Research Article | Open Access

# A Numerical Scheme to Solve Fuzzy Linear Volterra Integral Equations System

**Academic Editor:**Said Abbasbandy

#### Abstract

The current research attempts to offer a new method for solving fuzzy linear Volterra integral equations system. This method converts the given fuzzy system into a linear system in crisp case by using the Taylor expansion method. Now the solution of this system yields the unknown Taylor coefficients of the solution functions. The proposed method is illustrated by an example and also results are compared with the exact solution by using computer simulations.

#### 1. Introduction

Many mathematical formulations of physical phenomena contain integral equations. These equations appear in physics, biological models, and engineering. Since these equations are usually difficult to solve explicitly, so it is required to obtain approximate solutions. In recent years, numerous methods have been proposed for solving integral equations. For example, Tricomi, in his book [1], introduced the classical method of successive approximations for nonlinear integral equations. Variational iteration method [2] and Adomian decomposition method [3] were effective and convenient for solving integral equations. Also the Homotopy analysis method (HAM) was proposed by Liao [4] and then has been applied in [5]. Moreover, some different valid methods for solving this kind of equations have been developed. First time, Taylor’s expansion approach was presented for solution of integral equations by Kanwal and Liu in [6] and then has been extended in [7–9]. In addition, Babolian et al. [10] by using the orthogonal triangular basis functions solved some integral equation systems. Jafari et al. [11] applied Legendre’s wavelets method to find numerical solution system of linear integral equations. Also Sorkun and Yalçinbaş [12] approximated a solution of linear Volterra integral equations system with the help of Taylor’s series.

In this paper, we want to propose a new numerical approach to approximate the solution of a fuzzy linear Volterra integral equations system. This method converts the given fuzzy system that supposedly has a unique fuzzy solution, into crisp linear system. For this scope, first, the Taylor expansions of unknown functions are substituted in parametric form of the given fuzzy system. Then we differentiate both sides of the resulting integral equations of the system times and also approximate the Taylor expansion by a suitable truncation limit. This work yields a linear system in crisp case, so the solution of the linear system yields the unknown Taylor coefficients of the solution functions. An interesting feature of this method is that we can get an approximate of the Taylor expansion in arbitrary point to any desired degree of accuracy. Here is an outline of the paper. In Section 2, the basic notations and definitions of the integral equation and the Taylor polynomial method are briefly presented. Section 3 describes how to find an approximate solution of the given fuzzy Volterra integral equations system with using proposed approach. Finally in Section 4, we apply the proposed method by an example to show the simplicity and efficiency of the method.

#### 2. Preliminaries

In this section, the most basic used notations in fuzzy calculus and integral equations are briefly introduced. We started by defining the fuzzy number.

*Definition 2.1. *A fuzzy number is a fuzzy set such that (i)is upper semicontinuous, (ii) outside some interval , (iii)there are real numbers , for which:(1) is monotonically increasing on , (2) is monotonically decreasing on ,(3)*. *The set of all fuzzy numbers (as given by Definition 2.1) is denoted by [13, 14]. An alternative definition which yields the same is given by Kaleva [15] and Ma et al. [16].

*Definition 2.2. *A fuzzy number is a pair of functions and , which satisfy the following requirements: (i) is a bounded monotonically increasing left continuous function on and right continuous at , (ii) is a bounded monotonically decreasing left continuous function on and right continuous at , (iii). A popular fuzzy number is the triangular fuzzy number where denotes the modal value and the real values and represent the left and right fuzziness, respectively. The membership function of a triangular fuzzy number is defined as follows:
Its parametric form is
Triangular fuzzy numbers are fuzzy numbers in representation where the reference functions and are linear.

##### 2.1. Operation on Fuzzy Numbers

We briefly mentioned fuzzy number operations that have had been defined by the extension principle [17, 18]. where and are fuzzy numbers, denotes the membership function of each fuzzy number, is the minimum operator, and is a continuous function.

The above operations on fuzzy numbers are numerically performed on level sets (i.e., -cuts). For , a -level set of a fuzzy number is defined as and . Since level sets of fuzzy numbers become closed intervals, we denote by where and are the lower and the upper limits of the -level set , respectively. From interval arithmetic [19], the above operations on fuzzy numbers are written for the -level sets as follows: For arbitrary and , we define addition and multiplication by as [13, 14]:

*Definition 2.3. *For arbitrary fuzzy numbers the quantity
is the distance between and . It is shown that is a complete metric space [20].

