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Journal of Applied Mathematics

Volume 2012, Article ID 325473, 17 pages

http://dx.doi.org/10.1155/2012/325473
Research Article

Formulation and Solution of th-Order Derivative Fuzzy Integrodifferential Equation Using New Iterative Method with a Reliable Algorithm

Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt

Received 11 July 2012; Revised 9 September 2012; Accepted 10 September 2012

Academic Editor: Fazlollah Soleymani

Copyright © 2012 A. A. Hemeda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The nth-order derivative fuzzy integro-differential equation in parametric form is converted to its crisp form, and then the new iterative method with a reliable algorithm is used to obtain an approximate solution for this crisp form. The analysis is accompanied by numerical examples which confirm efficiency and power of this method in solving fuzzy integro-differential equations.

1. Introduction

In 1975 Zadeh and then Dubois and Prade [1] introduced fuzzy numbers and fuzzy arithmetic. This concept propagated widely to various problems, for example, fuzzy linear systems [2, 3], fuzzy differential equations [4, 5], fuzzy integral equations [68], and fuzzy integro-differential equations [912]. Additional topics can be found in [13, 14]. Recently, several numerical methods were suggested to solve integro-differential equations, for example, Sine-Cosine wavelet used by Tavassoli Kajani et al. to obtain a solution of linear integro-differential equations [15] and variational iteration method used by Abbasbandy and Hashemi to formulate and solve fuzzy integro-differential equation [12], by Saberi-Nadjafi and Tamamgar for solving system of integro-differential equations [16], and by Shang and Han for solving th-order integro-differential equations [17]. Some other worthwhile works can be found in [18, 19]. Recently, Daftardar-Gejji and Jafari [20, 21] proposed the new iterative method. This method has proven useful for solving a variety of linear and nonlinear equations such as algebraic equations, integral equations, ordinary and partial differential equations of integer and fractional order, and system of equations as well. The new iterative method is simple to understand and easy to implement using computer packages and yields better results [21] than the existing Adomian decomposition method [22], homotopy perturbation method [23], or variational iteration method [24]. For more details, see [2540].

In the present work we apply the new iterative method with a reliable algorithm to solve the th-order derivative fuzzy integro-differential equation in its crisp form.

2. Preliminaries

In this section we set up the basic definitions of fuzzy numbers and fuzzy functions.

Definition 2.1 (see [2, 3, 12]). A fuzzy number in parametric form is an ordered pair of functions , which satisfy the following requirements:(1) is a bounded left continuous nondecreasing function over ,(2) is a bounded left continuous nonincreasing function over ,(3) , .

Remark 2.2 (see [6, 12]). Let , be a fuzzy number, and we can take It is clear that , and , and also a fuzzy number is said symmetric if is independent of for all .

Let for each partition of and for arbitrary , suppose that and , . The definite integral of over is provided that this limit exists in the metric . If the fuzzy function is continuous in the metric , its definite integral exists. Also we have and .

Definition 2.3 (see [33]). Let and . We say that is Hukuhara differentiable at if there exists an element , such that for all sufficiently small, and where is Hukuhara difference, is the set of fuzzy numbers, and the limit is in the metric .

In parametric form, if , then , where are Hukuhara differentiable of , respectively.

From (2.2), the th-order Hukuhara differentiable, , of at can be defined as in the following definition.

Definition 2.4. Let and where is th-orderHukuhara differentiable of at for all . We say that is th-order Hukuhara differentiable at if there exists an element such that for all sufficiently small, , and

. In parametric form, as above, if , then , where are th-order Hukuhara differentiable of , respectively. From (2.3a), in case , we obtain (2.2).

3. Fuzzy Integro-Differential Equation

Consider the th-order derivative integro-differential equation [12] as follows: where , , , , , is a known function and the kernel with the initial conditions as follows:

in the fuzzy case; that is, and be fuzzy functions. Let where all derivatives are, with respect to , fuzzy functions. Therefore, related fuzzy integro-differential equation of (3.1a) can be written as follows:

and its two crisp equations can be written as follows:

4. New Iterative Method

For simplicity, we present a review of the new iterative method [20, 21, 3439, 41], and then we introduce a suitable algorithm of this method for solving the th-order derivative fuzzy integro-differential equations.

Consider the following general functional equation: where is a nonlinear operator from a Banach space , and is a known function.

We are looking for a solution of (4.1) having the series form: The nonlinear operator can be decomposed as follows: From (4.2) and (4.3), (4.1) is equivalent to We define the recurrence relation: Then The -term approximate solution of (4.1) is given by .

4.1. Reliable Algorithm

After the above presentation of the new iterative method, we introduce a reliable algorithm of this method for solving any th-order derivative fuzzy integro-differential equation. Consider the th-order derivative integro-differential equation (3.1a) and (3.1b) defined as with the following initial conditions:

The initial value problem (4.7a) and (4.7b) is equivalent to the following integral equation: where is the solution of the th-order differential equation:

and is an integral operator of order . Also, the two crisp equations (3.4a) and (3.4b) are equivalent to the two integral equations: where and are the solutions of the th-order differential equations: and are the following two integral equations:

We get the solution of (4.8) or the two (4.10a), (4.10b) by employing the recurrence relation (4.5).

