Advances in Mathematical Physics

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

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

## On Fuzzy Fractional Laplace Transformation

^{1}Department of Mathematics, Urmia Branch, Islamic Azad University, P.O. Box 969, Oromiyeh, Iran^{2}Department of Physics, Urmia Branch, Islamic Azad University, P.O. Box 969, Oromiyeh, Iran^{3}Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia^{4}Department of Mathematics and Computer Science, Çankaya University, 06530 Ankara, Turkey^{5}Institute 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 [1–15]. 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 [19–26]. 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 [28–35]. 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 [1–13]. 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 [19–26].

*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.

*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.

*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.

*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.

*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.

#### 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.

#### References

- K. B. Oldham and J. Spanier,
*The Fractional Calculus*, Academic Press, New York, NY, USA, 1974. View at MathSciNet - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives*, Gordon and Breach Science Publishers, New York, NY, USA, 1993. View at MathSciNet - R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in
*Fractals and Fractional Calculus in Continuum Mechanics*, pp. 223–276, Springer, New York, NY, USA, 1997. View at Google Scholar · View at MathSciNet - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, New York, NY, USA, 1999. View at MathSciNet - R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific, 2000. View at MathSciNet - B. J. West, M. Bologna, and P. Grigolini,
*Physics of Fractal Operators*, Springer, New York, NY, USA, 2003. View at MathSciNet - G. M. Zaslavsky,
*Hamiltonian Chaos and Fractional Dynamics*, Oxford University Press, 2008. View at MathSciNet - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier, Amsterdam, The Netherlands, 2006. View at MathSciNet - A. K. Golmankhaneh,
*Investigations in Dynamics: with Focus on Fractional Dynamics*, Lap Lambert Academic Publishing, 2012. - R. Herrmann,
*Fractional Calculus an Introduction for Physicists*, World Scientific Publishing, 2011. - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, World Scientific Publishing, 2012. - D. Baleanu, J. A. T. Machado, and A. C. Luo,
*Fractional Dynamics and Control*, Springer, 2012. - A. K. Golmankhaneh, N. A. Porghoveh, and D. Baleanu, “Mean square solution of second-order random differential equations by using Homotopy analysis method,”
*Romanian Reports in Physics*, vol. 65, no. 2, pp. 350–361, 2013. View at Google Scholar - H. Jafari, H. Tajadodi, D. Baleanu, A. A. AL-Zahrani, Y. A. AL Hamed, and A. H. Zahid, “Exact solution of boussinesq and Kdv-mKdv equations by fractional sub-equation method,”
*Romanian Reports in Physics*, vol. 65, no. 4, pp. 1119–1124, 2013. View at Google Scholar - A. Parvate and A. D. Gangal, “Calculus on fractal subsets of real line—II: conjugacy with ordinary calculus,”
*Fractals*, vol. 19, no. 3, pp. 271–290, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science, New York, NY, USA, 2012. - X.-J. Yang,
*Local Fractional Functional Analysis and Its Applications*, Asian Academic Publisher Limited, Hong Kong, 2011. - R. Goetschel Jr. and W. Voxman, “Elementary fuzzy calculus,”
*Fuzzy Sets and Systems*, vol. 18, no. 1, pp. 31–43, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Lakshmikantham and R. N. Mohapatra,
*Theory of Fuzzy Differential Equations and Inclusions*, vol. 6 of*Series in Mathematical Analysis and Applications*, CRC Press, 2003. View at MathSciNet - P. Diamond and P. Kloeden,
*Metric Spaces of Fuzzy Sets: Theory and Applications*, World Scientific, Singapore, 1994. View at MathSciNet - J. J. Buckley, E. Eslami, and T. Feuring,
*Fuzzy Mathematics in Economics and Engineering*, vol. 91 of*Studies in Fuzziness and Soft Computing*, Physica, Heidelberg, Germany, 2002. View at MathSciNet - B. Bede, I. J. Rudas, and A. L. Bencsik, “First order linear fuzzy differential equations under generalized differentiability,”
*Information Sciences*, vol. 177, no. 7, pp. 1648–1662, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 6, pp. 2859–2862, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - O. Kaleva, “Fuzzy differential equations,”
*Fuzzy Sets and Systems*, vol. 24, no. 3, pp. 301–317, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Khastan and J. J. Nieto, “A boundary value problem for second order fuzzy differential equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 9-10, pp. 3583–3593, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Salah, M. Khan, and M. A. Gondal, “A novel solution procedure for fuzzy fractional heat equations by homotopy analysis transform method,”
*Neural Computing and Applications*, vol. 23, no. 2, pp. 269–271, 2013. View at Publisher · View at Google Scholar · View at Scopus - S. Arshad and V. Lupulescu, “Fractional differential equation with the fuzzy initial condition,”
*Electronic Journal of Differential Equations*, vol. 34, pp. 1–8, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Salahshour, T. Allahviranloo, and S. Abbasbandy, “Solving fuzzy fractional differential equations by fuzzy Laplace transforms,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 3, pp. 1372–1381, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. A. Anastassiou,
*Fuzzy Mathematics: Approximation Theory*, vol. 251 of*Studies in Fuzziness and Soft Computing*, Springer, 2010. View at MathSciNet - G. A. Anastassiou, “Fuzzy fractional calculus and the Ostrowski integral inequality,” in
*Intelligent Mathematics: Computational Analysis*, vol. 5 of*Intelligent Systems Reference Library*, pp. 553–574, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar - G. A. Anastassiou, “Fuzzy Ostrowski inequalities,” in
*Fuzzy Mathematics: Approximation Theory*, vol. 251 of*Studies in Fuzziness and Soft Computing*, pp. 65–73, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar - L. Kexue and P. Jigen, “Laplace transform and fractional differential equations,”
*Applied Mathematics Letters*, vol. 24, no. 12, pp. 2019–2023, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Arshad and V. Lupulescu, “On the fractional differential equations with uncertainty,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 11, pp. 3685–3693, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mazandarani and A. V. Kamyad, “Modified fractional Euler method for solving fuzzy fractional initial value problem,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 18, no. 1, pp. 12–21, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet