Chapter 3
Analytical Methods for Fuzzy Fractional Differential Equations (FFDES)

Several numerical and/or analytical methods are available to solve fuzzy fractional differential equations (FFDEs). Analytical solutions play a significant role in proper understanding of qualitative features of various science and engineering problems. But it is not always possible to obtain the analytical solution of the same. We have discussed in the introduction that few researchers have developed analytical methods to obtain solution of n-term FFDEs. But it has been seen that the existing methods always convert the FFDEs to two crisp differential equations (Arshad and Lupulescu, 2011; Mohammed et al., 2011; Prakash et al., 2015; Salah et al., 2013) or sometime coupled differential equations, depending upon the sign of the coefficients. Accordingly, more computational time is needed to solve the coupled or system of FFDEs. This motivates the study to develop new analytical methods with less computation time. In this regard, first, we will briefly discuss the concept of n-term fuzzy fractional linear ordinary differential equations. In general, three cases may arise according to sign of the coefficients in the FFDEs. Finally, this chapter includes three analytical methods by considering three cases to solve n-term linear FFDEs as follows.

1. Method 1:Fuzzy-Center-Based Method (FCM) (Tapaswini and Chakraverty, 2014a).
2. Method 2:Method Based on Addition and Subtraction of Fuzzy Numbers (ASFM).
3. Method 3:Double-Parametric-Based Method (DPM) (Tapaswini and Chakraverty, 2013, 2014a).

First, we discuss the general form of n-term linear FFDEs along with the possible three cases in detail.

3.1 n-Term Linear Fuzzy Fractional Linear Differential Equations

Let us consider n-term FFDEs in general form as

subject to fuzzy initial conditions

where and are real constants, are fuzzy numbers, and denotes the fractional derivative in Caputo sense. Here, is the solution to be determined.

Through r-cut approach, we may write the given fuzzy differential equation (Eq. (3.1)) as

subject to fuzzy initial condition

One may note that we have three cases arising with respect to the sign of the coefficients.

As such, we discuss these three cases in the following.

Case 1 Coefficients an - 1(t), an - 2(t), ., a1(t), a0(t) are all positive

By using the definition of Hukuhara derivative (Bede, 2008), one may write Eq. (3.2) as

and

subject to fuzzy initial condition

Case 2 Coefficients an - 1(t), an - 2(t), ., a1(t), a0(t) are all negative

From Eq. (3.2), we have

subject to fuzzy initial condition

Case 3 Coefficients an - 1(t), ., an - m(t) are positive and an - m - 1(t), an - m - 2(t),., a1(t), a0(t) are negative

From Eq. (3.2), we have

and

subject to fuzzy initial condition

Here, three analytical methods have been proposed to obtain the solution of n-term FFDEs.

3.2 Proposed Methods

Method 1 Fuzzy-Center-Based Method (FCM) (Tapaswini and Chakraverty, 2014a)

Here, fuzzy center has been used to solve n-term fuzzy linear fractional differential equations with respect to three cases. First, the fuzzy center solution is obtained and then the lower bound is written in terms of fuzzy center, from which we may find the upper bound of the fuzzy solution. Similarly, the lower bound can be obtained.

Case 1 Coefficients an - 1(t), an - 2(t), ., a1(t), a0(t) are all positive

First, we will write Eq. (3.1) in terms of fuzzy center as

with initial condition

Equation (3.9) may easily be solved to obtain by applying Laplace transform of Caputo derivative. Now, solving Eq. (3.3) or (3.4), one may get or , respectively. Next, substituting the aforementioned value of and or to the definition of fuzzy center, we may find the solution as or .

Case 2 Coefficients an - 1(t), an - 2(t), ., a1(t), a0(t) are all negative

Equation (3.1) may be written in terms of fuzzy center as

with initial condition

may be obtained by solving Eq. (3.10).

Using the definition of fuzzy center, one may write Eqs. (3.5) and (3.6) as

and

It may be seen that the aforementioned differential equations are now crisp differential equations. Hence, solving one of the aforementioned crisp differential equations, one may get the solution as or . Applying the definition of fuzzy center, one may get or .

Case 3 Coefficients an - 1(t), ., an - m(t) are positive and an - m - 1(t), an - m - 2(t),., a1(t), a0(t) are negative

In this case, we may write Eq. (3.1) in terms of fuzzy center as

with initial conditions

Similarly, we may have the solutions for .

Next, Eqs. (3.7) and (3.8) are written as

and

Similarly to the previous cases, one may obtain the lower and upper bounds of the solution by solving Eqs. (3.14) and (3.15), respectively, using the value of fuzzy center . Otherwise, only one equation, namely Eq. (3.14) or (3.15), may be solved, and using the expression or , one may have the solution bounds.

In the following paragraphs, example problems are solved using the proposed Method 1 (FCM) and are also compared with the existing crisp solutions.

Example 3.1

Let us consider the following (Case 1) linear FFDE:

subject to fuzzy initial conditions

For , the crisp solutions is obtained by the method of Kazem (2013) as

According to Eq. (3.9), the differential equation (Eq. (3.16)) can be written as

Solving Eq. (3.17), one may obtain . Proceeding as Eq. (3.3) or (3.4) with the aforementioned value of and solving any one of those equations, we obtain the value of and .

Hence, one may obtain the final solution as , where

and

Example 3.2

We consider the following (Case 2) linear FFDE:

subject to the fuzzy initial condition

We can find the center solution as . Subsequently, by applying Method 1 (FCM), we get the solution

and

It is interesting to note that the lower and upper bounds are the same for of Examples 3.1 and 3.2, which show complete agreement with the crisp solution obtained by Kazem (2013) and Odibat and Momani (2008a). Also, the fuzzy solutions of both the examples exactly match with that of the method of Prakash et al. (2015).

Method 2 Method Based on Addition and Subtraction of Fuzzy Numbers (ASFM)

This method is based on addition and subtraction of fuzzy numbers to solve n-term fuzzy fractional linear differential equations, and the methods for each of the three cases are given as follows.

Case 1 Coefficients an - 1(t), an - 2(t), ., a1(t), a0(t) are all positive

First, one may write Eq. (3.2) as

and

with initial condition

and

respectively.

Let us now denote

and

Substituting these values in Eqs. (3.18)-(3.21), respectively, we get

and

subject to initial condition

and

respectively.

It may be seen that the aforementioned differential equations are now crisp differential equations. Hence, solving Eqs. (3.22) and (3.23) by any standard method, one may get the solution and , where and . Now solving the aforementioned system of equation, one may...