View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Institute of Advanced Engineering and Science International Journal of Electrical and Computer Engineering (IJECE) Vol. 10, No. 6, December 2020, pp. 6521~6530 ISSN: 2088-8708, DOI: 10.11591/ijece.v10i6.pp6521-6530  6521 On solving fuzzy delay differential equation using bezier curves Ali F. Jameel1, Sardar G. Amen2, Azizan Saaban3, Noraziah H. Man4 1,2,3,4 School of Quantitative Sciences, College of Art and Sciences, Universiti Utara Malaysia (UUM), Malaysia 2Department of Financial and Banking, Collage of Business Administration and Financial Science, Al-Kitab University, Iraq Article Info ABSTRACT Article history: In this article, we plan to use Bezier curves method to solve linear fuzzy delay differential equations. A Bezier curves method is presented and modified to solve fuzzy delay problems taking the advantages of the fuzzy set theory properties. The approximate solution with different degrees is compared to the exact solution to confirm that the linear fuzzy delay differential equations process is accurate and efficient. Numerical example is explained and analyzed involved first order linear fuzzy delay differential equations to demonstrate these proper features of this proposed problem. Received Apr 14, 2020 Revised May 27, 2020 Accepted Jun 12, 2020 Keywords: Bezier control points Delay differential equations Fuzzy differential equations Fuzzy numbers Fuzzy set theory Residual function Copyright © 2020 Institute of Advanced Engineering and Science. All rights reserved. Corresponding Author: Ali F. Jameel, School of Quantitative Sciences, College of Art and Sciences, Universiti Utara Malaysia (UUM), Sintok, 06010 Kedah, Malaysia. Email: alifareed@uum.edu.my 1. INTRODUCTION Fuzzy set theory is a powerful instrument for modeling uncertainty in a wide range of real issues and for processing vague or subjective information in mathematical patterns. DDEs are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at a previous time. Often called DDEs time–delay systems with or with dead impact–time, inherited process equations with deviating argument [1, 2]. The fundamental theory of steady works and key theory variables such as unique solutions are found in [1-3]. Next, a large number of the Delay Differential Equation have been extensively investigated in the novel, and monographs were published, including considerable on in [4], and so forth. The research advantage of the differential delays is because many systems