Differential equation

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Integrability of Riccati Equations

Delivered by Letian Chen on Friday January 27, 2017

In 1841, 3 years before Joseph Liouville discovered transcendental numbers, Joseph Liouville showed that the Riccati equation $y' = ay^2 + bx^m$ has a quadrature solution if and only if $m = 0, -2, -\frac{4n}{2n±1}$. Nowadays, we see Liouville's results as the foundation to qualitative analysis to differential equations, a fascinating subarea of DEs, which studies for example the existence and uniqueness of different kinds of equations. In my talk, I will (hopefully) prove the classical result of Liouville after a quick review of history and related linear theory.

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Differential Equations on Fractals

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analysis differential equation fractal

Gillespie Algorithm

The Gillespie Algorithm for stochastic equations is used heavily in a number of fields of applied mathematics, in particular computational systems biology.

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algorithm applied mathematics biology computational biology computational mathematics computer science differential equation probability stochastic equation

The Stochastic Lotka-Volterra Equation

The famous Lotka-Volterra equations model the population of a predator and a prey, which are of course closely related. Stochastic variations on the Lotka-Volterra equations can be an illuminating introduction to the methods used to solve stochastic differential equations.

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applied mathematics differential equation ecology stochastic equation