Generalized Hukuhara conformable fractional derivative and its application to fuzzy fractional partial differential equations

The main focus of this paper is to develop an efficient analytical method to obtain the traveling wave fuzzy solution for the fuzzy generalized Hukuhara conformable fractional equations by considering the type of generalized Hukuhara conformable fractional differentiability of the solution. To achieve this, the fuzzy conformable fractional derivative based on the generalized Hukuhara differentiability is defined, and several properties are brought on the topic, such as switching points and the fuzzy chain rule. After that, a new analytical method is applied to find the exact solutions for two famous mathematical equations: the fuzzy fractional wave equation and the fuzzy fractional diffusion equation. The present work is the first report in which the fuzzy traveling wave method is used to design an analytical method to solve these fuzzy problems. The final examples are asserted that our new method is applicable and efficient.


Introduction
During the last decade, the interest of mathematicians in fuzzy differential equations has been rapidly increasing. The main reason for this development is that using these problems will lead to a much more effective and elegant way of treating real-life issues. A particular subgroup of fuzzy differential equations is described with operators of fractional nature. Fractional calculus is a set of methods and hypotheses that extend the concept of a derivative operator from integer-order n to arbitrary order α. Modeling like biological population models, the predator-prey models and infectious diseases models, etc., are generalized to fractional order. Fractional calculus is not only a productive and emerging field, but it also represents a new philosophy, how to construct and apply a certain type of nonlocal operator to real-world problems (Ghobaei-Arani et al. 2019; Ghobaei-Arani and Souri 2019; Ghobaei-Arani et al. 2021;Khorsand et al. 2018;Mazandarani and Xiu 2021;. The interest in fractional fuzzy differential equations aroused in 2012 with a paper by Agarwal et al. (2012). The existence and uniqueness of a fuzzy solution for fractional differential equations are proved in Arshad and Lupulescu (2011). The concepts of fuzzy fractional integral and Caputo partial differentiability based on generalized Hukuhara differentiability for the fuzzy multivariable functions have been introduced by Viet Long et al. (2017). Hoa et al. (2019) introduced the fuzzy Caputo-Katugampola fractional differential equations in fuzzy space, and under generalized Lipschitz condition, the existence and uniqueness of the solution are proved. The analytical solutions to some linear fractional partial fuzzy differential equations under certain conditions were investigated in Shaha et al. (2020). The perturbation-iteration algorithm was developed for numerical solutions of some types of fuzzy fractional partial differential equations with generalized Hukuhara derivative Senol et al. (2019). Zureigat et al. (2020) analyzed the compact Crank-Nicholson scheme to solve the fuzzy time diffusion equation with fractionalorder 0 < α ≤ 1. Some new methods and useful materials concerning fuzzy fractional differential and fuzzy fractional partial differential equations are introduced in Allahviranloo (2021).
Recently, a new well-behaved simple fractional derivative called "the conformable fractional derivative" is defined by Khalil et al. (2014); Arqub and Al-Smadi (2020). This new definition seems to be a natural extension of the usual derivative. Researchers started to combine this new definition with fuzzy calculus (Harir et al. 2020(Harir et al. , 2021Martynyuk et al. 2020). They used the concept of H-differentiability or strongly generalized differentiability. But it is well-known that the usual Hukuhara difference between two fuzzy numbers exists only under very restrictive conditions (Diamond 2002;Bede and Gal 2005). To overcome this shortcoming, we will introduce the fuzzy conformable fractional derivative under generalized Hukuhara differentiability and prove some important properties for this kind of differentiability.
Consider the following generic form of second-order fuzzy fractional partial differential equation defined based on generalized Hukuhara conformable fractional derivative with 0 < α ≤ 1. The main contribution of this paper is to find the wave traveling solutions for problem (1). For this purpose, the concept of generalized Hukuhara conformable fractional differentiability is introduced thoroughly in the fuzzy functions. Next, the fuzzy fractional wave equation and fuzzy diffusion equation are introduced based on the generalized Hukuhara conformable fractional differentiability. Finally, we discuss the fuzzy traveling wave solutions for these equations by considering the type of α gH -differentiability. We now give a brief outline of the main sections of the paper and state the aims and objectives of each section. Section 3 deals with aspects of background knowledge in fuzzy mathematics and fuzzy derivatives with emphasis on the generalized Hukuhara differentiability. In Sect. 4, generalized Hukuhara conformable fractional differentiability is studied. Some properties for this concept of differentiability are proved. A fuzzy fractional wave equation and a fuzzy fractional diffusion equation under generalized Hukuhara conformable fractional differentiability are introduced in Sects. 5 and 6, respectively. Moreover the fuzzy traveling wave solutions of these equations are investigated in different scenarios. Finally in Sect. 7, the conclusions are given.

