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Xiaobin Guo, Dequan Shang, "Approximate Solution of th-Order Fuzzy Linear Differential Equations", Mathematical Problems in Engineering, vol. 2013, Article ID 406240, 12 pages, 2013. https://doi.org/10.1155/2013/406240
Approximate Solution of th-Order Fuzzy Linear Differential Equations
The approximate solution of th-order fuzzy linear differential equations in which coefficient functions maintain the sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results.
Fuzzy differential equations (FDEs), which are utilized for the purpose of the modeling problems in science and engineering, have been studied by many researchers. Most practical problems require the solutions of fuzzy differential equations (FDEs) which are satisfied with fuzzy initial conditions, and therefore a fuzzy initial problem occurs and needs to be solved. However, most fuzzy initial value problems could not be solved exactly. So it is necessary to consider their approximating methods.
Prior to discussing fuzzy differential equations and their associated numerical algorithms, it is necessary to present an appropriate brief introduction to derivative of the fuzzy-valued function. The concept of a fuzzy derivative was first introduced by Chang and Zadeh , and it was followed up by Dubois and Prade  who used the extension principle in their approach. Other fuzzy derivative concepts have been proposed by Puri and Ralescu  and Goetschel Jr. and Voxman  as an extension of the Hukuhara derivative of multivalued functions. In recent years, many works have been produced in the aspects of theories and applications on fuzzy differential equations; see [5–15]. The notation of fuzzy differential equation was initially introduced by Kandel and Byatt [16, 17] who later applied the concept of fuzzy differential equation to the analysis of fuzzy dynamical problems [18, 19]. A thorough theoretical research of fuzzy Cauchy problems was given by Kaleva [20, 21], Seikkala , Ouyang and Wu , Kloeden , and Wu . A generalization of fuzzy differential equation was given by Aubin [26, 27], Baĭdosov , Leland , and Colombo and Kivan . Some numerical methods for solving fuzzy differential equations were introduced in [30–33].
For an th-order linear differential equation with fuzzy initial conditions where are continuous functions, Buckley and Feuring  presented two analytical methods for solving them. The first was to fuzzify the crisp solution and then check to see if it satisfies the fuzzy differential equations with fuzzy initial conditions. The second method was the reverse of the first one, in that they firstly solved the fuzzy initial value problem and then checked to see if it defines a fuzzy function. In 2008, Allahviranloo et al.  utilized the collocation method to transfer (1) and (2) into a crisp system of linear equations for three specious cases; that is, all coefficient functions are positive, negative, , are negative are positive, respectively. By computing (3), they obtained the approximate solution of (1) and (2). However, their methods will be restricted for the general case.
In this paper, the th-order linear differential equation with fuzzy initial conditions is further investigated. It shows that the result obtained in this paper is an extension of Allahviranloo's conclusions. In addition, considering that the case of the order of the fuzzy differential equation and the number of basic functions in assumed solution are not always equal, we obtain the approximate solution of the original equations (1) and (2) by calculating the minimal norm least squares solution of crisp system of linear equations. Three illustrating examples are given, and one of them is compared with Allahviranloo's work and is shown be more accurate. The structure of this paper is organized as follows.
In Section 2, we recall some basic definitions and results about fuzzy numbers and the undetermined fuzzy coefficients method. In Section 3, a class of th-order fuzzy linear differential equations is investigated by converting it to a crisp system of linear equations, and some corollaries for the special cases are given. The proposed algorithms are illustrated by solving some examples in Section 4, and the conclusion is drawn in Section 5.
2.1. Fuzzy Number
Definition 1 (see ). A fuzzy number is a fuzzy set like which satisfies the following.(1)is upper semicontinuous.(2) is fuzzy convex; that is, for all .(3) is normal; that is, there exists such that .(4)Supp is the support of , and its closure cl(supp ) is compact.
Let be the set of all fuzzy numbers on .
Definition 2 (see ). A fuzzy number in parametric form is a pair of functions , , , which satisfies the following requirements.(1) is a bounded monotonic increasing left continuous function.(2) is a bounded monotonic decreasing left continuous function.(3), .
For arbitrary fuzzy number , , and real number ,(1) if and only if and ;(2);(3);(4)
Definition 3 (see ). For arbitrary , the quantity is the distance between fuzzy numbers and .
2.2. The Undetermined Fuzzy Coefficients Method
The undetermined fuzzy coefficients method is to seek an approximate solution as where , are positive basic functions whose all differentiations are positive. We compute the fuzzy coefficients in (6) by setting the error to zero as follows:
Lemma 4 (see ). Let basic functions , and all of their differentiations be positive; without loss of generality, then and .
3. Basic Results
3.1. New Models
Theorem 5. Suppose that each , in (1) is either nonnegative or negative over ; then, the th-order fuzzy linear differential equations
can be extended into a system of linear equations
By setting , and by solving (10), the fuzzy approximate solution of the original fuzzy linear differential equations is obtained as follows:
Proof. Let (5) be substituted in (1) and (2); we have
We express the pervious equations in parametric forms; then,
thus we have the corresponding systems as follows: being linear systems.
By setting , and by solving (10), we obtain the values of parameters therefore, we get the fuzzy approximate solution of the original fuzzy equation as follows:
Now, we consider another special case; that is, coefficient functions , are positive and negative alternately.
Corollary 6. Suppose that the coefficients functions are positive and , are negative, and is an odd number. Then, the th-order fuzzy linear differential equations (1) and (2) can be extended into the following linear equations:
(19) is a system of linear equations such that where
When is an even number, then the th-order fuzzy linear differential equations (1) and (2) can be extended into a system of linear equations where
Three special cases in Allahviranloo et al.’s paper  are viewed as corollaries from Theorem 5.
Corollary 9. Suppose that coefficients functions are nonnegative and are negative; then, the th-order fuzzy linear differential equations (1) and (2) can be extended into a system of linear equations where
3.2. Method to Solve Linear Equations
In the process of solving (32) by setting , whether it is consistent or inconsistent, we obtain the minimal norm least squares solution  by using the generalized inverse of the coefficient matrix ; that is,
Thus, we get
Therefore, we obtain the fuzzy approximate solution of the original fuzzy equation as follows:
4. Numerical Examples
Example 1. Consider the following second-order fuzzy linear differential equation:
The exact solution of equation is
Let ; then,
From (10), the extended linear equation is as follows:
By setting , the parameters are obtained, and by putting them into (38), we have
Example 2. Consider the following three-order fuzzy linear differential equation:
The exact solution of equation is
Let ; then,
From (19), the extended linear equation is as follows:
By setting , the parameters are obtained, and by putting them into (43), we have