Applications of group analysis to integro-differential, delay and stochastic differential equations
Group analysis is one of the methods for constructing exact solutions of differential equations. The algorithm developed for differential equations does not allow applying it to equations with nonlocal terms. Hence, the group analysis method has to be modified. The presentation consists of reviewing results developed by the author with his colleagues for solving this problem for integro-differential, delay differential and stochastic differential equations.
The first step on the way of modifying consists of defining a Lie group admitted by the original system of equations. For defining an admitted Lie group the proposed approach is similar to the approach, developed for differential equations: first, one has to construct determining equations, then to split these equations with respect to parametrical elements, and then to find the general solution of these equations.
Practical construction of determining equations is performed by using the canonical Lie-Bäcklund's representation of an infinitesimal generator and acting by it on the original equation. The derivatives with respect to the dependent variables should be understood in terms of the Frechet derivatives.
The developed method is illustrated by several applications.
The proposed approach can also be applied for defining a Lie group of equivalence, contact and Lie-Bäcklund transformations for equations with nonlocal terms.