Related works
The concept of the fuzzy derivative was first introduced by Chang and Zadeh (1972). The starting point of the topic in the set-valued differential equation and also fuzzy differential equation is Hukuhara's paper Hukuhara (1967). Puri and Ralescu (1986) suggested the fuzzy differential equations modeling with uncertainty under the concept of H-differentiability. Subsequently, Kaleva in Kaleva (1987) proposed fuzzy differential equations using the Hukuhara derivative, and some other authors developed it. But for some fuzzy differential equations in this framework, the diameter of the solution is unbounded as the time t increases (Diamond 2002).
To overcome this shortcoming, Bede and Gal introduced the weakly generalized differentiability and the strongly generalized differentiability for the fuzzy functions (Bede and Gal 2005). Moreover, they presented a more general definition of derivatives for the fuzzy functions and their applications for solving fuzzy differential equations (Bede and Gal 2005). Stefanini and Bede, by the concept of generalization of the Hukuhara difference of compact convex set, introduced generalized Hukuhara differentiability Stefanini and Bede (2009) for interval-valued functions. They showed that this concept of differentiability has relationships with weakly generalized differentiability and strongly generalized differentiability. The disadvantage of the strongly generalized differentiability of a function compared to Hdifferentiability is that the fuzzy differential equation has no unique solution (Bede and Gal 2005). Also, in Chalco-Cano et al. (2011) the authors studied relationships between the strongly generalized differentiability and the generalized Hukuhara differentiability, showing the equivalence between these two concepts when the set of switching points of the interval-valued function is finite (Table 1).
In this way, they use the LU-parametric representation of fuzzy numbers and fuzzy-valued functions to obtain valid approximations of fuzzy generalized Hukuhara derivative and solve fuzzy differential equations (Bede and Stefanini 2011). Allahviranloo et al. (2015) introduced the fuzzy generalized Hukuhara partial differentiability and solved the fuzzy heat equation under generalized Hukuhara differentiability. Moreover, in Moghaddam and Allahviranloo (2018) the authors obtained the fuzzy solutions of the fuzzy Poisson equation under generalized Hukuhara differentiability. Recently, Chalco-Cano et al. (2020) provided a new characterization of the switching points for generalized Hukuhara differentiability and shown that the set of all switching points is at most countable.

Preliminaries
In the following, we focus on the basic definitions and the necessary notation which will be used throughout the paper. Let E is the set of fuzzy numbers and T ⊂ E shows the set of all triangular fuzzy numbers.  Chang and Zadeh (1972) Define a fuzzy function and its inverse, fuzzy parametric functions, fuzzy observation, and control. Hukuhara (1967) The first definitions of Hukuhara difference, and Hukuhara derivative Hukuhara difference between two fuzzy numbers is not always a fuzzy number. Puri and Ralescu (1986) Prove the Rȧdström embedding theorem and define the concept of the differential of a fuzzy function The diameter of the solution is unbounded as the time t increases. Kaleva (1987) Define a fuzzy differential equations using the Hukuhara derivative The diameter of the solution is unbounded as the time t increases. Bede and Gal (2005) Define the strongly generalized differentiability and the weakly generalized differential of a fuzzy function using the Hukuhara derivative The fuzzy differential equations may not have a unique. Stefanini and Bede (2009) Define the generalized Hukuhara difference and generalized Hukuhara differentiability for interval-valued functions Allahviranloo et al. (2015) Define the fuzzy generalized Hukuhara partial differentiability and solve the fuzzy heat equation Harir et al. (2020) Define the fuzzy Generalized Conformable Fractional Derivative using the Hukuhara derivative Hukuhara difference between two fuzzy numbers is not always a fuzzy number.