have been the prototype of better differential delays in engineering, economics, science, etc. The difference equations of delays are delayed. Nevertheless, they are not realistic to regulate problems. Most of these equations obviously cannot be precisely solved. Efficient numerical methods must therefore be designed to approach their solutions. Ishiwata et al. used the rational approximation method and the collocation method [5-7] to compute numerical solutions of DDEs with proportional delays. Hu et al. [8] applied linear multi-step methods to compute numerical solutions for neutral DDEs. Other method obtained approximate solutions for variety of DDEs such as Runge Kutta methods, block methods and one- leg θ-methods in [8-12]. Moreover, the DDEs solved approximately via some approximation methods in many fields of mathematics using approximation methods: for example, the homotopy analysis method [13, 14], Adomian decomposition method [15] and homotopy perturbation Journal homepage: http://ijece.iaescore.com/index.php/IJECE 6522  ISSN: 2088-8708 method [16]. Fuzzy DDE problem will model when the crisp model is not complete and its premasters or conditions under fuzzy properties. FDDEs were solved by multiple researchers in recent years with an approximate solution in [17, 18]. We will present in this article new plans for the approximate solution of FDDEs by means of the curves of Bezier method in fuzzy domain and analyzed the fuzzy solutions in different degree of approximations. The outline of this paper will be as follows: FDDEs will be introduced in section 2. In section 3, Problem of Fuzzy Delay System will be declared. Introduced proportional delay with FDDEs in section 4. In sections 5 and 6 respectively degree elevation and Bezier curves and will be declared. Using Bezier control points for Solving FDDE aforementioned method and suggested will be implemented on it in section 7. In section 8 solved numerical problems, appeared the accuracy and adequacy of the method. Lastly, the conclusion briefly will be given in section 9. 2. DESCRIPTION OF DELAY FUZZY DIFFERENTIAL EQUATIONS Many DDEs are increasing, fundamentally optimistic in the models of epidemiology and population dynamics. It is therefore worth noting that positive initial data lead to positive solutions [15]. Consider the following FDDE: 𝐷𝑣̃(𝑥) = ̃𝑦(𝑥, 𝑣̃(𝑥), 𝑣̃(𝑥 – 𝑘)) (1) where for all fuzzy level sets 𝑟 ∈ [0,1] we have the following defuzzifications: - The fuzzy functions 𝑣̃(𝑥) [19] is denoted as 𝑣̃(𝑥; 