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Advances in Mathematical Physics

Volume 2014 (2014), Article ID 295432, 9 pages

http://dx.doi.org/10.1155/2014/295432
Research Article

On Fuzzy Fractional Laplace Transformation

1Department of Mathematics, Urmia Branch, Islamic Azad University, P.O. Box 969, Oromiyeh, Iran

2Department of Physics, Urmia Branch, Islamic Azad University, P.O. Box 969, Oromiyeh, Iran

3Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia

4Department of Mathematics and Computer Science, Çankaya University, 06530 Ankara, Turkey

5Institute of Space Sciences, P.O. Box, MG-23, 76900 Magurele-Bucharest, Romania

Received 12 February 2014; Accepted 3 March 2014; Published 30 March 2014

Academic Editor: Xiao-Jun Yang

Copyright © 2014 Ahmad 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.

Abstract

Fuzzy and fractional differential equations are used to model problems with uncertainty and memory. Using the fractional fuzzy Laplace transformation we have solved the fuzzy fractional eigenvalue differential equation. By illustrative examples we have shown the results.

1. Introduction

Fractional calculus is the generalization of the standard calculus. That involves the derivative of functions to arbitrary orders. But the fractional derivatives are nonlocal so they provided the mathematical models to non-Markov processes and memory processes. Fractional calculus has found many applications in science, engineering, and so forth [115]. Fractional dynamics has been introduced and it can be one of the models for the nonconservative systems. Fractional Newtonian with the memory is modeled by heterogeneous liquid [10]. Recently, fractional local derivative has been studied and generalized so that it can be applied on fractals [16]. Local fractional calculus and application in science and engineering have been suggested on Cantor sets [17, 18]. The uncertainty is important subject in measurement of quantities in physics. Fuzzy number can be used to show the uncertainty in measurement. Fuzzy sets have been introduced by Lotfi Zadeh in 1965 and since then they have been used in many applications [1926]. As a consequence, there is vast literature on the practical applications of fuzzy sets, while theory has a more modest coverage. Fuzzy fractional heat and wave equation has been solved by using homotopy analysis transform method [27]. This paper adopted fuzzy Laplace transforms method to solve problems of fuzzy fractional differential equations. Our motivation in this paper is due to two reasons. Firstly, one of the important and interesting transforms in the problems of fuzzy equations is Laplace transforms. The fuzzy Laplace transform method solves fuzzy fractional differential equations and fuzzy boundary and initial value problems [2835]. Secondly, this method is practically the most important operational method and also has advantage that it solves problems directly without determining a general solution in the first step and developing nonhomogeneous differential equation in the second step.

This paper is arranged in the following manner.

After an introduction to the present work, in Section 2, we recall some basic tools that involve the fractional calculus and the fuzzy numbers. In Section 3, the fuzzy fractional Laplace transformation is discussed. Finally, we present the conclusions in Section 4.

2. Basic Tools

2.1. Fractional Calculus

Fractional calculus deals with generalizations of integer order derivatives integrals to arbitrary order. In this section we present basic definitions and properties which will be used in the subsequent sections [113]. If and , then are called the left sided Riemann-Liouville (RL), fractional integral Riemann-Liouville, fractional derivative of order , and left sided Caputo fractional derivatives, respectively.

2.2. Fuzzy Numbers

Definition 1. A fuzzy number is a fuzzy set such that (i) is upper semicontinuous;(ii) outside some interval ;(iii)there are real numbers and , , for which(1) is monotonically increasing on ,(2) is monotonically decreasing on ,(3) , .

The set of all the fuzzy numbers (as given in Definition 1) is denoted by [1926].

Definition 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.

3. Fuzzy Fractional Laplace Transformation

Initial value problems are considered in fractional differential equations and solved by analytical and numerical methods [12]. In recent works, dynamical processes are considered the randomness and uncertainty. Stochastic and fuzzy differential equations are mathematical model for such dynamical processes, respectively. Suppose , and is fuzzy real number [24]. Then, the fractional fuzzy differential is where is continuous in the case of and so (4) reduces to a fractional differential equation. And if one chooses in (4), we have a fuzzy differential equation.

