Fuzzy Linear and Nonlinear Integral Equations: Numerical MethodsView this Special Issue
Research Article | Open Access
Approximating the Solution of the Linear and Nonlinear Fuzzy Volterra Integrodifferential Equations Using Expansion Method
The present research study introduces an innovative method applying power series to solve numerically the linear and nonlinear fuzzy integrodifferential equation systems. Finally, it ends with some examples supporting the idea.
Fuzzy integrodifferential equations have attracted great interests in recent years since they play a major role in different areas of theory such as control theory. For the first time, Chang and Zadeh have introduced fuzzy numbers as well as the related arithmetic operations [1, 2]. Furthermore, applying the operators on fuzzy numbers has been developed by Mizumoto and Tanaka . It should be mentioned that the concept of LR fuzzy numbers was expressed by Dubois and Prade . In this regard, they made a significant contribution by providing a computational formula for operations on fuzzy numbers. After that, the notation of fuzzy derivative was presented by Seikkala . However, Goetschel, Jr., and Voxman have proposed the Riemann integral-type approach . Some mathematicians have separately worked on the existence and having a unique solution of fuzzy Volterra integrodifferential equation [7–9]. Recently, numerical methods have been applied to solve the linear as well as nonlinear differential equation fuzzy integral equation and fuzzy integrodifferential equation [7, 8, 10–13].
In this paper, we use the power series method of the exact solution of linear or nonlinear fuzzy integrodifferential equations, which is obtained by recursive procedure as follows.
We consider the following system of fuzzy integrodifferential equations: with initial condition , and
2. Basic Concepts
Let be a set of all triangular fuzzy numbers.
Definition 1. An arbitrary fuzzy number in the parametric form is represented by an ordered pair of functions which satisfy the following requirements.(i) is a bounded left-continuous nondecreasing function over .(ii) is a bounded left-continuous nonincreasing function over .(iii).
Definition 3. A triangular fuzzy number is defined as a fuzzy set in , which is specified by an ordered triple with such that are the endpoints of -level sets for all , where and . Here, , which is denoted by .
Definition 4. A fuzzy number is of type if there exist shape functions (for left), (for right), and scalar with the mean value of , is a real number, and , are called the left and right spreads, respectively. is denoted by .
Definition 5. Let , , and . Then,
Definition 6. The integral of a fuzzy function was defined in  by using the Riemann integral concept.
Let , for each partition of and for arbitrary , , and 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 , and also
Definition 7 (see ). Consider . If there exists such that , then is called the H-difference of and and is denoted by .
Proposition 8 (see ). If is a continuous fuzzy-valued function, then is differentiable, with derivative .
Definition 9 (see ). Let be a fuzzy valued function. If, for arbitrary fixed and , a such that is said to be continuous.
Definition 10 (see ).
Let be and . One says that is differentiable at if
there exists an element such that, for all sufficiently near to , there are , , and the limits or
there exists an element such that, for all , sufficiently near to 0, there are , and the limits
Lemma 11 (see ). For the fuzzy differential equation
where is supposed to be continuous, if equivalent to one of the integral equations:
On some intervalunder the differentiability condition, (i) or (ii), respectively.
Definition 12 (see ). For fuzzy number , one writes (1) , if , (2) , if , (3) , if , and (4) , if .
Theorem 13 (see ). Let and . If is differentiable on , then the function , defined by ,, is differentiable on and one has .
Corollary 14 (see ). Let and . And define by ,. If is differentiable on and is differentiable on , then is differentiable on and twice differentiable on , with .
Remark 15 (see ). In general, if the above is times differentiable on and is differentiable on , then is differentiable of order on and .
Theorem 16 (see ). Let be a function and denote , for each . Then one has the following.(i)If is differentiable in the first form (Definition 10), then , are differentiable functions and (ii)If is differentiable in the second form (Definition 10), then , are differentiable functions and
3. Approximation Based on the Expansion Method
Suppose the solution of the system of fuzzy integrodifferential equations (1) is as follows: where , for all . By using initial conditions, we have
where is constant matrix, is fuzzy vector, , and are polynomials of order equal or greater than 1. If by neglecting , we have fuzzy linear equations system of . By solving this system, the coefficient of in (20) can be determined.
In the second step, we assume that
where and are known and is unknown. With derivative of (23) we have case (1),; case (2),,
and by substituting , (23) into (1), we have or where by integration and sort of terms of above equation we obtain the following system: where is constant matrix, is fuzzy vector , and ,, and are polynomials of order greater than unity, where by neglecting , we have again fuzzy system of linear equations of and by solving this system, coefficients of in (23) can be determined. This procedure can be repeated till the arbitrary order coefficients of power series of the solution for the problem are obtained.
The following theorem shows convergence of the method. Without loss of generality, we prove it for .
Theorem 17. Let be the exact solution of the following fuzzy integrodifferential equation:
Assume that has a power series representation. Then,
Proof. According to the proposed method, we assume that the approximate solution to (27) is as follows:
Hence, it is sufficient that we only prove for .
