Width-asymptotics of the solution to the diffusion equation with Stieltjes initial condition
We obtain the asymptotic behavior of a particular solution to the diffusion equation in one-dimension for arbitrarily narrow and broad initial distributions. The particular solution is a Stieltjes transform of a separable solution of the same diffusion equation. The parameter of transformation happens to be inversely proportional to the width of the initial distribution, so that the desired asymptotic behavior for narrow (broad) initial distribution corresponds to large (small) parameter asymptotics of the Stieltjes transform. Consideration of the large parameter regime leads to a divergent infinite series which falls under the classical theory of asymptotics. On the other hand, consideration of the small parameter regime leads to an infinite series of divergent integrals which is outside the scope of the classical methods of asymptotics. Here we demonstrate how to solve the small parameter asymptotics by means of the method of finite-part integration.