A method of solution of the weighted heat equation

In this paper, a method of finding solutions of the heterogeneous heat equation is developed. Taking Fourier transforms, then Laplace transforms, of the heterogeneous heat equation yields an equation that involves a convolution with the solution. In order to make sense of what it would mean to "invert" the convolution (so that the solution is isolated), the '"mollified"' Fourier transform is introduced. This transform is used in the development of the method of inverting the convolution.