Width-asymptotics of the solution to the diffusion equation with Stieltjes initial condition

Authors

  • Rexcel F. De Peralta ⋅ PH National Institute of Physics, University of the Philippines Diliman
  • Philip Jordan D. Blancas ⋅ PH Department of Physics, Ateneo de Manila University and National Institute of Physics, University of the Philippines Diliman
  • Eric A. Galapon ⋅ PH National Institute of Physics, University of the Philippines Diliman

Abstract

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.

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Issue

Article ID

SPP-2023-PC-24

Section

Poster Session C (Mathematical Physics, Optics, and Interdisciplinary Topics)

Published

2023-07-08

How to Cite

[1]
RF De Peralta, PJD Blancas, and EA Galapon, Width-asymptotics of the solution to the diffusion equation with Stieltjes initial condition, Proceedings of the Samahang Pisika ng Pilipinas 41, SPP-2023-PC-24 (2023). URL: https://proceedings.spp-online.org/article/view/SPP-2023-PC-24.