Semi-Analytical Scheme for Solving Intuitionistic Fuzzy System of Differential Equations

The aim of this article is to implement the Generalized Modified Adomian Decomposition Method to compute the semi-numerical solution of the linear system of intuitionistic fuzzy initial value problems. Here, we consider the initial values as generalized trapezoidal intuitionistic fuzzy numbers. The technique is applied to brine tanks problem and coupled mass spring systems.Theoretically, different approaches to solving a system of generalized trapezoidal intuitionistic fuzzy differential equations are discussed in this study under the presumption that the coefficients of the system of the differential equations are associated to generalized trapezoidal intuitionistic fuzzy numbers. The approximate results are compared with exact solutions which shows good efficiency. The corresponding graphs at different levels of uncertainty show the example’s numerical outcomes. The graphical representations further demonstrate the effectiveness and accuracy of the proposed method in comparison to existing semi-numerical methods in the literature.


I. INTRODUCTION
System of differential equation plays a significant role in modeling and studying many naturally occurring phenomenon such as population models, economic models, friction model, bacteria culture model, predator-prey model, weight loss and oil production model, bank account and drug concentration problem, human immunodeficiency virus (HIV) model. In classical set theory, the variables or parameters are taken as crisp numbers. But in actual case, these variables or parameters are usually uncertain or vague. So, these variables may be considered as a fuzzy numbers. In other words, to overcome uncertainty we use The associate editor coordinating the review of this manuscript and approving it for publication was Longzhi Yang . fuzzy numbers. So the system of differential equations are converted to system of fuzzy differential equations (SFDE).
The concept of fuzzy set theory was firstly introduced by Zadeh in 1965, as the extension of classical set theory [1]. The concept of fuzzy set theory has been applied to various fields of science and engineering to handle vagueness and uncertainty. In 1987, Kandel and Byatt [2] introduced the fuzzy differential equations. The fuzzy differential equations have been applied in numerous daily life problems [3], [4], [5]. Vasavi et al. [6] discussed fuzzy differential for cooling problems. Devi and Ganesan used fuzzy differential equations in modelling electric circuit problem [7]. Ahmad et al. [8] studied a mathematical method to find the solution of fuzzy integro differential equations. Sadeghi et al. [9] studied the system of fuzzy differential equation. Buckley et al find the solution of system of first order linear fuzzy differential equations by extension principle [10]. Hashemi et al. find the series solution of SFDE [11]. In 1986, Atanassov [12] introduced an extension of fuzzy set theory known as intuitionistic fuzzy set. The intuitionistic fuzzy set [13] not only provides the information about membership values but also the non-membership values respectively, and so that the sum of both values is less than one. Intuitionistic fuzzy differential equations are being studied widely and being used in various fields of Physics, Chemistry, Biology as well as among other fields of science and engineering. Melliani and Chadli obtained the approximate and numerical solutions of intuitionistic fuzzy differential equations with linear differential operators [14], [15]. Gulzar et al. worked on fuzzy algebra [16], [17], [18]. Akin and Bayeg [19], [20] studied a method to find general solution of second order intuitionistic fuzzy differential equation and to solve the system of intuitionistic fuzzy differential equations with intuitionistic fuzzy initial values. Mondal and Roy [21], [22], [23] studied the generalized intuitionistic fuzzy Laplace transform method and to solve the system of differential equations with initial value as triangular intuitionistic fuzzy number. Saw et al introduced a method for solving system of linear intuitionistic fuzzy equations [24].
The Adomian Decomposition Method (ADM) which is a semi analytical method was first presented by Adomian in 1980's [25], [26]. This method is very efficient in finding the solutions of differential equations, algebraic equations as well as integral equations. In this article, we will propose the Generalized Modified Adomian Decomposition Method (GMADM) to find the solutions of the system of linear intuitionistic fuzzy differential equations with initial values as generalized trapezoidal intuitionistic fuzzy number. This modification was proposed by Wazwaz [27]. He presented a reliable modification to the ADM. In this modification Wazwaz divides the original function into two parts, one part assigned to the initial term of the series and the other to the second term. This modification results in a different series being generated. The efficiency of this method depends only on the choice of the parts into which the original function is to be divided.
First order system of fuzzy differential equations is important among all the fuzzy differential equations. There are many approaches to solve the SFDEs. Buckley and Feuring [28] solving the linear system of first order ordinary differential equations with fuzzy initial conditions by extension principle using triangular fuzzy number. The geometric approach is developed by Gasilova et al. [29] and series solution is developed by Hashemi et al. [30]. Mondal and Roy [31] studied strong and weak solution of first order homogeneous intuitionistic fuzzy differential equation, subsequently and studied system of differential equation in literature. Melliani,et al. [32] discussed the existence and uniqueness of the solution of the intuitionistic fuzzy differential equation and its system using the analytical technique. Therefore, finding an efficient and accurate algorithm for investigating FIE has been one the hot areas of research in recent time. To achieve these goals, various methods and procedures were used to handle differentia equations, using triangular fuzzy number, for details, see [9], [33].
In this study, motivated by the aforementioned work, we solve the system of differential equations using a GMADM and a more generalized fuzzy system of differential equations, namely a trapezoidal intuitionistic fuzzy system of intuitionistic differential equations.
The main contributions of this research work are summarized below.
• GMADM is used to solve a system of differential equations using initial conditions as a Generalized trapezoidal intuitionistic fuzzy number.
• In order to solve a system of fuzzy intuitionistic differential equations that have not before been explored, the computational complexity of the suggested GMADM is discussed.
• Applications of system of Generalized trapezoidal intuitionistic fuzzy differential equations in mechanical engineering are taken into consideration in a Generalized trapezoidal intuitionistic fuzzy environment.
• Computational tools are used to evaluate the effectiveness and applicability of the suggested analytical scheme.
This paper is organized as follows: In section II, we recall some basic definitions which we will use in further sections. In sections III, we introduced our proposed method. In section IV, the efficiency of this method has been illustrated by applications. In the last section, we give conclusions.

