Solution and Interpretation of NeutrosophicHomogeneous Difference Equation

: In this manuscript, we focus on the brief study of finding the solution to and analyzingthe homogeneous linear difference equation in a neutrosophic environment, i


Uncertainty Theory and Neutrosophic Sets
The uncertainty theory becomes a very helpful tool for real life modeling in discrete and continuous systems. The different theories of the fuzzy uncertainty theory have been given a new direction since the setting of the fuzzy set, invented by Professor Zadeh [1]. This is generalized representation of [1] is established as an intuitionistic fuzzy set theory by Atanassov [2]. Atarasov gave a novel designusing the intuitionistic fuzzy theory, where he demonstrated the idea of a membership function and non-membership function by which degree of belongingness and nonbelongingness, respectively, can be measured in a set. Liu and Yuan [3] ignited the perception of a triangular intuionistic fuzzy set, which is the affable blend of a triangular fuzzy number and a intuionistic fuzzy set theory. Ye [4] set up the idea for a trapezoidal intuionistic fuzzy set. Smarandache [5] found his more generalized idea as a neutrosophic set, considering terms of the truth membership function, the indeterminacy membership function, and the falsity membership function. This theory become more beneficial and germane, rather than the common fuzzy and intuitionistic fuzzy theory settings.
Several researchershave already worked in the neutrosophic field, some of which have developed the theory [6,7], while some have applied the related theories in an applied field [8,9].Various kinds of forms and extensions of the Neutrosophic set, such as the triangular neutrosophic set [10], the bipolar neutrosophic sets [11][12][13][14], and the multi-valued neutrosophic sets [15],were also found.

Difference Equation in an Uncertain Environment
There exist some works associated with difference equation and uncertainty. Mostly, researchers have worked on the difference equation allied with fuzzy and intuitionistic fuzzy environments. We are now giving details descriptions of some related published work. In the literature [16], Deebaet al. found a strategy for solving the fuzzy difference equation with an interesting application. The model involving CO2 levels in blood streamflow is thinkingin the view ofthe fuzzy difference equation by Deebaet al. [17]. Lakshmikantham and Vatsala [18] talk abou tdifferent basic theories and properties of fuzzy difference equations. Papaschinopoulos et al. [19,20] and PapaschinopoulosandSchinas [21] discuss more findings in a similar context. Papaschinopoulos and Stefanidou [22] provide an explanation on boundedness with asymptotic behavior of a fuzzy difference equation. Umekkanet al. [23] give a finance application based on discrete system modeling in a fuzzy environment. Stefanidou et al. [24] treat the exponential-type fuzzy difference equation. The asymptotic behavior of a second order fuzzy difference equation is considered by Din [25]. The fuzzy non-linear difference equation is considered by Zhang et al. [26], where Memarbashi and Ghasemabadi [27] corporate with a volterra type rational form by Stefanidou and Papaschinopoulos [28]. The economics application is considered by Konstantinos et al. [29]. Mondal et al. [30] solve the second-order intuitionistic difference equation. Non-linear interval-valued fuzzy numbers and their relevance to difference equations are shown in [31]. National income determination models with fuzzy stability analysis in a discrete system are elaborately discussed by Sarkar et al. [32]. The fuzzy discrete logistic equation is taken and stability situations are found in the literature [33]. Zhang et al. [34] show the asymptotic performance of a discrete time fuzzy single species population model. On discrete time, a Beverton-Holt population replica with fuzzy environment is illustrated in [35]. Additionally, a different view of the fuzzy discrete logistic equation is taken under uncertainty in [36]. The existence and stability situation of the difference equation with a fuzzy setting is found by Mondal et al. [37]. Important results are also found for fuzzy difference equations by Khastan and Alijani [38] and Khastan [39].

Novelties of the Work
In this connection of the above idea, few advances can still be prepared, which include: (1) The homogeneous difference equation, solved and analyzed with a neutrosophic initial condition, neutrosophic coefficient, and neutrosophic coefficient and initial together as a different section, which was not done earlier.
(2) Establishment of the corresponding characterization theorem for the neutrosophic set with a difference equation.
(3) Different theorems, lemmas, and corollary drawn for the purpose of the study. (4) Numerical examples of the difference equation with a neutrosophic number, solved and illustrated for better understanding of our observations.
(5) An application in actuarial science, illustrated in a neutrosophic environment for better understanding of the practical application of the proposed theoretical results.