Harir et al. (2021)
Prove the existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability The diameter of the solution is unbounded as the time t increases.

Martynyuk et al. (2020)
Define the fractional-like Hukuhara-type derivatives Hukuhara difference between two fuzzy numbers is not always a fuzzy number.

Chalco-Cano et al. (2020)
A new characterization of the switching points for generalized Hukuhara differentiability Let a = a 1 , a 2 , a 3 and b = b 1 , b 2 , b 3 are two triangular fuzzy numbers, so the generalized Hukuhara difference, a gH b, is defined as follows (Bede 2013) Actually In this article, we assume that a gH b ∈ T.
Here, D is the Hausdorff distance. Then, we say that L ∈ E is limit of f in t 0 , which is denoted by lim t→t 0 f (t) = L. Also the fuzzy function f is said to be fuzzy continuous if defines a triangular fuzzy number.
is a triangular fuzzy number.

Lemma 3.8 (See Moghaddam and Allahviranloo 2018)
where denotes Hukuhara difference and f (t) be a fuzzy function. Definition 3.9 (See Allahviranloo et al. 2015) Let (x 0 , t 0 ) ∈ D ⊆ R 2 , then the first generalized Hukuhara partial derivative ([g H − p]-derivative for short) of a fuzzy value function defines a triangular fuzzy number, and defines a triangular fuzzy number.
respect to x without any switching point on D and

Generalized Hukuhara conformable fractional derivative
In this section, we are going to introduce conformable fractional derivative based on the generalized Hukuhara derivative. Moreover, we will prove several properties for this kind of differentiability.
If the generalized Hukuhara conformable fractional derivative of f of order α exists, then we simply say f is α gH -differentiable.

Theorem 4.2 If a fuzzy function f
ε. By using Theorem 3.2, we conclude that Therefore, according to Definition 3.1, it can be concluded that the function f is fuzzy continuous.
provided that T β gH ( f )(t) ∈ E. If the generalized Hukuhara conformable fractional derivative of f of order β exists, then we simply say f is β gH -differentiable.
Definition 4.4 Let α ∈ (0, 1) and f is α gH -differentiable at a point t > 0. We can say that f (t) is defines a triangular fuzzy number.
be a triangular fuzzy number.

Example 4.8 Consider the fuzzy function
We have the following α gH -derivatives of f (t) Therefore, the fuzzy function . This function is switched to α ii.gHdifferentiable at t = π 2 . Hence, the point t = π 2 is a switching point of Type I for the differentiability of f (Fig. 1). Theorem 4.9 Let g : I → ζ is real-valued differentiable at t, and f : ζ → E be a fuzzy function such that f is g Hdifferentiable at the point g(t) without any switching points, and α ∈ (0, 1).
. We have the following cases i. If g (t) > 0. Hence by attention to Theorem 4.6 we have ii. If g (t) < 0, then We can use the same procedure when f (t) is [(ii) − g H]differentiable at g(t).