𝑟) = [𝑣(𝑥; 𝑟), 𝑣(𝑥; 𝑟)], - The fuzzy delay functions 𝑣̃(𝑥 − 𝑘) is denoted as 𝑣̃(𝑥 − 𝛼; 𝑟) = [𝑣(𝑥 − 𝛼; 𝑟), 𝑣(𝑥 − 𝛼; 𝑟)] - The fuzzy first order H-derivative, see [19] 𝐷𝑣̃(𝑥; 𝑟) = [𝐷𝑣(𝑥; 𝑟), 𝐷𝑣(𝑥; 𝑟)], Next, assume that the fuzzy function in (1) can be written as: 𝑦̃(𝑥, 𝑣̃(𝑥), 𝑣̃(𝑥 − 𝛼)) = 𝑦̃ (𝑥, 𝑉̃ (𝑥)) such that 𝑦̃ (𝑡, 𝑉̃ (𝑥)) = [𝑦 (𝑡, 𝑉̃ (𝑥)) , 𝑦 (𝑡, 𝑉̃ (𝑥))] By using Zadeh extension principles [20], we have the following membership function 𝐹 (𝑥, 𝑉̃ (𝑥; 𝑟)) = 𝑚𝑖𝑛{𝐷𝑣̃(𝑥; 𝑟): 𝜇|𝜇 ∈ [𝑉̃ (𝑥)]𝑟 }, where 𝐺 (𝑥, 𝑉̃ (𝑥; 𝑟)) = 𝑚𝑎𝑥{𝐷𝑣̃(𝑥; 𝑟): 𝜇|𝜇 ∈ [𝑉̃ (𝑥)]𝑟 }, { 𝑦 (𝑥, 𝑉̃ (𝑥; 𝑟)) = 𝐹 (𝑡, 𝑉(𝑥; 𝑟), 𝑉(𝑥; 𝑟)) = 𝐹 (𝑥, 𝑉̃ (𝑥; 𝑟)) 𝑦 (𝑥, 𝑉̃ (𝑥; 𝑟)) = 𝐺 (𝑡, 𝑉(𝑥; 𝑟), 𝑉(𝑡; 𝑟)) = 𝐺 (𝑥, 𝑉̃ (𝑥; 𝑟)) (2) with a single delay 𝑘 > 0. For each 𝑟 ∈ [0,1], suppose that [𝑦̃(𝑥, 𝑉̃ )]𝑟 and[𝑦̃𝑣 (𝑥, 𝑉̃ )]𝑟 , are continuous on ℝ3 . Let 𝜑: [𝑧 − 𝑘, 𝑧] → ℝ be continuous where 𝑧 ∈ ℝ be given. Require the solution 𝑣(𝑥) of (1) satisfying 𝑣̃(𝑥; 𝑟) = 𝜑̃(𝑥; 𝑟),𝑧 − 𝑘 ≤ 𝑥 ≤ 𝑧 (3) and satisfying (1) on 𝑧 ≤ 𝑥 ≤ 𝑧 + 𝛼 for some 𝛼 > 0. Note: should be explain 𝐷𝑣̃(𝑥; 𝑟) as the right-hand derivative at z. Now demonstrate a material system design problem that shows phenomenon of time delay. The question picked in this section is exactly the right one in the test (1). Int J Elec & Comp Eng, Vol. 10, No. 6, December 2020 : 6521 - 6530 Int J Elec & Comp Eng ISSN: 2088-8708  6523 3. PROBLEM OF TIME FUZZY DELAY SYSTEM The existence of lags in economic systems is completely normal since a decision for the results of ̃(𝑥) be article should be given a fixed period after. In one sample [21] of total economy and suppose 𝑈 the proceeds which can divide into autonomous expenditure, consumption 𝐴̃(𝑥), and investment 𝐵̃ (𝑥). From section 2, we have: ̃(𝑥; 𝑟) = 𝐴̃(𝑥; 𝑟) + 𝐵̃ (𝑥; 𝑟) + 𝐶̃ (𝑥; 𝑟) 𝑈 (4) ̃(𝑥; 𝑟) 𝐴̃(𝑥; 𝑟) = 𝑏̃ 𝑈 where 𝑏̃ is a consumption fuzzy coefficient following the properties of triangular fuzzy number [17]. From (4), ̃(𝑥; 𝑟) = 𝑈 𝐵(𝑥;𝑟) + 𝐶(𝑥;𝑟) 1 − 𝑏(𝑟) 𝐵̃ (𝑥;𝑟) + 𝐶̃ (𝑥;𝑟) {𝐵(𝑥;𝑟) + 𝐶(𝑥;𝑟) 1 − 𝑏̃ (5) 1 − 𝑏(𝑟) ̃ (𝑥) there is limited time between the production and Assuming that, following a decision to run 𝐷 ordering of capital instruments. In expression of the paper of capital savings 𝐽̃(𝑥), we have ̃ (𝑥 − 𝑘; 𝑟) 𝐽̃′ (𝑥; 𝑟) = 𝐷 (6) 1 𝑥 ̃ (𝑇; 𝑟)𝑑𝑇 𝐵̃ (𝑥; 𝑟) = ∫𝑥−𝑘 𝐷 𝑘 (7) ̃ (𝑥; 𝑟) is For each fuzzy level set 𝑟 ∈ [0,1] in crisp domain the economic rationale suggests that 𝐷 ̃(𝑥; 𝑟) and by the capital paper 𝐽̃(𝑥; 𝑟) such that given by rate of saving proportionate to 𝑈 ̃ (𝑥; 𝑟) = 𝛾(1 − 𝑏̃) 𝑈 ̃(𝑥; 𝑟) − 𝛿 𝐽̃(𝑥; 𝑟) + 𝜌 𝐷 where 𝛾 > 0, 𝛿 > 0 