Structure of the Paper
In Section 1, we recall the related work and write the novelties of our study. The preliminary concepts are addressed in Section 2. The difference equation with a neutrosophic variable is defined and corresponds with a necessary theory, for which a lemma is prepared for the study in Section 3. Section 4 shows the solution of the neutrosophic homogeneous difference equation. Two numerical examples are shown in Section 5. In Section 6, we take an appliance of an actuarial science problem in the neutrosophic data and solve it. The conclusion and future research scope are written in Section 7.

Definition 2. Neutrosophicfunction:
If we take the set of all real numbers as notation ℛ and real valued fuzzy numbers as notationℛ ℱ , then the function : ℛ → 0,1 is called a fuzzy number valued function if satisfies the subsequent properties.
(1) is the upper semi continuous. (2) is the fuzzy convex, i.e., is normal, i.e., ∃a ∈ ℛ, such that ( ) = 1 Definition 3.Triangular neutrosophic number: [40]If we consider the measure of the truth, for which indeterminacy and falsity are not dependent, then a Triangular Neutrosophic number is taken as = ( , , ; , , ; , , ), where the truth membership, falsity, and indeterminacy membership function is treated as follows: Here, 0 < , , ≤ 1 and 0 < + + ≤ 3 The verbal phrase with the number can be written as in Table 1: We consider an autonomous linear homogeneous difference equation of the form: with the initial condition = .The solution of Equation (2) can then be written as: Theorem 1. [41] Let ∈ ℕ, ≥ 2. A linear homogeneous system of the first order difference equation is given in matrix form as: where, = ( , , … , ) and = × , , = 1, 2, … , The solution of Equation (3) can then be written as: The difference Equation (1) is considered as the neutrosophic difference equation if any one of the following conditions are added: (i) The initial condition or conditions are the neutrosophic number (Type I); (ii) The coefficient or coefficients are the neutrosophic number (Type II); (iii) The initial conditions and coefficient or the coefficients are both neutrosophic numbers (Type III).

Theorem 2.Characterization theorem:
Let us consider the neutrosophic difference equation problem: with initial value = as a neutrospohic number, where : * × ℤ → * , such that (1)The parametric form of the function is:

Note 2.
For some cases, the solution may not become strictly strong or weak solution type. In this scenario,a specific time interval or specific interval of α, β, or γ becomes the strong or weak solution.
The main objective is to find the strong solutions. For scenariosin which neitherthe strong nor weak solutions occur, we call them non-recommended neutrosophic solutions. We strongly recommended taking strong solutions.

Solution of Neutrosophic Homogeneous Difference Equation
Considering linear homogeneous difference equations: In a neutrosophic sense, another inequivalent form of is (7) taken as: Remarks 1. Equations (7) and (8) are equivalent in a crisp sense, but in fuzzy sense they are not equivalent.
Proof 1. If we take the fuzzy difference Equation (7), it becomes Theorem 1. i.e., Clearly, from (9) and (10), we conclude that they are different. Therefore, in a crisp sense, (7) and (8) are the same, but not in a neutrosophic sense.
Theorem 3. Suppose a and u are positive neutrosophic numbers, then ∃ is a unique positive solution for Equation (7).
Therefore, we obtain the following: The first pairs of equations can be written in the matrix form as: From Equation (22), let the co-efficient matrix be = 0 −µ −µ 0 Therefore, Therefore, the solution of (4.1.6), using Theorem (3.1), is given by: When is an even natural number, the general solutions are: In the matrix form, the first pairs of equations of Equation (28) can be written as: The solution of (29) is given by: where, The solution of Equation (30) when is even is: In this case, solutions become crisp numbers, i.e., ( ) = µ( ) µ( ) . The solution of Equation (16), which follows from Equation (26), is thengiven by:

The solution of Equation (30) when is odd:
The above equations show that the solution for is even only. When is odd, the solutions, which follow from Equation (32), are as follows:

The Solution When = 1and the Initial Condition is a Neutrosophic Number
Taking the( , , )-cut of Equation (8) In the matrix form, the first pairs of Equation (36) can be written as: The solution of Equation (37) is, when is even: When is odd, the solutions are: For both cases, when either is even or odd, , ( )and , ( )leads to a convergent solution. In a similar way, solutions of remaining equations are as follows: when is even: The solutions of (43) are, when is even: The solutions of (44) and (45) In the matrix form, among the above equations, the first pair of Equation (50) can be written as: When is odd: In this case, the solutions are given, following from Equation (50): when is even: when is odd: Remarks 2.We plot the solution for n = 2. From the above Table 2 and Figure 1, we see that u , (α) is an increasing function and u , (α) is a decreasing function, with respect to α. On the other hand,u , (β) is a decreasing function and u , (β) is an increasing function, with respect to β. Additionally,u , (γ) is a decreasing function and u , (γ) is an increasing function, with respect to γ. Therefore, using the concept of Definition 3.2, we call the solution a strong solution.