*Definition 2.4. *Let . For each partition of and for arbitrary ), suppose
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 [13]. Also,
More details about properties of the fuzzy integral are given in [13, 15].

##### 2.2. System of Integral Equations

The basic definition of integral equation is given in [21].

*Definition 2.5. *The Fredholm integral equation of the second kind is
where
In (2.12), is an arbitrary kernel function over the square and is a function of . If the kernel function satisfies , we obtain the Volterra integral equation
In addition, if be a crisp function, then the solution of the above equation is crisp as well. Also if be a fuzzy function, we have Fredholm’s fuzzy integral equation of the second kind which may only process fuzzy solutions. Sufficient conditions for the existence and uniqueness of the solution of the second kind equation, where is a fuzzy function, are given in [22, 23].

*Definition 2.6. *The second kind fuzzy linear Volterra integral equations system is in the form
where and are real constants. Moreover, in system (2.15), the fuzzy function and kernel are given and assumed to be sufficiently differentiable with respect to all their arguments on the interval . Also we assume that the kernel function and is the solution to be determined.

Now let and be parametric form of and , respectively. To simplify, we assume that . In order to design a numerical scheme for solving (2.15), we write the parametric form of the given fuzzy integral equations system as follows:
where

##### 2.3. Taylor’s Series

Let us first recall the basic principles of the Taylor polynomial method for solving Fredholm’s fuzzy integral equations system (2.15). Because these results are the key for our problems, therefore, we explain them. Without loss of generality, we assume that With above supposition, the system (2.16) is transformed to the following form: Now we want to obtain the solution of the above system in the form of which are the Taylor expansions of degree at for the unknown functions and , respectively. For this scope, we calculate th derivative of each equation in the system (2.19) with respect to and get For brevity, we define symbols as below: The aim of this study is to determine the coefficients and , in system (2.21). For this intent, we expanded and in Taylor’s series at arbitrary point and substituted its th truncation in (2.21). Now we can write where Consequently, the matrix form of expression (2.23) can be written as follows: where Parochial matrices are defined with the following elements: where

#### 3. Convergence Analysis

In this section, we proved that the above numerical method converges to the exact solution of fuzzy system (2.15).

Theorem 3.1. *Let the kernel be bounded and belong to and and for be Taylor polynomials of degree that their coefficients are produced by solving the linear system (2.25). Then these polynomials converge to the exact solution of the fuzzy Volterra integral equations system (2.15), when .*

*Proof. *Consider the system (2.15). Since, the series (2.20) converge to and , respectively, then we conclude that
and it holds that
We defined the error function by subtracting (2.19) and (3.1) as follows:
where
We must prove when , the error function becomes zero. Hence, we proceed as follows:
Since is bounded, therefore, and imply that and proof is completed.

#### 4. An Example

In this section, we present an example of fuzzy linear Volterra integral equations system and results will be compared with the exact solution.

*Example 4.1. *Consider the system of fuzzy linear Volterra integral equations with
kernel functions
and , . The exact solution in this case is given by
In this example, we assume that . Using (2.23)–(2.25), the coefficients matrix is calculated as follows:
where
With using above matrices, we can rewrite the linear system (2.25) as follows:
The vector solution of the above linear system is
After propagating this vector solution in (2.20), we have
As shown in Figures 1 and 2, the present method gives the analytical solution for this kind of fuzzy equations system, if the exact solution be polynomial.

#### 5. Conclusions

Fuzzy integral equations systems, which have a very important place in physics and engineering, are usually difficult to solve analytically. Therefore, it is required to obtain approximate solutions. In this study, mechanization of solving fuzzy linear Volterra integral equations system of the second kind by using the Taylor expansion method have proposed. This Taylor method transforms the given problem to a linear algebraic system in crisp case. The solution of the resulting system is used to compute unknown Taylor coefficients of the solution functions. Consider that to get the best approximating solutions of the given fuzzy equations, the truncation limit must be chosen large enough. An interesting feature of this method is finding the analytical solution for given fuzzy system, if the exact solution be polynomials of degree or less than . The results of the example indicate the ability and reliability of the present method.