and 𝜌 is direction factor. Combining (6) and (7) to obtain the following 𝐵̃ (𝑥; 𝑟) = 1 𝑘 [ 𝐽̃(𝑥 + 𝑘; 𝑟) − 𝐽̃(𝑥; 𝑟)] (8) (9) From (6) and (9), we get ̃(𝑥; 𝑟) = 𝑈 1 𝑘(1− 𝑏̃ ) ̃ 𝐶 (𝑥;𝑟) [ 𝐽̃(𝑥 + 𝑘; 𝑟) − 𝐽̃(𝑥; 𝑟)] + ̃ 1− 𝑏 By combining (8)-(10), we can yield ̇ 𝑟) = 𝛾 𝐽̃′ (𝑥; 𝑟) − (𝛿 + 𝛾 ) 𝐽̃(𝑥 + 𝑘; 𝑟) + [𝛾𝐶̃ (𝑥; 𝑟) + 𝜌] 𝐽̃(𝑥; 𝑘 𝑘 (10) (11) Express acceptance rate new appointment information. It is a delayed-type template operational FDDE. 4. FDDES WITH PROPORTIONAL DELAY In this research, the Bezier control point method can finish off approximate analytical solutions with a high level of reliability. Consider the following neutral functional FDEE with proportional delays [21, 22], 𝑛 ̃ ̃ (𝑘) (𝑝𝑘 𝑥; 𝑟) + 𝑦̃(𝑥; 𝑟) (𝑣̃(𝑥; 𝑟) + 𝑎̃(𝑥; 𝑟)𝑣̃(𝑝𝑘 𝑥; 𝑟)) = 𝛽̃ 𝑣̃(𝑥; 𝑟) + ∑𝑛−1 𝑘=0 𝑏𝑘 (𝑥; 𝑟)𝑣 (12) with the fuzzy initial conditions ∑𝑛−1 ̃ (𝑘) (0; 𝑟) = 𝛿̃𝑖 (𝑟), 𝑘=0 𝑐̃𝑖𝑘 𝑣 𝑖 = 0, 1, … , 𝑛 − 1. (13) On solving fuzzy delay differential equation using bezier curves (Ali F. Jameel) 6524  ISSN: 2088-8708 Here, 𝑎̃(𝑥; 𝑟) and𝑏̃𝑘 (𝑥; 𝑟) (𝑘 = 0, 1, … , 𝑛 − 1) k are given analytical fuzzy functions, and 𝛽̃ , 𝑝̃𝑘 , 𝑐̃𝑖𝑘 , and 𝛿̌𝑖 denote given fuzzy constants with 0 < 𝑝𝑘 < 1, (𝑘 = 0, 1, … , 𝑛). The presence and singularity of the multi pantograph equation analytical solution is demonstrated [21], the solution Dirichlet sequence is constructed and the asymptotic stability of the analytical solution is sufficiently defined.. It is proved that 1 the θ-methods with a variable step size are asymptotically stable if < 𝜃 ≤ 1. There are several examples 2 that show the properties of the θ-methods. In order to apply the Bezier control point method, we rewrite (12) as 𝑛 𝑛 (𝑘) (𝑝𝑘 𝑥) + 𝑦(𝑥), 𝑥 ≥ 0. (𝑣̃(𝑥; 𝑟)) = 𝛽𝑣̃(𝑥; 𝑟) − (𝑎(𝑥)𝑣(𝑝𝑘 𝑥)) + ∑𝑛−1 𝑘=0 𝑏𝑘 (𝑥)𝑣 A particular class of crisp DDE represents neutral functional DEEs with proportional delays. The mathematical modeling of real-world phenomena takes such functioning DEEs on a significant role [12]. 5. BEZIER CURVES IN FUZZY DOMAIN From the definition of Bezier curve polynomial of 𝑚 degree [23] and according to sections 2-4, we have the following fuzzy analysis ̃ ̃ 𝑚 𝑥−𝑏1 ; 𝑟) , 𝑥 ∈ [𝑏1 , 𝑏2 ]. 𝐵̃ (𝑥; 𝑟) = ∑𝑚 𝑗=0 𝑃𝑗 𝐵𝑗 ( 𝑥−𝑏1 ; 𝑟) = 𝐵̃𝑗𝑚 ( 𝑏2 −𝑏1 𝑚! 𝑏2 −𝑏1 ( 𝑥−𝑏1 𝑗!(𝑚−𝑗)! 𝑏2 −𝑏1 𝑗 ; 𝑟) ( 𝑏2 −𝑥 𝑏2 −𝑏1 ; 𝑟) (14) 𝑚−𝑗 𝑃𝑗 is control points of Bezier coefficient and 𝐵̃𝑗𝑚 are the polynomial of Bernstein on interval [a1, a2] per each fuzzy level set 𝑟 ∈ [0,1], see Figure 1. In particular ̃𝑚 𝐵̃ (𝑥; 𝑟) = ∑𝑚 𝑗=0 𝑃𝑗 𝐵𝑗 (𝑥; 𝑟) , 𝑥 ∈ [0,1]. 𝐵̃𝑗𝑚 (𝑥; 𝑟) = 𝑚! 𝑗!