Temperature distribution in a uniform medium heated by linear absorption of a Gaussian light beam

The linear-diffusion equation is considered for a positive half-space with heat sources represented by Gaussian functions in the transverse plane and by exponential decay along the longitudinal axis. The exact solution is presented as a single quadrature of the complementary error function (erfc). The approximate solution is suggested in the form of the product of two Gaussian functions and the hyperbolic secant function. Comparison with the exact solution shows that the error of this approximation is near 10%. The approximation may be used in different medical applications, e.g., laser angioplasty.