SURVEY PAPER THE DERIVATION OF THE GENERALIZED FUNCTIONAL EQUATIONS DESCRIBING SELF-SIMILAR PROCESSES Raoul R. Nigmatullin 1 , Dumitru Baleanu 2,3 Abstract The generalized functional equations describing a wide class of different self-similar processes are derived. These equations follow from the obser- vation that microscopic function describing an initial self-similar process increases monotonically or even cannot have a certain value. The last case implies the behavior of trigonometric functions cos(zξ n ), sin(zξ n ) at ξ> 1 and n >> 1 that can enter to the microscopic function and when the lim- its of the initial scaling region are increasing and becoming large. The idea to obtain the desired functional equations is based on the approxi- mate decoupling procedure reducing the increasing microscopic function to the linear combination of the same microscopic functions but having smaller scales. Based on this idea the new solutions for the well-known Weierstrass- Mandelbrot function were obtained. The generalized functional equations derived in this paper will help to increase the limits of applicability in de- scription of a wide class of self-similar processes that exist in nature. The procedure that is presented in this paper allows to understand deeper the relationship between the procedure of the averaging of the smoothed func- tions on discrete self-similar structures and continuous fractional integrals. MSC 2010 : Primary 28A80; Secondary 26A33 Key Words and Phrases: self-similar (fractal) processes; Weierstrass- Mandelbrot function; fractional calculus; solutions of functional equations c  2012 Diogenes Co., Sofia pp. 718–740 , DOI: 10.2478/s13540-012-0049-5