(𝑚−𝑗)! (15) (𝑥; 𝑟) 𝑗 (1 − 𝑥; 𝑟)𝑚−𝑗 (16) where 𝐵̃ (𝑥; 𝑟) is a fuzzy parametric Bezier curve when it’s polynomial of vector valued. Figure 1 shows the comprise line segments with control polygon of a Bezier curve 𝐶𝑗 − 𝐶𝑗+1 , j = 0, 1, … , m − 1. If 𝐵̃ (𝑥; 𝑟) polynomial of a scalar valued, the function is call 𝑦̃ = 𝐵̃ (𝑥; 𝑟) then from [23, 24] an explicit Bezier curve denoted by (𝑥, 𝐵̃(𝑥; 𝑟)). P2 P1 P3 P4 P5 P0 Figure 1. Degree 5 bezier curve with control polygon 6. SOLUTION OF FDDE USING BEZIER CONTROL POINTS Consider the following boundary value problem 𝐿(𝑣̃(𝑥; 𝑟), 𝑣̃(𝑝0 𝑥; 𝑟), … , 𝑣̃(𝑝𝑛 𝑥; 𝑟))𝑣̃ (𝑛) (𝑥; 𝑟) − 𝛽𝑣̃(𝑥; 𝑟) + (𝑎(𝑥)𝑣̃(𝑝𝑛 𝑥; 𝑟)(𝑛) − ̃ ∑𝑛−1 ̃ (𝑘) (𝑝𝑘 𝑥; 𝑟) + 𝑦̃(𝑥; 𝑟) , , 𝑥 ≥ 0 𝑘=0 𝑏𝑘 (𝑥; 𝑟)𝑣 𝑑 𝑗 𝑣̃(0;𝑟) 𝑑𝑥 𝑗 = 𝑎𝑗 , 𝑑 𝑗 𝑣̃(1;𝑟) 𝑑𝑥 𝑗 = 𝛽𝑗 , 𝑗 = 0,1, … , 𝑛 − 1, Int J Elec & Comp Eng, Vol. 10, No. 6, December 2020 : 6521 - 6530 (17) (18) Int J Elec & Comp Eng ISSN: 2088-8708  6525 where 𝐿 is differential operator with proportional delay, is 𝑦̃(𝑥; 𝑟) also a polynomial in x, 0 < 𝑝𝑘 < 1 and (k = 0, 1, ···, m) [24, 25]. We propose to represent the approximate solution of eq. (18) 𝑣̃(𝑥; 𝑟) in fuzzy Bezier form. The preference between Bezier and B-Spline is that the Bezier form is easier to carry out multiplication, contrast and degree elevation operations symbolically than B-Spline. We choose the sum of squares of the Bezier control points of the residual to be the measure quantity. Minimizing this quantity gives the approximate solution. Therefore, the obvious spotlight is in the following, if the minimizing of the quantity is zero, so the residual function is zero, which implies that the solution is the exact solution. We call this approach the control point based method. By following [25] detailed steps of the method are as follows: Step 1. Choose a degree n and symbolically express the solution 𝑣̃(𝑥; 𝑟)in the degree m (𝑚 ≥ 𝑛) Bezier form 𝑣̃ = 𝑣̃(𝑥; 𝑟) = ∑𝑚 ̃𝑗 (𝑥; 𝑟)𝐵̃𝑗𝑚 (𝑥; 𝑟) 𝑗=0 𝛼 (19) where the control points are 𝛼0, 𝛼1 , … , 𝛼𝑚 to be de-termined. Step 2. Substituting the approximate solution 𝑣 = 𝑣(𝑥) into the (19), we obtain the residual function 𝑅̃ (𝑥; 𝑟) = 𝐿(𝑣̃(𝑥; 𝑟), ̃𝑣 (𝑝0 𝑥), ̃𝑣 (𝑝1 𝑥; 𝑟), … , ̃𝑣 (𝑝𝑛 𝑥; 𝑟)) − 𝑦̃(𝑥; 𝑟). This is a polynomial in 𝑥 with degree ≤ h, where ℎ = max{𝑚 − 𝑛 + deg (𝛼̃(𝑥; 𝑟), 𝑚 + deg (𝑏̃0 (𝑥; 𝑟)) , 𝑚 − 1 + deg (𝑏̃1 (𝑥; 𝑟)) , … , 𝑚 − 𝑛 + 1 + deg (𝑏̃𝑛−1 (𝑥; 𝑟)) , deg(𝑦̃(𝑥; 𝑟))}. So the residual function 𝑅̃ (𝑥; 𝑟) can be expressed in fuzzy Bezier form as well, 𝑅̃ = 𝑅̃(𝑥; 𝑟) = ∑ℎ𝑗=0 𝑏̃𝑗 𝐵̃𝑗ℎ (𝑥) (20) where for each fuzzy level set 𝑟 ∈ [0,1]the control points 𝑏̃0 , 𝑏̃1 , … , 𝑏̃ℎ are linear functions in the unknowns 𝛼̃𝑗 . These functions are derived using the operations of multiplication, degree elevation and differentiation for Bezier form. Step 3. Construct the objective function 𝐹̃ (𝑥; 𝑟) = ∑ℎ𝑗=0 𝑏̃𝑗2 (𝑥; 𝑟). Then 𝐹̃ (𝑥; 𝑟) is also a fuzzy function of 𝑎0 , 𝑎1 , … , 𝑎𝑚 . Step 4. Solve the constrained optimization problem: min 𝐹̃ (𝑥; 𝑟) = ∑ℎ𝑗=0 𝑏̃𝑗2 (𝑎0 , 𝑎1 , … , 𝑎𝑚 ; 𝑟), 𝑑 𝑗 𝑣̃(0;𝑟) 𝑑𝑥 𝑗 = 𝑎𝑗 , 𝑑 𝑗 𝑣̃(1;𝑟) 𝑑𝑥 𝑗 = 𝛽̃𝑗 ,𝑗 = 0,1, … , 𝑛 − 1, (21) by some optimization techniques, such as Lagrange multipliers method, we can be used to solve (21). Step 5. Substituting the minimum solution back into (19) arrives at the approximate solution to the differential equation. 7. NUMERICAL EXAMPLE In this part, we used the mentioned control-point-based method on Bezier control points to solve DDE’s and system of DDE’s. As a practical example, we consider Evens and Raslan [6] the following pantograph delay equation in fuzzy form: 1 𝑥 𝑥 1 𝑢̃′ (𝑥) = exp ( ) 𝑢̃ ( ) + 𝑢̃(𝑥), 0 ≤ 𝑥 ≤ 1, 2 2 𝑢̃(0) = [𝑟. 2 − 𝑟]. 2 2 (22) On solving fuzzy delay differential equation using bezier curves (Ali F. Jameel) 6526  ISSN: 2088-8708 The exact solution is given by u( x; r )  re x , u( x; r )  (2  r )e x . According to [5] (21) can be written in defuzzfication form 1 𝑥 𝑥 1 𝑥 𝑥 1 𝑢′ (𝑥; 𝑟) = exp ( ; 𝑟) 𝑢 ( ; 𝑟) + 𝑢(𝑥; 𝑟), { ′ 2 1 2 2 𝑢(0; 𝑟) = 𝑟 2 𝑢 (𝑥; 𝑟) = exp ( ; 𝑟) 𝑢 ( ; 𝑟) + 𝑢(𝑥; 𝑟), 2 2 2 𝑢(0; 𝑟) = 2 − 𝑟 (23) 2 For numerical implementation, we consider the approximate solution using Bezier curves of degree 3 (m=3) and 8 (m=8) respectively as given in (15). In order to obtain the residual function, we also approximate ex in Taylor polynomial of order 6. The detail results are as follows. 7.1. Degree-3 Bezier curve Let, 3  a i Bi3 ( x) u ( x; r )   i 1 ,  3 u ( x; r )  3 a i Bi ( x )  i 1   (24)  where 0  x  1 and a i and a i , i  0, ..., 3 are the Bezier control points need to be determined. Substitute into (23) and the residual functions can be obtained, i.e. 3  1 d 3 x 3 x 3  R( x; r )  ( a i Bi ( x))  exp( ; r )( a i Bi ( )) 2 2 2 dx i 1  i 1  3    a i Bi3 ( x))  i 1  3 1 x x d 3  3 exp( ; )( ( ; ) ( ( ) ) a B x r ai Bi3 ( )) R x r   i i    2 2 2 dx i 1 i 1  3    ai Bi3 ( x))   i 1 (25) The right-hand side of (25) is a polynomial of degree 8 and therefore the residual function can be represented in the form of (20) with ℎ = 8 as follows.   