Dyadic harmonic analysis is a branch of Fourier analysis which, instead of classic trigonometric series, studies series based on orthonormal systems composed by locally constant functions. Due to its apparent simplicity, the most well-known example of them is the Walsh-Paley system. This system is formed by Walsh functions which we know that they only take values 1 and -1. This property offers a wide range of applications in the world of digital technology. There are another known arrangements of Walsh functions, the original Walsh systems and the Walsh-Kaczmarz system. In the recent decades, an increasing number of mathematicians supports the view that the appropriate environment for the development of Fourier analysis is the theory of locally compact groups. Walsh functions can be represented on the complete direct product of cyclic groups of order 2, as a system of characters. The theory of dyadic analysis was enriched by the work of Vilenkin, when he generalized the Walsh-Paley system in 1947. He studied the commutative cases, the so-called Vilenkin systems which are formed by the character system of complete direct product of arbitrary cyclic groups. Similarly, Vilenkin systems can be generalized considering the complete products of finite groups, where these finite groups are not necessarily commutative, and we follow the way of harmonic analysis to obtain the appropriate orthonormal systems. The properties of representative product systems can be very different relative to the properties of the Walsh-Paley and Vilenkin systems. For example they consist of not necessarily uniformly bounded functions, which may even take 0 as a value at some points.

Walsh functions can be applied in solving differential equations numerically. The method consists in discretizing the equivalent integral equation, substituting the functions by the 2^n-th partial sums of Walsh-Fourier series of them and approaching the solution by a Walsh polynomial. In that way, the problem is reduced to solve a large linear system. This method was implemented in 1975 by C. F. Chen and C. H. Hsiao for the solution of system of ordinary linear differential equations of first-order with constant coefficients. We deal with the analysis of this method and we extend it for systems with non-constant coefficients. Moreover, we propose an iterative method to speed up the computations and we modified the numerical solution to obtain more accuracy in the approximation. We implemented this method to solve different types of differential equations, for instance Riccati equations.