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Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 6380978, 13 pages
http://dx.doi.org/10.1155/2016/6380978
Research Article

Generalization of Fuzzy Laplace Transforms of Fuzzy Fractional Derivatives about the General Fractional Order

Department of Mathematics, Faculty of Education for Girls, Kufa University, Najaf, Iraq

Received 17 July 2015; Accepted 13 October 2015

Academic Editor: Zhen-Lai Han

Copyright © 2016 Amal Khalaf Haydar and Ruaa Hameed Hassan. 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 main aim in this paper is to use all the possible arrangements of objects such that of them are equal to 1 and (the others) of them are equal to 2, in order to generalize the definitions of Riemann-Liouville and Caputo fractional derivatives (about order ) for a fuzzy-valued function. Also, we find fuzzy Laplace transforms for Riemann-Liouville and Caputo fractional derivatives about the general fractional order under H-differentiability. Some fuzzy fractional initial value problems (FFIVPs) are solved using the above two generalizations.

1. Introduction

Fuzzy Fractional Differential Equations (FFDEs) can offer a more comprehensive account of the process or phenomenon. This has recently captured much attention in FFDEs. As the derivative of a function is defined in the sense of Riemann-Liouville, Grünwald-Letnikov, or Caputo in fractional calculus, the used derivative is to be specified and defined in FFDEs as well [1].

Many researchers have worked on the field of Fuzzy Fractional Differential Equations (FFDEs); for example, Salahshour et al. [2] dealt with the solutions of FFDEs under Riemann-Liouville H-differentiability by fuzzy Laplace transforms; Mazandarani and Kamyad [1] presented the solution to FFIVP under Caputo-type fuzzy fractional derivatives by a modified fractional Euler method; Wu and Baleanu [3] proposed a novel modification of the variational iteration method (VIM) by means of the Laplace transform; they extended the method successfully to fractional differential equations; Ahmadian et al. [4] reveal a computational method based on using a Tau method with Jacobi polynomials for the solution of fuzzy linear fractional differential equations of order , and Allahviranloo et al. [5] introduced the fuzzy Caputo fractional differential equations under the generalized Hukuhara differentiability.

This paper is arranged as follows. Basic concepts are given in Section 2. In Section 3, the general formula of the fuzzy Riemann-Liouville fractional derivatives and the general formula of Laplace transforms of the fuzzy Riemann-Liouville fractional derivatives for a fuzzy-valued function are found. In Section 4, the general formula of the fuzzy Caputo fractional derivatives and the general formula of Laplace transforms of the fuzzy Caputo fractional derivatives for a fuzzy-valued function are found. In Section 5, conclusions are drawn.

2. Basic Concepts

In this section, we give the basic concepts which are needed in the next sections. We denote as the space of all continuous fuzzy-valued functions on . Also, we denote the space of all Lebesgue integrable fuzzy-valued functions on the bounded interval by .

Theorem 1 (see [6]). Let be a positive integer. Let be continuous on ; is the class of piecewise continuous functions on and integrable on any finite subinterval of and let Then, one finds the following:
If is of class C, thenand
if is continuous on , then for where

Definition 2 (see [7]). A fuzzy number in parametric form is a pair of functions ,  , which satisfy the following requirements.(1) is a bounded nondecreasing left continuous function in and right continuous at 0.(2) is a bounded nonincreasing left continuous function in and right continuous at 0.(3)Consider ,  .We denote the set of all real numbers by and the set of all fuzzy numbers on is indicated by .

Definition 3 (see [8]). Let . If there exists such that , then is called the H-difference of and , and it is denoted by . The sign “” always stands for H-difference and also note that .

Definition 4 (see [9]). Let be continuous fuzzy-valued function; suppose that is improper fuzzy Rimann-integrable on then is called fuzzy Laplace transforms and is denoted as , .
We havealso by using the definition of classical Laplace transform:then, we follow:

Definition 5 (see [2]). Let . The fuzzy Riemann-Liouville integral of fuzzy-valued function is defined as follows:

3. Generalization of Fuzzy Laplace Transforms of the Fuzzy Riemann-Liouville Fractional Derivatives of Order

In this section, we define Riemann-Liouville fractional derivatives of the general fractional order and we find fuzzy Laplace transforms for Riemann-Liouville fractional derivatives of the general fractional order for fuzzy-valued function under H-differentiability.

