Seminar “Nonpotential dynamical systems and neural network technologies”

The representation of the solution of a mathematical physics problem in the form of a series or integral is not always possible, and for nonlinear problems it is possible only in exceptional cases. Therefore, approximate methods are often used in applications that widely use modern computer technology. One of the most effective methods of approximate solution of a large class of problems for differential equations is the finite difference method (or the grid method). Its idea is as follows. The domain of continuous variation of the arguments of the desired function is replaced by a discrete set of points (nodes), called a grid; instead of functions of a continuous argument, discrete argument functions defined in the nodes of the grid and called grid functions are considered. The derivatives included in the DE are replaced by difference relations, and the DE is replaced by a system of algebraic equations. The initial and boundary conditions are also replaced by the corresponding conditions for the grid function. As a result, a difference scheme is constructed — a finite system of algebraic equations, put in accordance with this problem, containing a differential equation and additional conditions. The solution of the difference scheme is called an approximate (or grid) solution of the differential problem. At the same time, it is natural to demand that the constructed difference scheme be uniquely solvable and the corresponding grid solution, with the fineness of the grid tending to zero, indefinitely approaches the solution of the problem for the DE. Here such important concepts as convergence, approximation and stability of the difference scheme arise.