem will involve the error function, erf(x). 

The error function, erf (z) and its complement, erfc (z) — 1 — erf (z) are frequently encountered in the solution of diffusion problems. They can be handled numerically using their tabulated values [42], or graphical representations, or preferably by means of computer algorithms as, for example, recently published by Oldham [43]. 

To gain quantitative information on the profile characteristics, the profile shape must be evaluated mathematically. The parameter Dt (D, diffusion constant t, exposure time) that describes the depth of the diffusion front that penetrated into the sample was determined by fitting the data with an error function (erf). The resulting curve describes the result of an undisturbed diffusion process. If the exposure time t is known, e.g. by radiocarbon dating, the diffusion constant D, a material constant, can be derived from this data. 

An attempt was made to correlate the slope of the sensor response curve to the initial diffusible hydrogen concentration in the sample. The steady state portion of the curve could be assumed to be proportional to the flux of hydrogen from the weld metal. To investigate this possibility theoretical curves were generated using an equation derived from the error function erf(x). 

In test programs, where the numerical solution is compared with the analytical solution, the latter often involves the error function erf or the complementary error function erfc. The latter could be obtained simply by subtracting erf from unity but a better approximation is obtained by the direct algorithm. The two routines, ERF and ERFC, were given to the author by a colleague, who probably obtained them from an IBM collection. They have been adapted to Fortran 90/95 by the author, and coupled to the above module. The comments in capitals are the original comments. 

This equation describes how rapidly (positive or negative) excess neutral electrolyte is created. As discussed before, this model is representative of ion transport in double layers as occurs in electrophoresis and electro-osmosis. It is recalled that the error function erf b is defined as... 

Many recent diffusion experiments have relied on various microanalytical techniques to quantify diffusion gradients produced by heating under hydrothermal conditions or in a vacuum (e.g., Cherniak, 1993 Cherniak and Watson, 2000). Assuming simple, one-dimensional diffusion, these gradients are related to Dj through the inverse error function (erf ) ... 

The ratio (f integrals appearing here is tabulated and is known as the error function, erf. 

Table B.14 Values of the error function, erf, and error function complement, erfc...

We frequently have occasion to use integrals of the type of Case II above in which the upper limit is not extended to infinity but only to some finite value. These integrals are related to the error function (erf). We define... 

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