Remarks 3.
We plotted the solution for n = 5. From the above Table 3 and Figure 2, we see that u , (α) is an increasing function and u , (α) is a decreasing function, with respect to α. On the other hand,u , (β) is a decreasing function and u , (β) is an increasing function, with respect to β. Additionally,u , (γ) is a decreasing function and u , (γ) is an increasing function, with respect to γ. Therefore, using the concept of Definition 6, we call the solution a strong solution.  We interpret the solution for fixed , , = 0.4 and different in Table 4 and Figure 3.  As previous examples, we easily interpret the solutions in a different manner.

Application of the Method in Actuarial Science
Let us consider that a sum is invested at a compound interest of per unit amount and per unit of time and is the amount at the end of time . We then get the difference equation associated with the problem, which is: If, for some reason, may vary, we are interested to find the possible amount after a certain time interval.
For this problem, let us consider hypothetical data and solve it. Suppose a person has initially invested = 10000$in a firm, where they get about 4% interest (which may be considered a neutrosophic value).
As per Table 1, if we take the verbal phrase for a triangular neutrosophic number, we then set the interest rate as follows: For the truth part: low as 3%, medium as 4%, high as 5%; For the falsity portion: very low as 2%, medium as 4%, very high as 6%; For the indeterminacy part: between low and very low 2.5%, medium 4%, between high and very high 5.5%, i.e., we can takẽ= (3,4,5; 2,4,6; 2.5,4,5.5)% per annum rate. We wish to predict the amount of money after 10 years.
The solution of (66) can be written using the concept of (19), as follows:  Table 5 and Figure4, we see that S , (α) is an increasing function and S , (α) is a decreasing function, with respect to α. On the other hand,S , (β) is a decreasing function and S , (β) is an increasing function, with respect to β . Additionally,S , (γ) is a decreasing function and S , (γ) is an increasing function, with respect to γ. Therefore, using the concept of Definition 3.2, we call the solution a strong solution. (2) From Table   5, we can see that we find the crisp solution at α = 1 , β, γ = 0 (since, at α = 1 , β, γ = 0, the neutrosophic number becomes a crisp number) and fort = 10 is equal to 14802.4428. Therefore, we can say that after 10 years, the most probable chance to get the money is 14802.4428$.(3)If we consider α = 0 and β, γ = 1, i.e., in the case that we get the most uncertain solution interval, we observe that

Conclusion and Future Research Scope
In this paper, we find the solution strategy for solving and analyzing homogeneous linear difference equations with neutrosophic numbers, i.e., we found the solutions of the homogeneous difference equations with initial conditions and coefficients, both as neutrosophic numbers. We demonstrate the solution of different cases using the neutrosophic characterization theorem, which is established in this paper. The strong and weak solution concepts are also applied to different results.
Moreover, the outcomes of the study are as follows: (1) The difference type of the homogeneous difference equation is solved in a neutrosophic environment and the symmetric behavior between them is discussed.
(2) The characterization theorem for the neutrosophic difference equations are established.
(3) The strong and weak solution concept is applied for the neutrosophic difference equation. For some limitations, we did not study the different perspectives of related research in the theory of difference equations with uncertainty in this present work. From this work, anyone can take motivation and find a new theory and results in the following field, as follows: (1) The solution of difference equation can be found with different types of uncertainty, such as Type 2 fuzzy, interval valued fuzzy, hesitant fuzzy, rough fuzzy environment.
(2) Finding several methods (analytical and numerical) for solving non-linear first and higher order difference equations or system of difference equations with uncertainty.
(3) Solving the real-life model associated with the discrete system modeling with uncertain data. As a final argument, we can surely say that this research is very helpful to the research community who deals with discrete system modeling with uncertainty.