Homotopy Perturbation Method for Solving Linear Fuzzy Delay Differential Equations Using Double Parametric Approach

Delay differential equations (known as DDEs) are a broad use of many scientific researches and engineering applications. They come because the pace of the shift in their mathematical models relies all the basis not just on their present condition, but also on a certain past cases. In this work, we propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using the Homotopy Perturbation Method with double parametric form fuzzy numbers. The detailed algorithm of the approach to fuzzification and defuzzificationis analysis is provided. In the initial conditions of the proposed problem there are uncertainties with regard to the triangular fuzzy number. A double parametric form of fuzzy numbers is defined and applied for the first time in this topic for the present analysis. This method’s simplicity and ability to overcome delay differential equations without complicating Adomian polynomials or incorrect nonlinear assumptions. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show the features of this proposed method, a numerical example is illustrated, involving first order fuzzy delay differential equation. These findings indicate that the suggested approach is very successful and simple to implement.

Abstract Delay differential equations (known as DDEs) are a broad use of many scientific researches and engineering applications. They come because the pace of the shift in their mathematical models relies all the basis not just on their present condition, but also on a certain past cases. In this work, we propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using the Homotopy Perturbation Method with double parametric form fuzzy numbers. The detailed algorithm of the approach to fuzzification and defuzzificationis analysis is provided. In the initial conditions of the proposed problem there are uncertainties with regard to the triangular fuzzy number. A double parametric form of fuzzy numbers is defined and applied for the first time in this topic for the present analysis. This method's simplicity and ability to overcome delay differential equations without complicating Adomian polynomials or incorrect nonlinear assumptions. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show the features of this proposed method, a numerical example is illustrated, involving first order fuzzy delay differential equation. These findings indicate that the suggested approach is very successful and simple to implement.

Introduction
The variety of life experiments in science and engineering can be formulated in the form of ordinary or partial differential equation models. Models that involve delaying differential equations (DDEs) are defined as type of differential equations where the time derivatives at the current time depend on the solution and possibly its derivatives at previous times. A class of such equations, which involve derivatives with delays as well as the solution itself, has been called neutral DDEs [1,2]. For instance, drives, sensors and field networks involved in feedback loops can involve delays. In epidemic dynamics, time delay system is also used for designing multiple different mechanisms [3]. Most DDEs that arise in population dynamics and epidemiology model intrinsically have nonnegative quantities. Therefore, it is important to establish that nonnegative initial data which give rise to nonnegative solutions.
For modeling a dynamic system where the information regarding its behavior is insufficient, Fuzzy differential equations (FDEs) are known as a useful tool [4]. When these experiments are not modelled completely and their nature is uncertain or vague, fuzzy models can be used. FDEs are suited to modeling such dynamical systems that have been applied in many applications such as in population modelling [5,6], mathematical physics [7] and medical sciences [8]. Approximate methods were generally used to solve fuzzy delay differential equations (FDDEs) with single parametric form fuzzy numbers [9,10].
Many researchers in the field of Science and Engineering have used Homotopy Perturbation Method (HPM) to achieve an approximate solution for different kinds of linear and nonlinear models [10][11][12][13], even the fuzzy partial differential equation models as in Sarmad et al [14]. Solving problems with HPM often help to better understand a physical problem, and may help improve future procedures and designs used to solve these problems. Also, this method has a useful feature in that it provides the solution in a rapid convergent power series with the elegantly computable convergence of the solution without any need for discretization and linearization as in numerical methods [15]. However, in using a single parametric form for HPM, a n x n fully fuzzy system has to be converted to 2n x 2n crisp system. On the other hand, for the double parametric form, the n x n fully fuzzy system is converted to the same order of crisp system, hence requiring a less amount of computation. The double parametric form, which has been employed in fuzzy differential equation, is more general and straightforward [16].
Our aim here is to construct a new form of HPM based on the approach of double parametric form of fuzzy numbers to solve first order DDEs using fuzzy-set theory properties.
The outline of this research is as follows: Section 2 presents some tools and definitions of the fuzzy number for the fuzzy analysis of the fuzzy model. Section 3 introduces the defuzzification of the general FDDE in new fuzzy number form. In the Section 4, we modified the standard HPM into double parametric form fuzzy numbers form. Section 5, we implement the HPM in the Section 4 on first order linear FDDE to show the capability of the method. Finally section 6 presents the conclusions of this research.

