Heat Equation Revisited: General Solutions to BoundaryValue Problems
CFM Seminars
 Speaker

MooYoung Choi, Seoul National University
 When

2022/11/09
13:00  Place
 Auditorium, Centro de Fisica de Materiales
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We first present a general method to solve the heat equation in terms of the
Koopmanâ€“Darmois family of exponential functions, which leads to a new
closedform solution in addition to the â€œfundamentalâ€ 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 heteromixture distribution, which generalizes the
Gaussian distribution function to a skewed and heavytailed distribution, and
thus provides a candidate for the empirical distribution functions. We then
consider the Feynmanâ€“Kac formula, which establishes a link between parabolic
partial differential equations and stochastic processes in the context of the
SchrÃ¶dinger 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 FeynmanKac formula does
not produce a unique solution but carry infinitely many solutions of the
correspondingpartial differential equation.
Host: Ivo Souza