Nonstandard Methods for Solving the Heat Equation

2018/06/05 by Tristram de Piro, de Piro, Tristram
Mathematics · #FOS: Mathematics #Mathematical and Theoretical Analysis #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1806.02333

openalex publication_date 2018/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We apply convergence results for discrete Markov chains, to prove the existence of an equilibrium limit in the nonstandard heat equation. We construct a nonstandard backward martingale from a nonstandard solution, and show, using the Feynman-Kac method, how to derive an explicit formula for such solutions, when the initial condition is S-continuous. Finally, we prove that that the nonstandard solution to the heat equation, with a smooth initial condition, specialises to the classical solution.

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