Fractional differential equations are mathematical models which are appeared in the vast areas of science and engineering and great attention has been directed toward them over the last few decades. To be more specific, FDEs are the generalizations of classical differential equations which play a significant role in the mentioned areas. Fortunately, it is possible to establish a traveling wave transformation for a fractional differential equation which can convert it to an ordinary differential equation (ODE) of integer order such that the resulting ODE can be easily solved using a variety of robust methods.^[1–15]

One of well-designed methods which may be employed to solve nonlinear fractional differential equations is the method. This method has a fairly great performance in handling nonlinear FDEs. For example, Mohyud-Din and Ali^[16] adopted the method to construct the solitary wave solutions of the fractional generalized Sawada-Kotera equation; and Zahran^[17] established the exact solutions of some nonlinear FDEs using the method.

Another mathematical method which is truly robust to solve nonlinear FDEs is a modified form of Kudryashov method. Fundamental of this method is described in detail in the next sections, so here just some applications of this technique are reviewed. Korkmaz^[18] utilized the modified Kudryashov method to construct the exact solutions of a family of the conformable time-fractional Benjamin-Bona-Mahony equations; and Hosseini et al.^[19] constructed a series of new exact solutions of the conformable time-fractional Klein-Gordon equations using the modified Kudryashov method. More articles may be found in Refs. [20–28].

In this paper, the -expansion and modified Kudryashov methods are adopted to obtain the exact solutions of the DDCFDR equation as^[29]

The rest of this work is as follows: In sec. 2, we define the conformable fractional derivative and list some of its properties. In sec. 3, we explain the ideas of the -expansion and modified Kudryashov methods. In sec. 4, we employ the methods to solve the density-dependent conformable space-time fractional diffusion-reaction equation. Finally, we give a brief conclusion in sec. 5.

There are various definitions for the fractional derivatives. Among these, the conformable fractional derivative has gained a special interest during the last years. The αth order of the conformable fractional derivative of f can be defined as^[30]

A series of the properties of conformable derivative may be listed as follows^[30,32]

Using the transformation

3.1 Exp Method

We look for an explicit solution for the Eq. (2) as the following form

where the constants a_n, are determined later, N is a positive number, which is computed by the technique of homogeneous balance, and is an explicit function that satisfies the following ODE

Now, we have the following cases:

3.2 Modified Kudryashov Method

The initial steps of the modified Kudryashov method are as before. With the same transformation, the original fractional differential equation can be converted to a nonlinear ODE, which its solution is supposed to be in the form

where the constants a_n, are determined later, N is a positive number which is computed by the technique of homogeneous balance, and is an explicit function that satisfies the following ODE

By inserting Eq. (4) into Eq. (2) with the help of MAPLE and equating the coefficients of like powers of , we will get an algebraic system for obtaining a_nʼs, l₁, and l₂. Setting the results into Eq. (4), at the end gives new exact solutions of Eq. (1).

In this section, the exact solutions of density-dependent conformable space-time fractional diffusion-reaction equation will be extracted using the -expansion and modified Kudryashov methods. Some of the solutions for the mentioned equation are new and have been reported for the first time.

Using the transformation

4.1 Applying the exp Method

From the technique of homogeneous balance, we find N = 1. This offers a series as the following form

By inserting Eq. (6) into Eq. (5) with the help of MAPLE and equating the coefficients of like powers of , we will gain an algebraic system for obtaining a_nʼs, l₁, and l₂ as follows

After solving the above system, we find

4.2 Applying the Modified Kudryashov Method

It is obvious that N = 1. Therefore, a series can be derived as follows

By inserting Eq. (7) into Eq. (5) with the help of MAPLE and equating the coefficients of like powers of , we will get an algebraic system for obtaining a_nʼs, l₁, and l₂ as

After solving the above system, we gain

The exact solutions of density-dependent conformable space-time fractional diffusion-reaction equation have been successfully extracted using the newly well-organized techniques called the -expansion and modified Kudryashov methods. Although the both methods result in a number of exact solutions for the governing model, the modified Kudryashov method has some clear advantages over the method. For example

(i) The modified Kudryashov method provides more straightforward solution procedure.

(ii) The modified Kudryashov method considers an arbitrary constant as the base of the exponential function; therefore, this method can generate new exact solutions of FDEs.

(iii) The modified Kudryashov method can be easily applied to handle the high order differential equations as illustrated by Zayed and Alurrfi.^[21]

Accordingly, it is fair to say that the modified Kudryashov method can be considered as one of the best techniques to extract new exact solutions of FDEs.