Fuzzy traveling wave solution of the fuzzy fractional wave equation
We want to find traveling wave fuzzy solution of the fuzzy one-dimensional homogeneous fractional wave equation. Consider this problem as follows where α ∈ ( 1 2 , 1) and D α t gH is the generalized Hukuhara conformable fractional partial derivatives with respect to t and υ x gH is the generalized Hukuhara partial derivative with respect to x. Here, f (x), g(x) are given continuous fuzzy functions. We will find the triangular analytical fuzzy solutions of Eq. (7) by using traveling wave method provided that the types of α gH -differentiability of υ(x, t) with respect to t and [g H − p]-differentiability with respect to x are the same. By considering the type of α gH -differentiability of υ(x, t) with respect to t, we have different cases as follows: 1-1-gH . Let υ(x, t) and D α t gH υ are α i.gH -differentiable with respect to t, and υ(x, t) and υ Here, we outline the main steps of traveling wave method function Step 1 Consider fuzzy one-dimensional homogeneous fractional wave Eq. (7) υ( which can be analyzed through a change of variables υ(x, t) = U (ξ ). Here, U is a continuous function and g H-differentiable in ξ and γ is a positive real constant.
Step 2 We have therefore, by using Theorem 4.9, the fuzzy multivariate chain rule (Moghaddam and Allahviranloo 2018), we have Hence Eq. (7) is reduced to the following fuzzy ordinary differential equations of ξ Step 3 To find fuzzy solutions for ordinary differential Eqs. (8) and (17), we need some auxiliary boundary conditions, which in this article, we consider the following auxiliary boundary conditions By using Proposition 3.4, Eq. (17) can also be written as follows One possibility is for where C 1 and C 2 are fuzzy integral constants. But boundary conditions (9) cannot be satisfied unless C 2 = 0. Thus the only traveling solution is a fuzzy constant. Another possibility is for γ 2 = κ 2 . In this case are traveling wave solutions of the fuzzy fractional wave equation and U can be any two g H-differentiable function. In general, it follows that any solution to the fuzzy fractional wave equation can be obtained as a superposition of two traveling waves, Since Eq. (12) is a fuzzy solution for Eq. (7), then it must apply to the initial conditions of Eq. (7) υ( Hence, the initial condition υ( By considering Theorem 4.9 we have By the initial condition D α t gH υ(x, 0) = g(x), we can write After integration by using Lemma 3.7 The following system of equations is obtained by Eqs. (19) and (15) such that this system of equations has the following fuzzy solutions On the other hand, according to Lemma 3.8, G (x) can be rewritten as follows By substituting these equations for F and G into general solution (12), the fuzzy traveling wave solution is obtained as follows Here, υ(x, t) and D α t gH υ are α i.gH -differentiable with respect to t, and υ(x, t) and υ x gH are [(i) − g H]differentiable with respect to x. 2-2-gH. Let υ(x, t) and D α t gH υ are α ii.gH -differentiable with respect to t and υ(x, t) and υ x gH are [(ii) − g H]differentiable with respect to x then U (ξ ) is a [(ii) − g H]-differentiable fuzzy function. In this case, the main steps of the fuzzy traveling wave method are as follows Step 1 Let we can analyze fuzzy one-dimensional homogeneous fractional wave Eq. (7) through the following change variables where U is a continuous function and g H-differentiable in ξ and γ is a positive real constant.
Step 2 We have therefore, by using Theorem 4.9, the fuzzy multivariate chain rule (Moghaddam and Allahviranloo 2018), we have Hence Eq. (7) is reduced to the following fuzzy ordinary differential equations of ξ Step 3 As we explained in case 1-1-gH, any solution of the fuzzy fractional wave equation can be obtained as follows Equation (18) is a fuzzy solution for Eq. (7); then, the initial condition υ( On the other hand, using Theorem 4.9 and the initial value Integrate each side of the above equation by using Lemma 3.7, therefore Consequently, we find that By solving this system and using Lemma 3.8, the following results are obtained Hence Eq. (21) is reduced to the following fuzzy ordinary differential equation of ξ Step 3 To find fuzzy solutions for ordinary differential Eq. (23), we need some auxiliary boundary conditions, which in this article, we consider the following auxiliary boundary conditions We integrate both sides of Eq. (23). According to the auxiliary boundary conditions expressed in Eq. (24), the integration constants are zero and This equation has the following fuzzy solution (Armand and Gouyandeh 2017) which satisfies the condition U (ξ ) = 0 when ξ → ∞. Therefore υ(x, t) = C e −(x−K t α α ) .
Using initial condition (22), we can write and finally the fuzzy solution for the fuzzy linear diffusion equation is equal The other case of differentiability can be examined in a similar way.

Conclusion
In this paper, we have defined the generalized Hukuhara conformable fractional derivative and the type of differentiability of this derivative is studied in detail, and we have proved some novel properties for it. The fuzzy traveling wave solution of the fractional wave equation and fractional diffusion equation was obtained by considering the type of α gH -differentiability. To demonstrate the efficiency of the method, the fuzzy traveling wave solutions of some examples were obtained. All results show that this method is a compelling and efficient method for obtaining an analytical solution for the fuzzy linear fractional partial differential equation.