3.1. Fractional Differential Equations with Uncertainty

Let us consider the fractional equation with fuzzy condition; that is, where is continuous [24]. For example, if and with which is continuous, then the solution for (5) is where

As a pursuit of fractional fuzzy differential in the following section we generalized fuzzy Laplace transformation method to fractional fuzzy Laplace method. Now, we solve illustrated examples in the subsequence sections.

Example 3. Consider the following fuzzy fractional eigenvalue differential equations as where is the number of fuzzy triangular which is called fuzzy Riemann-Liouville initial condition. Then, the above equation is extended based on its lower and upper functions as follows:

Now, we solve these equations according to the two following cases, using the generalized fractional fuzzy Laplace transform (FFLT). The equation with lower functions is and with upper functions is

Now, we use the FFLT for solving (9):

After using Laplace transform on (12), we get

Thus, we have

Taking the fuzzy inverse Laplace transform we obtain

In a similar manner we are led to

Therefore, the general solution will be

In Figures 1, 2, 3, and 4 we have plotted the solutions for the case of fractional, fractional fuzzy, and fuzzy, respectively.

295432.fig.001
Figure 1: and .
295432.fig.002
Figure 2: and .
295432.fig.003
Figure 3: and .
295432.fig.004
Figure 4: and .

Example 4. Let us consider the following fuzzy fractional differential equation as where and (18) is called fuzzy Caputo initial condition. Then, (18) will be equivalent to

Applying Laplace transform on (19), we obtain

In view of (21) we arrive at

Also, by taking inverse Laplace transform of (22) we deduce that

Likewise, by doing the same calculation (20) will be

Therefore, we have the final solution

In Figures 5, 6, 7, and 8 we have sketched the solutions for the case of fractional, fractional fuzzy, and fuzzy, respectively.

295432.fig.005
Figure 5: and .
295432.fig.006
Figure 6: and .
295432.fig.007
Figure 7: and .
295432.fig.008
Figure 8: and .

Example 5. Consider the following fuzzy fractional differential equations with fuzzy Caputo initial condition as

So (26) will become two equations with lower and upper functions such as

Applying Laplace transform and inverse Laplace transform on (27) one is led to

Therefore, the general solution will be as follows:

Figures 9, 10, 11, and 12 show the graphs of the solutions for the cases of fractional, Fuzzy fractional, and fuzzy, respectively.

295432.fig.009
Figure 9: and .
295432.fig.0010
Figure 10: and .
295432.fig.0011
Figure 11: and .
295432.fig.0012
Figure 12: and .

Example 6. Suppose the following fuzzy fractional differential equation with fuzzy initial Riemann-Liouville condition:

So its lower and upper functions equations are

Using the same manner we get the solutions

Finally, we obtain general solution

Figures 13, 14, 15, and 16 indicate the graphs of the solutions for the cases of fractional, fuzzy fractional, and fuzzy, respectively.

295432.fig.0013
Figure 13: and .
295432.fig.0014
Figure 14: and .
295432.fig.0015
Figure 15: and .
295432.fig.0016
Figure 16: and .

Example 7. Let us consider the fuzzy fractional differential equation involving fuzzy Riemann-Liouville initial condition:

Equation (34) will be system of two equations such as

The solutions for (34) are

And the general solution will be as

Figures 17, 18, 19, and 20 present the graphs of the solutions for the cases of fractional, fuzzy fractional, and fuzzy, respectively.

295432.fig.0017
Figure 17: and .
295432.fig.0018
Figure 18: and .
295432.fig.0019
Figure 19: and .
295432.fig.0020
Figure 20: and .

4. Conclusion

In this work, we have generalized the fractional Laplace transformation to the fuzzy fractional Laplace transformation. Then, we have solved the fractional fuzzy differential equation using suggested fuzzy fractional Laplace transformation. Riemann-Liouville and Caputo fractional derivatives were used in the fractional fuzzy differential equations. Moreover, Liouville and Caputo fractional initial condition is chosen in the example to show the difference. The illustrated graphs present the difference between fuzzy, fractional, and ordinary differential equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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