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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 thisdoi:10.1364/ao.33.003831 pmid:20935723 fatcat:q7jiavribzgpzg4rtjkjcnp4na