Triangular Fuzzy Number [17]
Fuzzy numbers are a subset of the real numbers set, and represent uncertain values. Fuzzy numbers are linked to degrees of membership referring to how true it is to say if something belongs or does not belong to a determined set. A fuzzy number µ is called a triangular fuzzy number if it is defined by three numbers  <  <  where the graph of ( ) is a triangle with the base on the interval [, ] and vertex at =  as illustrated in Figure 1 and where its membership function is given by the following Since the r-level set is the link between the fuzzy domain and the crisp domain, we can use the advantages of the theories in crisp domain and in the fuzzy domain such that

Single Parametric form of Fuzzy Number [19]
The class of all fuzzy subsets of ℝ is denoted by ̃ and the solution of Fuzzy Initial Value Problem (FIVP) satisfy the following properties: ̃ is called the space of fuzzy numbers and ℝ is a proper subset of ̃.

Double Parametric Form of Fuzzy Number [20]
Using the parametric form as defined in definition ̃= and if = 1 then ̃ ( ; 1) = ( ) (upper fuzzy number). In this way, the double parametric form provides less computational work than single parametric form.

Fuzzy Delay Differential Equation in Double Parametric Form
Consider the FDDEs [9].

Fuzzy HPM in Double Parametric Form
This section presents the analysis of HPM for approximately solving first order FDDE as in Eq. (1) under double parametric form fuzzy number. According to the fuzzy analysis by Ali et al in [24], to obtain the approximate solution of the first order fuzzy initial value problem by HPM we rewrite Eq.
where ∈ [0, 1] is an embedding parameter to deform Eq. (5) and initial guessing functions can be defined as follows: where changes from 0 to 1 . Eqs. (6-7) are called deformation homotopy. The usage of as a small parameter is to construct HPM series function from Eq. (7) such that each ̃( ; ) in Eq. (3) In the standard HPM for solving fuzzy differential equation with fuzzy numbers with single parametric form, it is required to analyze the HPM for the lower and upper bound solution of Eq.(1) and hence would demand more computational work. The advantage of the parameter that deforms from 0 to 1 reduces the computational and analysis work to obtain the solution of fuzzy differential equations for illustration, at = 0 in Eq. (10) and this is applicable for all the HPM analysis in double parametric form fuzzy numbers. Details and illustration are provided in the next section.

Results and Discussion
In this section, we present a numerical problem involving FDDE with an approximate solution obtained by HPM described in Section 4. This is followed by fuzzy analysis under double parametric form fuzzy number. Consider the linear FDDE with fuzzy initial conditions [9]: where the fifth order HPM approximate solution in double parametric for fuzzy number for all ∈ [0,1] and ∈ [0,1] is illustrated in Tables (1-6) and Figure 2 as follows:    Tables 1 and 6, the lower and upper bound solution of Eq. (13) is obtained with fifth-order HPM solution when = 0 and 1 respectively that satisfies the fuzzy differential equations solution for all level sets ∈ [0,1] for = 1. Additionally, for each = 0.2,0.4, 0.6 and 0.8 also follow the fuzzy differential equations solution for all level sets ∈ [0,1] for = 1 and other results can be obtained for all values of ∈ [0,1].
This analysis is illustrated in Figure 2 in the form of triangular fuzzy number, presenting and comparing the results under double parametric form fuzzy number of fifth order HPM and the exact solution of Eq. (13) for different values of ∈ [0,1].

Conclusions
A new form of HPM has successfully been constructed based on the approach of double parametric form of fuzzy numbers to solve first order DDEs using fuzzy-set theory properties. In this work, homotopy perturbation method (HPM) has successfully been analyzed, altered and applied to the approximate solution of first order FDDE using a double parametric form of fuzzy numbers. The proposed double parametric form approach is found to be easy and straightforward with less computational work. The performance of the method is shown using a triangular fuzzy number. It is interesting to note of the values of ∈ [0,1] satisfy the fuzzy differential equation solution in the form of the triangular fuzzy number of all fuzzy level set.