Note that, for , the initial condition gives
Moreover, for , if we set and in (27), we obtain
On the other hand, from (29) and (31), we have
By substituting (33) into (27) and setting ,we get
For , differentiating (27) with respect to s, we have
Setting in (35), we get
According to (29), (31), and (34), let
By substituting (37) into (35) and setting , we obtain
So, with comparison (36) and (38), we conclude that or By constituting the above procedure, we can easily prove (30) for .
4. Numerical Illustrations
Example 1. Consider the following system of fuzzy linear Volerra integrodifferential equations:
with initial conditions
From the initial conditions
Let the solution of (41)be
For obtaining ,, we substitute (44) into (41); then we will have
where are and by neglecting them, we have
We go to next step. Let Similar to previous step, by substituting (50) into (41), we have So,
By neglecting , , , which are O and solve system , we obtain
and then in a similar way go to next step and we have
Example 2. As second example we consider the following nonlinear fuzzy integrodifferential equation:
Typically, we use the power series method for obtaining the solution of problem. From the initial condition, , let the solution of (56) be the form
By neglecting ,, which are , we obtain and then
For the next step, we assume that
From above relation and by neglecting , we have .
By repeating this method, we can compute more coefficients of the solution.
Example 3. Consider the following nonlinear fuzzy integrodifferential equation:
Again, we use the power series method for obtaining the solution of the problem. From the initial condition, . Assume that the solution of (63) is the form
And by substituting it into (63), we have From the above relation and by neglecting , we have By continuing this procedure, more coefficients of the solution can be computed.
In summary, this study has exploited power series to find a numerical solution for linear as well as nonlinear fuzzy Volterra integrodifferential equations. In effect, using power series can provide an approximate solution for the mentioned integral equations. Since there are challenging issues to solve the nonlinear integrodifferential equations, the presented method can be simply applied to find an appropriate solution for this kind of equations that is regarded as a considerable benefit of this method undoubtedly.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
- S. S. L. Chang and L. A. Zadeh, “On fuzzy mapping and control,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 2, pp. 30–34, 1972.
- L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning,” Information Sciences, vol. 8, no. 3, pp. 199–249, 1975.
- M. Mizumoto and K. Tanaka, “The four operations of arithmetic on fuzzy numbers,” Systems Computers Controls, vol. 7, no. 5, pp. 73–81, 1976.
- D. Dubois and H. Prade, “Operations on fuzzy numbers,” International Journal of Systems Science, vol. 9, no. 6, pp. 613–626, 1978.
- S. Seikkala, “On the fuzzy initial value problem,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 319–330, 1987.
- R. Goetschel Jr. and W. Voxman, “Elementary fuzzy calculus,” Fuzzy Sets and Systems, vol. 18, no. 1, pp. 31–43, 1986.
- R. Alikhani, F. Bahrami, and A. Jabbari, “Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 1810–1821, 2012.
- T. Allahviranloo, M. Khezerloo, O. Sedaghatfar, and S. Salahshour, “Toward the existence and uniqueness of solutions of second-order fuzzy volterra integro-differential equations with fuzzy kernel,” Neural Computing and Applications, vol. 22, no. 1, pp. 133–141, 2013.
- S. Hajighasemi, T. Allahviranloo, M. Khezerloo, M. Khorasany, and S. Salahshour, “Existence and uniqueness of solutions of fuzzy Volterra integro-differential equations,” in Information Processing and Management of Uncertainty in Knowledge-Based Systems, vol. 81 of Communications in Computer and Information Science, part 2, pp. 491–500, 2010.
- B. Bede and L. Stefanini, “Generalized differentiability of fuzzy-valued functions,” Fuzzy Sets and Systems, vol. 230, pp. 119–141, 2013.
- S. Salahshour and S. Abbasbandy, “A comment on “Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations”,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 5, pp. 1256–1258, 2014.
- T. Allahviranloo and S. Hashemzehi, “The homotopy perturbation method for fuzzy Fredholm integral equations,” Journal of Applied Mathematics, Islamic Azad University of Lahijan, vol. 19, pp. 1–13, 2008.
- M. Gachpazan, “Numerical scheme to solve integro-differential equations system,” Journal of Advanced Research in Scientific Computing, vol. 1, no. 1, pp. 11–21, 2009.
- B. Bede and S. G. Gal, “Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 581–599, 2005.
- L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.
- H. J. Zimmermann, Fuzzy Sets Theory and Its Applications, Kluwer, Dordrecht, The Netherlands, 1991.
- D. Dubois and H. Prade, Fuzzy Sets and Systems, Academic Press, London, UK, 1980.
- 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.
- M. L. Puri and D. A. Ralescu, “Fuzzy random variables,” Journal of Mathematical Analysis and Applications, vol. 114, no. 2, pp. 409–422, 1986.
- 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.
- Y. Chalco-Cano and H. Román-Flores, “On new solutions of fuzzy differential equations,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 112–119, 2006.
- T. Allahviranloo, N. Mikaeilvand, N. A. Kiani, and R. M. Shabestari, “Signed decomposition of fully fuzzy linear systems,” Applications and Applied Mathematics, vol. 3, no. 1, pp. 77–88, 2008.
Copyright © 2014 T. Allahviranloo 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.