4.2. Convergence of the New Iterative Method

Now, we introduce the condition of convergence of the new iterative method, which is proposed by Daftardar-Gejji and Jafari in (2006) [20], also called (DJM) [39]. From (4.3), the nonlinear operator is decomposed as follows [39]: Let and Then .

Set Then is a solution of the general functional equation (4.1). Also, the recurrence relation (4.5) becomes Using Taylor series expansion for , , we have In general: In the following theorem we state and prove the condition of convergence of the method.

Theorem 4.1. If N is in a neighborhood of and for any and for some real and , , then the series is absolutely convergent, and, moreover,

Proof. In view of (4.18), Thus, the series is dominated by the convergent series , where . Hence, is absolutely convergent, due to the comparison test.

For more details, see [39].

5. Numerical Examples

Example 5.1. Consider the following fuzzy integro-differential equation: where is the exact solution for (5.1), , , , and , with the initial conditions , . Taking , . The exact solution for related crisp equations is , . At first, we identify and . From (3.4a) and (3.4b), (5.1), where , we have

From (4.10c), (4.10d), we obtain Therefore, from (4.10a), (4.10b) the fuzzy integro-differential equations (5.2a) and (5.2b) are equivalent to the following integral equations: Let . Therefore, from (4.5), we can obtain easily the following first few components of the new iterative solution for (5.1): and so on. The 6-term approximate solution is It is clear that the iterations converge to the exact solution of the two crisp equations as the number of iteration converges to , that  is, , and . From the relations between , and in Section 2, it follows immediately that this solution is the solution of the fuzzy integro-differential equation (5.1).

Example 5.2. Consider the following fuzzy integro-differential equation: In this example is the exact solution for (5.7), , , , and the initial conditions are , , , . Also, the exact solution for related crisp equations is , with , . At first, we identify , . From (3.4a) and (3.4b), (5.7), where , we have

From (4.10c), (4.10d), we obtain Therefore, from (4.10a), (4.10b), the fuzzy integro-differential equations (5.8a) and (5.8b) are equivalent to the following integral equations: Let , . Therefore, from (4.5), we can obtain easily the following first few components of the new iterative solution for (5.7): and so on. The 6-term approximate solution is It is clear that the iterations converge to the exact solution of as , that is, , . As the above example the solution of (5.7) follows immediately.

Example 5.3. The third example is the fuzzy integro-differential equation: where is the exact solution for (5.13), , , , and the initial conditions are , , , . The exact solution for related crisp equations is , with , . At first, we identify , , and from (3.4a) and (3.4b), (5.13), where , we have

From (4.10c), (4.10d), we obtain Therefore, from (4.10a) and (4.10b), the fuzzy integro-differential equations (5.14a) and (5.14b) are equivalent to the integral equations: Let , . Therefore, from (4.5), we can obtain easily the following first few components of the new iterative solution for (5.13): and so on. The 6-term approximate solution is It is clear that the iterations converge to the exact solution of as , that is, , . Therefore the solution of (5.13) follows immediately.

Example 5.4. Let us consider the following fuzzy integro-differential equation: with the exact solution , , , , and the initial conditions , , , . The exact solution for related crisp equations is , with , . From (3.4a), (3.4b), and (5.19), where , the two crisp equations are

From (4.10c), (4.10d) we obtain Therefore, from (4.10a), (4.10b), the fuzzy integro-differential equations (5.20a) and (5.20b) are equivalent to the following integral equations: Let Therefore, from (4.5), we can obtain the following first approximate solutions for (5.19):

and so on. In the same manner the rest of components can be obtained. The 5-term approximate solution is

Now, let us define the absolute value of the th-term error by , where is the exact solution and is the th-term approximate solution respectively. It is clear from previous above solutions that the smallest value of the absolute th-term error is at , that is, , and the largest value is at as shown in Table 1. Therefore the iterations converge to the exact solution of , that is, , , and the solution of (5.19) follows immediately.

tab1
Table 1

Example 5.5. The final example is the following fuzzy integro-differential equation: with the exact solution , , , , and the initial values , , , . The exact solution for related crisp equations is , with , . As above, where , the two crisp equations are

As the above examples, we obtain Therefore, the fuzzy integro-differential equations (5.27a) and (5.27b) are equivalent to the following integral equations: Let Therefore, we obtain

and so on. The 5-term approximate solution is

As the previous example, it is clear from the obtained results that the smallest value of the absolute th-term error is , , and the largest value is at as shown in Table 2. Therefore, the iterations converge to the exact solution of , that is, , . Therefore, the solution of (5.26) follows immediately.

tab2
Table 2

6. Conclusion

In this work, the new iterative method used with a reliable algorithm to solve the th-order derivative fuzzy integro-differential equations in crisp form and the cases of first- and second-order derivatives are taken into account. The obtained results concluded that the approximate solutions are in high agreement with corresponding exact solutions, which means that this method is suitable and effective to solve fuzzy integro-differential equations. Moreover, the solutions of the higher order fuzzy integro-differential equations can be calculated, as a future prospects, in a similar manner.

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