Definition 6. Let , and and are values of rounded up and down to the nearest integer number, respectively. One can see that , and the functions and are defined as:for , such that are all the possible arrangements of objects which have the number given in the rule:where of them equal 1 (meaning Riemann-Liouville type derivative in the first form) and of them equal 2 (meaning Riemann-Liouville type derivative in the second form) and . is the Riemann-Liouville type fuzzy fractional differentiable function of order , , at , if there exists an element such that for all and for sufficiently near zero. Then: If , then If , thenfor ,  , such that are all the possible arrangements of objects which have the number given by the rule:If the fuzzy-valued function is differentiable as in Definition 6 cases defined in (11), it is the Riemann-Liouville type differentiable in the first form and denoted by . If is differentiable as in Definition 6 cases defined in (12), it is the Riemann-Liouville type differentiable in the second form and denoted by .
We note that if we take in Definition 6 we get Definition  3.2 [2] which is introduced by Salahshour et al.

Theorem 7. Let be a fuzzy-valued function such that for , , and . Suppose that and is the number of repetitions of number 2 among for ,  , say, , such that ; that is, and . Then, one has the following:If is an even number, thenIf is an odd number, thenwhere

Proof. Suppose that is an even number, and then , . Now, we have two probabilities as follows.
The first probability is is the Riemann-Liouville type fuzzy fractional differentiable function in the first form (), and then from (11) of Definition 6, we have:Multiplying both sides by , , we obtain:By taking on both sides of the above equations, we get:Now, since is equal to the limits defined in (8) of Definition 6, then by applying (8) for times, we getSince is equal to the limits defined in (9) of Definition 6, then by applying (9) once, we get:Since is equal to the limits defined in (8) of Definition 6, then by applying (8) for times, we get:Since is equal to the limits defined in (9) of Definition 6, then by applying (9) once, we get:In other words, from (23) we note that, after applying (8) and (9) for any even number from , we will get an equation similar to (23). Therefore, for , we have:since is an even number.
Finally, since is equal to the limits defined in (8) of Definition 6, then by applying (8) for times, we get:Then,Substituting (26) in (19) yields the following:The second probability is is the Riemann-Liouville type fuzzy fractional differentiable function in the second form (), and then, by applying (12) of Definition 6, we can getSince is an even number, then by replacing by in (24), we get:Similarly, by applying (8) times for and (9) once for , we get:Finally, since is equal to the limits defined in (8) of Definition 6, then, by applying (8) for times, we get:Then,Substituting (32) in (28) yields:If is an odd number, the proof is similar.

We note that if we take in Theorem 7, we get Theorem  3.2 [2] which is found by Salahshour et al.

Corollary 8. Let be a fuzzy-valued function and for and . Suppose that and is the number of repetitions of number 2 among for ,  . Then, one has the following.
If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for If is -differentiable fuzzy-valued function, then for where

Theorem 9. Suppose that is fuzzy-valued function; for . One supposes that and is the number of repetitions of 2 among , say, , such that ; that is, and . Then, one has the following:If is an even number, we have:such thatIf is an odd number, we have:such that

Proof. By we mean . Suppose that is an odd number; then, from Theorem 7, when , we get:Therefore, we get:Then, from (54), we get:We know from Laplace transform of the Riemann-Liouville fractional derivative of order thatThe above equation can be written as:In a similar manner, we can get:Since and is an odd number, then we have the following equations:The last one of the above equations yields from Theorem 7 because is an odd number. Using (57), (58), and the above equations, (55) becomes:where is defined as in (52).
If is an even number, the proof is similar.

We note that if we take in Theorem 9, we get Theorem  4.4 [2] which is found by Salahshour et al.

Corollary 10. Suppose that . One supposes that . Then, one finds the following.
If is -differentiable fuzzy-valued function, thenIf is -differentiable fuzzy-valued function, thenIf is -differentiable fuzzy-valued function, thenIf is -differentiable fuzzy-valued function, thenIf is -differentiable fuzzy-valued function, thenIf is -differentiable fuzzy-valued function, thenIf is -differentiable fuzzy-valued function, thenIf is -differentiable fuzzy-valued function, then

Example 11. Consider the following FFIVP:We note thatBy taking fuzzy Laplace transform for both sides of (69), we getNow, by using Theorem 9 when we have cases as follows.

Case 1. Let be -differentiable. By using Theorem 9, when (even), (72) becomesThen, we getThe solution of FFIVP (69) is as follows:

Case 2. Let be -differentiable. By using Theorem 9, when (odd), (72) becomesThen, we get:The solution of FFIVP (69) is as follows:

Case 3. Let be -differentiable. By using Theorem 9, when (odd), (72) becomes:Then, we get:The solution of FFIVP (69) is as follows:

Case 4. Let be -differentiable. By using Theorem 9, when (even), (72) becomes:Then, we get: