Solving Fuzzy Differential Equations by Using Power Series

In this paper, the series solution is applied to solve third order fuzzy differential equations with a fuzzy initial value. The proposed method applies Taylor expansion in solving the system and the approximate solution of the problem which is calculated in the form of a rapid convergent series; some definitions and theorems are reviewed as a basis in solving fuzzy differential equations. An example is applied to illustrate the proposed technical accuracy. Also, a comparison between the obtained results is made, in addition to the application of the crisp solution, when the-level equals one.

In this paper, the RPS method was employed to solve third-order fuzzy differential equations. The approximate solution is represented in the form of power series. Moreover, the approximate solution and all its derivatives (if they exist) converge to the exact solution and all its derivatives, respectively. The suggested algorithm created a swiftly convergent series with an easily computable components using a symbolical calculation program. Series expansions are used in numerical calculations, especially for quick estimates that are made manually. Often, we express FDEs in terms of serial expansions. However, the RPS theory is an analytical method for solving different types of ordinary and partial differential equations [7]. The classical higher order, i.e. Taylor series method, is computationally expensive for large orders and proper for the linear problems. On the other hand, the suggested method is an alternate procedure for earning analytic Taylor series solutions of systems of FIVPs. The purpose of this paper is to develop the implementation of the residual power series method for earning an analytical solution for the first-order fuzzy DE of the following form [3]: ( ) ( ( )) ( ) ( ) and the third-order differential equations in the following form: -× → and , h : , -× → are fuzzy-number-valued functions, ( ) is an unknown function of variable (x) to be specified, are fuzzy numbers, and( ) , (a ) are real constants with a>0. The structure of the paper includes the following: In section 2, we provide some important definitions and basic results to be used in this paper. In section 3, we present the theory of fuzzy DEs. In section 4, the main idea of the Residual power series method is introduced. In section 5, we illustrate the proposed method by a numerical example.

Preliminaries
In this section, we present basic concepts for fuzzy calculus and concept of fuzzy derivative; we will adopt strongly generalized differentiability. Definition 2.1. [8]. A fuzzy number is a fuzzy set: R → [0, 1] which satisfies the following requirements: We will let denotes the set of fuzzy numbers on R. Obviously, R ⊂ , where R is understood as R Rχ = {χ * + : x ∈ R } ⊂ . The -level represents a fuzzy number , denoted by , -, is defined as: for n, m = 1, 2. The principle of the derivative properties is known and can be found in previous articles [ 11,12,13]. In this paper, we extend the theorem proved in two of those articles [11,12] then are .differentiable functions". and

3-Theory of third-order fuzzy differential equations
In this section, we study the theory of third-order fuzzy differential equations under strongly generalized differentiability. Furthermore, we present an algorithm to solve these types of problems, which consists of eight classical ODEs systems for fuzzy DE (1. The following definition of ( ( )) is a conclusion to the extension principle of Zadeh when ( ) is a fuzzy number [4]: ( ( ))( ) * ( )( ) ( ) ∈ + . Thus, according to the theory of Nguyen [14,15], it follows that: , where the two expression endpoint functions and are defined, respectively, as: Similarly, taking into account the type of differentiability, we can write The objective of the next algorithm is to implement a procedure to solve the fuzzy DE (1.1) in a parametric form, in terms of its -levels representation.

Algorithm 3.1[16]:
To find the solutions of fuzzy DE (1.1), we discuss the following two cases: Let be an ( ) -solution for fuzzy DE (1.2). To find it, we apply Theorem 2.2 and, considering the initial values, we can transform fuzzy DE (1.2) to a system of third-order ODEs. Therefore, the possible ODEs systems for this type of fuzzy problems are eight, as follows:

Algorithm 3.2:
To find the solutions of the fuzzy differential equation (1.2) in term of its -level representation, we consider the following cases: Case I. For (1, 1)-differentiable, consider the differentiability of y and ( ) in the sense (i) , (iii) and (v) of Theorem 2.2. Then we get the following (1, 1)-system of ODEs: in the sense (i) , (iv) and (vi) of Theorem 2.2. Then we get the following (1, 2)-system of ODEs: ( ) Case III. For (1, 1)-differentiable, consider the differentiability of y and ( ) in the sense (i), (iii) and (vi) of Theorem 2.2. Then we get the following (1, 1)-system of ODEs: in the sense (i), (iv) and (v) of Theorem 2.2. Then we get the following (1, 2)-system of ODEs: For (2, 1)-differentiable, consider the differentiability of y and ( ) in the sense (ii) , (vii) and (ix) of Theorem 2.2. Then we get the following (2,1)-system of ODEs: Case VI. For (2, 2)-differentiable, consider the differentiability of y and ( ) in the sense (ii) ,(viii) and (x) of Theorem 2.2. Then we get the following (2,2)-system of ODEs: subject to the initial conditions Case VII. For (2, 1)-differentiable, consider the differentiability of y and ( ) in the sense (ii) , (viii) and (x) of Theorem 2.2. Then we get the following (2,3)-system of ODEs: subject to the initial conditions Subject to the initial conditions

4-The Residual Power Series Method for Solving Fuzzy Deferential Equations
In this section, to get series solutions for systems of Initial Value Problems with initial conditions, we use our theorem of the RPS. At first, we interpreted and analyzed the RPS theorem for solving IVPs for fuzzy DE(1.1) with respect to (1) before applying the RPS method for solving the system of ODEs (3.1) and (3.2), we define the residual functions for this system as follows: ] and ∈ [0, 1]. In order to approximate the solution, substitute the expansion of ( ) and ( ) in Eq. (4.2) to get To obtain the first approximate solution put in Eq. 2) can be written as follows: And to find the second approximation, we differentiate Eq. (4.3) respect to( x ), and using (x 0 ) (x 0 ) = 0 to obtain the following results The second approximation of ODEs (3.1) and (3.2) using 2th-truncated series, can be written as:

Ibraheem
Iraqi Journal of Science, 2020, Special Issue, pp: 92-107 Similarly, to find the third approximation, we differentiate of Eq. (4.3) with respect to( x ), and using the fact ( ) ( ) = 0 to obtain the following results The third approximation for the system of ODEs (3.1) and (3.2) using 3th-truncated series is as follows : In order to find the k-th approximate solution for system of ODEs (3.1) and (3.2), it is enough to substitute the k-th -truncated series ( ) and ( ) instead of the expansion of ( ) and ( ), respectively, into the residual Eq. (4.3), and then apply the same procedure since it is easy to show that ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )for each s ≤ k.
The following theorem is an extension of theorem shown in a previous work [14], which shows the convergence of the RPS method in the sense of (1)differentiability. Theorem 4.1. Suppose that y (x) and y (x) are the exact solutions of ODEs (3.1) and (3.2) in the sense of (1)-differentiability. Then, the approximate solution obtained by the Residual Power Series method is just the Taylor expansion of that y (x) and y (x) Proof:-Suppose that the approximate solution of ODEs (3.1) and (3.2) be as follows: To prove the theory, we show that the coefficients c n and d n in Eq. (4.7) be as follows: c n ( ) , and for n=1 , substitute x x 0 into Eq.(3.1) to obtain f 1 (x 0 y (x 0 ) y (x 0 )) y (x 0 ) , and f 2 (x 0 y (x 0 ) y (x 0 )) y (x 0 ). On the other hand, from Eq. (3.1), we can write by substituting Eq. (4.9) in Eq. (3.1) and then putting x x 0 , we get Further, for 2 , by differentiating Eq. (3.1) with respect to x, we can get by substituting x x 0 in Eq. (4.11), we can obtain According to Eqs. (4.9) and (4.10), we can create the approximation system of ODEs (3.1) and (3.2) as follows: By substituting Eq. (4.13) in Eq. (4.11) and putting x x 0 , we can obtain: by comparing Eqs. (4.12) and (4.14), we can get c 2 ( ) Similarly, for 3 , by differentiating Eq. (3.1) with respect to x, we get According to Eqs. (4.9), (4.10) and (4.15), we can create the approximation system of ODEs (3.1) and (3.2) as follows: by comparing Eqs. (4.18) and (4.16) and putting x x 0 , we can conclude that c 3 ( ) Corollary 4.1. [16]. If either y (x) or y (x) is a polynomial, then the Residual Power Series method will access the exact solution.

CONCLUSIONS
From the present study, we may conclude the following: 1. The accuracy of the results may be checked with = 1, in which the upper and lower solutions must be equal. 2. The crisp solution or the solution of the nonfuzzy boundary value problem is obtained from the fuzzy solution by setting = 1, and, therefore, the fuzzy boundary value problems may be considered as a generalization to the nonfuzzy boundary value problems. 3. The Residual Power Series Method proved its validity and accuracy in solving Fuzzy Deferential Equations.