II. PRELIMINARIES
In this section, the fundamental definitions of fuzzy set and intuitionistic fuzzy set are presented.
Definition 1 [34]: Let Definition 6 [23]: An intuitionistic fuzzy set ⋆ I is said to be convex set for the membership function if it satisfy the following condition: Definition 7 [23]: An intuitionistic fuzzy set ⋆ I is said to be concave set for the non-membership function if it satisfy the following condition: Applying the ✠ L −1 operator on both sides of (3), (4), (5) and (6), we get; where, the above functions are found by using the initial conditions. Now by using the GMADM the solutions of the (7), (8), (9) and (10 ), can be expressed in the form of an infinite series for the unknown functions as follows: Using (11), (12), (13) and (14), into (7), (8), (9) and (10), we have: According to the GMADM the recursive relation for the (15), (16), ( 17) and (18), is as follows: 33210 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
The nth term approximation to the solution is defined as follows:  27) and (28), and using the initial conditions we obtain; Now by using GMADM the solution of (29), (30), (31) and (32), can be expressed as; Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
By taking (α, β)-cut of (38), (39), we get the following equations: Here ✠ L = d 2 dt 2 and by taking both sides of (40), (41), (42) and (40), and using the initial conditions we obtain; solutions, for the generalized trapezoidal intuitionistic fuzzy initial value problem used in example 2. Now by using GMADM the solution of (44), (45), (46) and (47), can be expressed as;  − 0.000225970018t 9 − 0.146155754t 8 1 500   56) and (57), and using the initial conditions we obtain; Now by using GMADM we get; The following is the mathematical and exact solution to Example 3 using the classical method, as shown in Figure 4(a-d) and Table 6:

