Heat Equation Revisited: General Solutions to Boundary-Value Problems
CFM Seminars
- Speaker
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MooYoung Choi, Seoul National University
- When
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2022/11/09
13:00 - Place
- Auditorium, Centro de Fisica de Materiales
- Add to calendar
-
iCal
We first present a general method to solve the heat equation in terms of the
Koopman–Darmois family of exponential functions, which leads to a new
closed-form solution in addition to the “fundamental†Gaussian solution.
Next, paying attention to the fact that empirical distribution functions often
differ from the Gaussian function, we construct a new solution, which
satisfies localized initial conditions like the Gaussian solution. The new
solution corresponds to a hetero-mixture distribution, which generalizes the
Gaussian distribution function to a skewed and heavy-tailed distribution, and
thus provides a candidate for the empirical distribution functions. We then
consider the Feynman–Kac formula, which establishes a link between parabolic
partial differential equations and stochastic processes in the context of the
Schrödinger equation in quantum mechanics. Specifically, the formula provides
a solution of the partial differential equation, expressed as an expectation
value for Brownian motion. Here it is shown that the Feynman-Kac formula does
not produce a unique solution but carry infinitely many solutions of the
correspondingpartial differential equation.
Host: Ivo Souza