#### References

- F. G. Tricomi,
*Integral Equations*, Dover Publications, New York, NY, USA, 1982. - L. Xu, “Variational iteration method for solving integral equations,”
*Computers & Mathematics with Applications*, vol. 54, no. 7-8, pp. 1071–1078, 2007. View at: Publisher Site | Google Scholar - E. Babolian, H. Sadeghi Goghary, and S. Abbasbandy, “Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method,”
*Applied Mathematics and Computation*, vol. 161, no. 3, pp. 733–744, 2005. View at: Publisher Site | Google Scholar - S. J. Liao,
*Beyond Perturbation: Introduction to the HomoTopy Analysis Method*, Chapman Hall/CRC Press, Boca Raton, Fla, USA, 2003. - S. Abbasbandy, “Numerical solutions of the integral equations: homotopy perturbation method and Adomian's decomposition method,”
*Applied Mathematics and Computation*, vol. 173, no. 1, pp. 493–500, 2006. View at: Publisher Site | Google Scholar - R. P. Kanwal and K. C. Liu, “A Taylor expansion approach for solving integral equations,”
*International Journal of Mathematical Education in Science and Technology*, vol. 20, pp. 411–414, 1989. View at: Publisher Site | Google Scholar - K. Maleknejad and N. Aghazadeh, “Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method,”
*Applied Mathematics and Computation*, vol. 161, no. 3, pp. 915–922, 2005. View at: Publisher Site | Google Scholar - S. Nas, S. Yalçınbaş, and M. Sezer, “A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations,”
*International Journal of Mathematical Education in Science and Technology*, vol. 31, no. 2, pp. 213–225, 2000. View at: Publisher Site | Google Scholar - S. Nas, S. Yalçınbaş, and M. Sezer, “A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations,”
*International Journal of Mathematical Education in Science and Technology*, vol. 31, no. 2, pp. 213–225, 2000. View at: Publisher Site | Google Scholar - E. Babolian, Z. Masouri, and S. Hatamzadeh-Varmazyar, “A direct method for numerically solving integral equations system using orthogonal triangular functions,”
*International Journal of Industrial Mathematics*, vol. 2, pp. 135–145, 2009. View at: Google Scholar - H. Jafari, H. Hosseinzadeh, and S. Mohamadzadeh, “Numerical solution of system of linear integral equations by using Legendre wavelets,”
*International Journal of Open Problems in Computer Science and Mathematics*, vol. 5, pp. 63–71, 2010. View at: Google Scholar - H. H. Sorkun and S. Yalçinbaş, “Approximate solutions of linear Volterra integral equation systems with variable coefficients,”
*Applied Mathematical Modelling*, vol. 34, no. 11, pp. 3451–3464, 2010. View at: Publisher Site | Google Scholar - R. Goetschel and W. Voxman, “Elementary fuzzy calculus,”
*Fuzzy Sets and Systems*, vol. 18, no. 1, pp. 31–43, 1986. View at: Publisher Site | Google Scholar - H. T. Nguyen, “A note on the extension principle for fuzzy sets,”
*Journal of Mathematical Analysis and Applications*, vol. 64, no. 2, pp. 369–380, 1978. View at: Google Scholar - O. Kaleva, “Fuzzy differential equations,”
*Fuzzy Sets and Systems*, vol. 24, no. 3, pp. 301–317, 1987. View at: Publisher Site | Google Scholar - M. Ma, M. Friedman, and A. Kandel, “A new fuzzy arithmetic,”
*Fuzzy Sets and Systems*, vol. 108, no. 1, pp. 83–90, 1999. View at: Publisher Site | Google Scholar - L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning. III,”
*Information Sciences*, vol. 9, no. 1, pp. 43–80, 1975. View at: Google Scholar - L. A. Zadeh, “Toward a generalized theory of uncertainty (GTU) an outline,”
*Information Sciences*, vol. 172, no. 1-2, pp. 1–40, 2005. View at: Publisher Site | Google Scholar - G. Alefeld and J. Herzberger,
*Introduction to Interval Computations*, Academic Press, New York, NY, USA, 1983. - M. L. Puri and D. A. Ralescu, “Fuzzy random variables,”
*Journal of Mathematical Analysis and Applications*, vol. 114, no. 2, pp. 409–422, 1986. View at: Publisher Site | Google Scholar - H. Hochstadt,
*Integral Equations*, John Wiley & Sons, New York, NY, USA, 1973. - W. Congxin and M. Ming, “On the integrals, series and integral equations of fuzzy set-valued functions,”
*Journal of Harbin Institute of Technology*, vol. 21, pp. 9–11, 1990. View at: Google Scholar - M. Friedman, M. Ma, and A. Kandel, “Numerical solutions of fuzzy differential and integral equations,”
*Fuzzy Sets and Systems*, vol. 106, no. 1, pp. 35–48, 1999. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2012 A. Jafarian et al. 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.