IV. RESULTS AND DISCUSSION
We discuss how the computing efficiency, stability, and residual error robustness of the proposed modified technique, GMADM, outperforms the ADM and TSM approaches.
• Tables 1 to 9 illustrate how GMADM, a recently created approach, is more reliable and consistent than ADM and TSM. While solving a system of fuzzy intuitionistic differential equations, it is observed that GMADM converges more quickly and accurately than ADM and TSM.
• Tables 3, 6, and 9 clearly demonstrate that GMADM is superior to ADM and TSM in terms of iterations, residual error, and CPU time.
• Figures 2-11 compare the numerical simulation of our recently modified family GMADM to a precise solution of the generalized trapezoidal intuitionistic fuzzy initial value problem used in Example 1-3 respectively.
• Figures 2-3, 4-11 illustrate the precise and approximate solutions for the membership and non-membership functions of the generalized trapezoidal intuitionistic   Illustrates the approximate solution iterations in column 2, residual error in column 3, and CPU time required by the numerical technique GMADM to determine the approximate solution of the system of generalized fuzzy intuitionistic differential equations used in Example 3 in column 4. Whenever the error of all methods is taken into account, we can conclude that the GMADM has a better convergence behavior and is more stable than the ADM and TSM, respectively.
fuzzy system of initial value problems used in example 1-3, respectively.
• The numerical results obtained in Tables 1-2,4-6,7-8, and Figures 1-11 clearly demonstrate that the exact and approximate solutions are matched up to 30 decimal places using GMADM, 7 decimal places using ADM, and 9 decimal places using TSM. The numerical simulation of our methods demonstrates unequivocally how much superior our method is to ADM and TSM.

V. CONCLUSION
In this work, Generalized Modified Adomian Decomposition Method have been utilized for computing the approximate solution of the linear system of generalized trapezoidal intuitionistic fuzzy initial value problems. We used the initial conditions as generalized trapezoidal intuitionistic fuzzy numbers. We have applied this procedure to brine tank problems and coupled oscillators. Moreover, by comparing the approximate results with exact solution, we have shown that this method is more reliable. Future studies will therefore focus on the solution of systems of higher order generalized tripezodial intuitionistic fuzzy system of differential equations as well as a system of nonlinear first order differential equations and their application [48], [49], [50], [51], [52], [53] in a more generalized fuzzy environment utilizing GMADM.
NASREEN KAUSAR received the Ph.D. degree in mathematics from Quaidi-Azam University, Islamabad, Pakistan. She is currently an Associate Professor of mathematics with Yildiz Technical University, Istanbul, Turkey. Her research interests include the numerical analysis and numerical solutions of the ordinary differential equations (ODEs), partial differential equations (PDEs), Volterra integral equations, as well as associative and commutative, nonassociative and noncommutative fuzzy algebraic structures and their applications.
KHULUD ALAYYASH received the B.Sc. degree (Hons.) in mathematics from King Khaled University, Saudi Arabia, the M.Sc. degree in applied computation and bio-mathematics from Heriot-Watt University, U.K., and the Ph.D. degree in mathematics of solid mechanics (multiscale modeling, limit states analysis, optimization, finite-element modeling, and structural analysis) from Cardiff University, U.K. She is currently a Treasurer and a member of the Society for Industrial and Applied Mathematics (SIAM) Chapter, Cardiff University (build cooperation between mathematics and the worlds of science and technology through our publications, research, and community). She is also an Assistant Professor.
MOHAMMED M. AL-SHAMIRI received the Ph.D. degree in geometric topological algebra from Menonufia University, Menonufia, Egypt, in 2008. He is currently an Associate Professor of mathematics with King Khalid University, Saudi Arabia. He is also an Associate Professor with Ibb University, Yemen. His research interests include topological spaces, topological geometry, fuzzy topology, graph and knot and fuzzy graph and fuzzy knot, geometrical transformations (folding, retraction, and deformation retract), fuzzy group, fuzzy ring, fuzzy module, fuzzy field, and fuzzy decision support systems.
NAYYAB ARIF thesis work focused on analytical techniques for solving generalized trapezoidal intuitionistic fuzzy initial value problems. She is about to enroll in the Ph.D. degree. Her research interests include fuzzy set theory and its applications. She researches analytical methods which solve fuzzy initial value problems.
RASHAD ISMAIL received the Ph.D. degree in scientific computing from Assiut University, Egypt, in 2011. He is currently an Assistant Professor with King Khalid University, Saudi Arabia. He is also an Assistant Professor of scientific computing with Ibb University, Yemen. His research interests include graph theory and its applications, graph domination, meta-heuristics, data mining, computer networks, fuzzy, fuzzy graph, and fuzzy decision support systems.