The Error Function and Related Functions

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 related function is the complimentary error function erfc(z), defined by 

Equations A2-8a and A2-8b always converge but for large absolute values of z (e.g., 5) the convergence is slow and truncation errors may dominate. Hence, in practice. Equation A2-8a is often applied for z 1, and Equation A2-8b is often applied for l z 4.5. Equation A2-8c is an asymptotic expression and must be truncated at or before the absolute value of the term in the series reaches a minimum. For large z (z 10), ze erfc(z) approaches 1/ /n. 

Integrated error functions are repeated integrations of the complementary error function. Define 

Integrated error functions can be expressed in terms of error functions. For example, integrating by part, we can find 

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Table 7.2 gives tabulated values of the error function and related functions in the solution of other semi-infinite conduction problems. For example, the more general boundary condition analogous to that of Equation (7.27), including a surface heat loss,... 

Note that neither initial nor boundary conditions have been applied yet. The above equation is the general solution for infinite and semi-infinite diffusion medium obtained from Boltzmann transformation. The parameters a and b can be determined by initial and boundary conditions as long as initial and boundary conditions are consistent with the assumption that C depends only on q (or ). Readers who are not familiar with the error function and related functions are encouraged to study Appendix 2 to gain a basic understanding. 

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