Modelling the Additive Functional Equations through RSM Matrices

This paper suggests one possible method to model additive type of functional equations using eigenvalues and eigenvectors of matrices with suitable numerical examples. The authors have defined a new type of Row Sum Matrix(RSM) and have discussed its eigenvalues and eigenvectors in order to model functional equations. The famous additive Cauchy functional equation and logical functional equation have also been modelled using identity matrix and logical matrix in this study.


Introduction
The study of functional equations becomes essential as it provides a powerful approach to work with important concepts and relationships in analysis and algebra with regard to symmetry, linearity and equivalence. Though the systematic study of such equations is a relatively recent area of mathematical study, they have been considered earlier in various forms by mathematicians such as Euler in the 18 th century and Cauchy in the 19 th century. Functional equations represent an alternative way of modelling problems in Physics. The interesting aspect in modelling physical problem through functional equation is that there is no necessity to assume the differentiability of the function f . Consequently, the functional equations lead often to other solutions than those given by partial differential equations, and these other solutions can be of interest to physicists. The most appealing characteristic of functional equation is its capacity to design mathematical models.
There have been so many researchers studying the solution and stability of different types of functional equations like additve, quadratic, cubic, quartic and mixed type of additive-quadratic, quadratic-cubic and so on. How ever, studies over the origin and formation of such functional equations are not convincing as they are only structure based trial and error methods. This fact has been influential for the authors to undergo a formal study on modelling functional equations.
The several functional equations and their stability problems have been extensively investigated by many mathematicians (see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]) This paper is organised as follows: In section-2, authors discuss the preliminaries and definitions of functional equations. In section-3, authors introduce and discuss the new type of Row Sum Matrix (RSM) and propose a new method namely Pasupathi Narasimman method to model the additive functional equations through eigenvalues and eigenvectors of RSM. In section-4 and 5, authors model the most famous additive Cauchy functional equation and logical functional equation respectively using RSM. Finally, section-6 ends with conclusion.

Preliminaries and Definition
A Hungarian Mathematician J. Aczel [1], an excellent specialist in functional equations, defines the functional equation as follows:

Definition 2.1 Functional Equation:
Functional Equations are equations in which both sides are terms constructed from the finite number of unknown functions and a finite number of independent variables.

Solutions of Functional Equation:
A solution of a functional equation is a function which satisfies the equation.

Definition 3.1 Row Sum Matrix(RSM)
A square matrix is said to be row sum matrix, if it has each row sum values are equal. A RSM-2 matrix is the row sum matrix of order 2.

Example 3.2 A general RSM-2 matrix is given by
The equation (

Definition 5.1 Logical matrix:
A matrix whose entries are all either 0 or 1.
In this section, authors model the additive functional equation corresponding to a matrix whose entries are all 1 which is of RSM-2 type. That is 11 = 11 L    a well known Logical matrix.
Using 1 M given in Remark 3.14, we get the following model matrix for L .
x y X  Here after, we may call the functional equation (12) is a Logical functional equation since it is derived from Logical matrix.

Conclusion
Thus, a new type of matrix RSM-2 is introduced and its eigenvalues and eigenvectors have been discussed and explained. We introduced a new and very first method called Pasupathi Narasimman method to model the additive type functional equations through eigenvalues and eigenvectors of matrices. This is the first attempt to model additive functional equations using eigenvalues and eigenvectors of matrices.
A famous Cauchy functional equation (10) and Logical functional equation (12) have also been modelled using Identity matrix and Logical matrix of order 2 respectively with the proposed Pasupathi Narasimman method.

Note 6.1 Open Problem
The modelled additive functional equation (4)