Method of fast expansions for solving nonlinear differential equations

A method is proposed for constructing fast converging Fourier series with the help of a special boundary function $M_q$. The convergence rate of the series is determined by the order $q$ of $M_q$, which makes it possible to use a small number of series terms. The general theory of constructing fast expansions is described, the error of the partial sum of a series is estimated, and an example of a non-linear integrodifferential problem is considered. Due to its remarkable properties, the fast expansion method can be effectively used in applications.