Generalized Gaussian error functions and their applications

Abstract

The Gaussian error functions play a significant role in analytical solutions for problems of thermomechanics and mass flow due to diffusion. These functions are extended to a family of generalized Gaussian error functions with an infinite number of members. The use of this new family of functions allows finding analytical solutions in form of absolutely converging series for a much wider class of problems compared to available analytical solutions. The Laplace transformation is an established tool to handle time-dependent problems. It is outlined in detail, how the generalized Gaussian error functions can be efficiently used for the inversion of the solution from the Laplace transform domain to the original domain by the extension of the so-called correspondence relation. As example, an analytical solution for the heat conduction from an inclusion into the outer space with material properties different to those of the inclusion is demonstrated.