| In spite great progress in numerical methods, closed form solutions attract attention of scientists and engineers, since such solutions are important, for instance, in optimal design problems and useful in qualitative analysis of advanced computations. The talk is devoted to a method of functional equations i two dimensions and its applications for finding closed form solutions of boundary value problems in domains with complex geometries. The classical Dirichlet and Neumann problems for circular multiply connected domains are reduced to a system of functional equations. The simple functional equation has the form $\varphi (z)=\varphi (sz)+g(z)$, $\left| z\right| <1$, where $g(z)$ is given and $\varphi (z)$ is unknown function analytic in the unit disk; the given constant $s$ satisfies the inequality $% \left| s\right| <1$. This functional equations can be solved by successive approximations. The main idea of the method is presented in the talk. The relations to the alternating Schwarz method, Poincar\'{e} series, Riemann-Hilbert problem and applications to mechanics of composite materials are discussed. |