Nonstandard Methods for Solving the Heat Equation

Tristram de Piro

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arXiv · 1806.02333

Nonstandard Methods for Solving the Heat Equation

Tristram de Piro

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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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Tristram de Piro. 2018-06-05. Nonstandard Methods for Solving the Heat Equation. https://arxiv.org/abs/1806.02333

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