A Study of Quadratic-Cubic Functional Equation (FES) In (-Fuzzy Spaces

The functional equations of mathematics are a relatively unpopular area. This is not because of insignificance: the expanding linear algebra which covers linear functions covers a much wider domain. Dynamic systems are generally described in a continuous time domain by differential equations. The dynamics are described by difference equation or by a map for discrete time systems, on the other hand. To create a trajectory or to determine certain device properties, functional equations must be dealt with. Functional equations are not often used to solve practical problems unlike differential calculus or linear algebra. This could be because of practical calculus technical difficulties. In the face of an engineering problem that needs these solutions. The author Lars Kindermann developed successfully methods to solve the equation with the help of neural networks at least numerically. In many fields of mathematics, functional equations exist, such as geometry, arithmetic, theory of probability, measurement theory, algebraic geometry and group theory. The apps are also used in information theory, code theory, fuzzy set theory, decision theory, game theory, artificial intelligence, cluster analysis, multi-value logic and many other fields. It also applies to other applications. Many new issues and hypotheses have inspired the development of new approaches and methods through functional equations. In the second half of the 20th century, the theory of functional equations was rapidly and productively developed. At the same time it was very fruitful to establish this theory. There are more and more mathematical texts and mathematicians working with functional equations every day.