Sergey P. Tsarev: Transformation and factorization of partial differential systems — applications to stochastic systems
Sadly, this tutorial will not take place.
Factorization of polynomials is a popular topic in lecture courses presenting modern algorithms of computer algebra. Factorization of linear ordinary differential operators with variable coefficients is less known but is an important method used for solution of the corresponding differential equations. Theory of factorization of linear partial differential operators (LPDOs) is even less popular due to a simple fact: a ``naive'' definition of factorization of a given LPDO $\hat L$ as its representation as a composition $\hat L=\hat L_1\circ \hat L_2$ of lower-order operators does not enjoy good algebraic properties and in general is not related to existence of a complete closed-form solution. An appropriate modification of the definition of factorization gives rise to a bunch of new methods of closed-form solution of a single linear partial differential equation and more advances techniques for systems of such equations. This approach is naturally related to differential transformations---another popular method of solution of differential equations.
In this tutorial we present the old but still fruitful Laplace cascade method and a number of its latest modifications and generalizations. A recent application to solution of an interesting system of linear partial differential equations describing the behavior of a simple nonlinear stochastic ordinary differential equation will be described.
The tutorial will present the subject in a form accessible for graduate students interested in algorithms of closed form solution of linear differential systems. It requires only basic knowledge of the theory of linear differential equations with variable coefficients.