R( x; r )     R ( x; r )    8  b i Bi8 ( x) i 0 8  (26) bi Bi8 ( x) i 0 To obtain the Bezier control points in (24), we follow the step 3 to step 5 as stated in section 6. The approximate solutions are available in Tables 1 and 2 and the comparsion of degree 3 bezier curve solution with exact solution of equation (22) is illustrated in Figure 2 such that: u ( x; r )  r [ (1  x) 3  4.0126515 (1  x) 2 x  5.4528665 (1  x) x 2  2.71821024 x 3 )] u ( x; r )  (2  r )(1  x) 3  (8.025303  4.0126516 r )(1  x) 2 x  (1.090573311  5.4528665 r )(1  x) x 2  (4.43642047  2.71821024 r ) x 3 )] Int J Elec & Comp Eng, Vol. 10, No. 6, December 2020 : 6521 - 6530 Int J Elec & Comp Eng Table 1. Appoximate and exact values for lower solution, u ( x; r ) (degree 3 bezier curve) r 0 0.2 0.4 0.6 0.8 1.0 approx 0 0.5436420472 1.0872840944 1.6309261416 2.1745681888 2.718210236  ISSN: 2088-8708 exact 0 0.5436563657 1.0873127314 1.6309690971 2.1746254628 2.7182818285 6527 Table 2. Approximate and exact values for upper solution, u ( x; r ) (degree 3 bezier curve) abs. error 0 1.4318499128x 10-5 2.8636998257x10-5 4.2955497386x 10-5 5.7273996514x 10-5 7.1592495642x 10-5 r 0 0.2 0.4 0.6 0.8 1.0 approx 5.436420472 4.892778425 4.349136378 3.805494330 3.261852283 2 .71821024 exact 5.4365636569 4.8929072912 4.3492509255 3.8055945598 3.2619381942 2.7182818285 abs. error 1.4318499128 x 10-4 1.2886649216x 10-4 1.1454799303 x 10-4 1.002294939 x 10-4 8.591099477 x 10-5 7.159249564 x 10-5 FDDE using Bezier at t = 1 1 approximate exact 0.9 0.8 0.7 r 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 u(r;1) 4 5 6 Figure 2. Approximate and exact solution of 𝑢(𝑡) at t = 1 (degree 3 bezier curves) 7.2. Degree-8 Bezier curve Let,  u ( x; r )    u ( x; r )    8  a i Bi8 ( x) i 0 8  a i Bi3 ( x) , (27) i 0 where 0  x  1 and a i and ai , i  0, ...,8 are the Bezier control points need to be determined. Substitute into (23) and the residual functions can be obtained, i.e. 8  x d 8 1 3 8 x  R ( x; r )  ( a i Bi ( x))  exp( ; r )( a i Bi ( )) dx 2 2 2  i 0 i 0  8    a i Bi8 ( x))  i 0  8 x d 8 1  8 x 8  R ( x; r )  dx ( a i Bi ( x))  2 exp( 2 ; r )( a i Bi ( 2 )) i 0 i 0  8    a i Bi8 ( x))  i 0  (28) The right-hand side of (25) is a polynomial of degree 13 and therefore the residual function can be represented in the form of (20) with ℎ = 13 as follows. On solving fuzzy delay differential equation using bezier curves (Ali F. Jameel) 6528  ISSN: 2088-8708 13  13  R( x; r )   b i Bi ( x)  i 0  13  R ( x; r )  bi Bi13 ( x)   i 0  (29) To obtain the Bezier control points in (27), we also use the step 3 to step 5 as stated in section 6. The approximate