## "Heat Equation Revisited: General Solutions to Boundary-Value Problems"
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 |