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General solutions of the heat equation

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  • Choi, ByoungSeon
  • Kim, Chansoo
  • Kang, Hyuk
  • Choi, M.Y.

Abstract

We derive a general solution of the heat equation through the use of the similarity reduction method. The obtained solution is expressed as linearly combined kernel solutions in terms of the Hermite polynomials, which appears to provide an explanation of non-Gaussian behavior observed in various cases. As examples, we consider a few typical boundary conditions and construct corresponding solutions, demonstrating the versatile applicability of our scheme. It is thus revealed that the heat equation carries many solutions under given boundary conditions. The entropy borne by a non-Gaussian solution is also computed and shown to approach in the long-time limit the maximum one corresponding to the fundamental (Gaussian) solution.

Suggested Citation

  • Choi, ByoungSeon & Kim, Chansoo & Kang, Hyuk & Choi, M.Y., 2020. "General solutions of the heat equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    • Handle: RePEc:eee:phsmap:v:539:y:2020:i:c:s037843711931653x
    DOI: 10.1016/j.physa.2019.122914
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    References listed on IDEAS

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    1. Jondeau, Eric & Rockinger, Michael, 2001. "Gram-Charlier densities," Journal of Economic Dynamics and Control, Elsevier, vol. 25(10), pages 1457-1483, October.
    2. Choi, ByoungSeon & Kang, Hyuk & Choi, M.Y., 2017. "Emergence of heavy-tailed skew distributions from the heat equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 470(C), pages 88-93.
    3. Choi, ByoungSeon & Roh, JeongHo, 2013. "On the trivariate joint distribution of Brownian motion and its maximum and minimum," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1046-1053.
    4. Choi, ByoungSeon & Choi, M.Y., 2018. "General solution of the Black–Scholes boundary-value problem," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 546-550.
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