solutions are in Tables 3 and 4 and the comparsion of degree 8 bezier curve solution with exact solution of equation (22) is illustrated n Figure 3 such that: u ( x; r )  r[(1  t ) 8  9t (1  t ) 7  35.5000002 t 2 (1  t ) 6  80.16666616 t 3 (1  t ) 5  113 .3750016 t 4 (1  t ) 4  102 .84166368 t 5 (1  t ) 3  58.44305288 t 6 (1  t ) 2  19.0279716 t 7 (1  t )  2.7182791t 8 ] u ( x; r )  (2  r )(1  t ) 8  (1.8  9r )t (1  t ) 7  (71.00000012  3.5.5000002 r )t 2 (1  t ) 6  (160.33333232  80.16666616 r )t 3 (1  t ) 5  (226.7500039  113.3750016 r )t 4 (1  t ) 4  (205.68332736  102.84166368 r )t 5 (1  t ) 3  (116.8861176  58.4430588 r )t 6 (1  t ) 2  (38.0559432  19.0279716 r )t 7 (1  t )  (5.4365582  2.7182791r )t 8 Table 3. Approximate and exact values for lower solution, u ( x; r ) (degree 8 bezier curve) r 0 0.2 0.4 0.6 0.8 1.0 approx 0 0.543655820 1.087311648 1.630967461 2.174623282 2.718279102 exact 0 0.5436563657 1.0873127314 1.6309690971 2.1746254628 2.7182818285 abs. error 0 5.45269859x10-7 1.090539719x 10-6 1.635809578x 10-6 2.181079437x 10-6 2.726349297x 10-6 Table 4. Approximate and exact values for upper solution, u ( x; r ) (degree 8 bezier curve) r 0 0.2 0.4 0.6 0.8 1.0 approx 5.4365582042 4.8929023838 4.3492465634 3.8055907430 3.2619349225 2.7182791021 exact 5.4365636569 4.8929072912 4.3492509255 3.8055945598 3.2619381941 2.7182818284 abs. error 5.452698593 x10-6 4.907428734 x10-6 4.362158874 x10-6 3.816889015 x10-6 3.271619156 x10-6 2.726349297 x10-6 FDDE using Bezier at t = 1 1 approximate exact 0.9 0.8 0.7 r 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 u(r;1) 4 5 6 Figure 3. Approximate and exact solution of u(t) at t = 1 (degree 8 bezier curves) Int J Elec & Comp Eng, Vol. 10, No. 6, December 2020 : 6521 - 6530 Int J Elec & Comp Eng ISSN: 2088-8708  6529 8. CONCLUSION This work has successfully implemented and applied Bezier control points to overcome linear and fuzzy DDEs. A general method framework has been successfully developed and evaluated using fuzzy sets properties to obtain rough solutions for fuzzy DDEs. Details have been provided regarding the BCP convergence mechanism related to the approximate first-order fuzzy DDEs solution. Studies of first-order linear fizzy DDEs by BCP have shown that the system is capable and reliable studies are obtained that match the properties of the solution of the fizzy differential equation in the form of the triangle fuzzy numbers with varying degrees of precision. 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