Abstract
The conformable double Sumudu decomposition method (CDSDM) is a combination of decomposition method (DM) and a conformable double Sumudu transform. It is an approximate analytical method, which can be used to solve linear and nonlinear partial differential equations. In this work, one-dimensional conformable functional Burger’s equation has been solved by applying conformable double Sumudu decomposition. Two examples are used to illustrate the method.
1. Introduction
Burgers’ equation is represented by nonlinear wave motion with linear diffusion. These equations have appeared in the area of applied sciences such as fluid mechanics and mathematical modeling. The importance of getting the exact or approximate solutions of Burgers’ equations in mathematics and physics is still a significant problem that needs new methods to discover exact or approximate solutions. Theoretical solution of Burger equations, based on Fourier series analysis, using the appropriate initial and boundary conditions was discussed in [1]. The authors in [2] studied different exact solutions of Burger’s like equations and their classifications. The author in [3] discussed ()-dimensional time-fractional Bogoyavlensky-Konopelchenko (BK) equation by using a subequation method that is predicated on the Riccati equation. The new wave solutions of the fractional Camassa-Holm equation are studied by applying a new extended direct algebraic method (see [4]). The wave solutions of the time-fractional Benjamin-Ono equation are established by using the Jacobi elliptic function expansion method for more details (see [5]). The solitary and traveling wave solutions for the time-fractional pKP equation have been obtained by applying the new direct algebraic method by means of the conformable derivative see [6]. Many powerful methods were used to solve nonlinear partial differential equations, such as the fractional Sumudu transform [7, 8]. The exact solutions for time-fractional Burgers’ equations were studied by the first integral method [9]. The authors in [10] employed the generalized two-dimensional differential transform method (DTM) and obtained the solution for the coupled Burgers’ equations with space- and time-fractional derivatives. In recent time, the new definition of the conformable derivative has been introduced for more specifics; we refer to [11–13]. Furthermore, the authors in [11, 14] are solved the fractional differential equations by conformable Laplace transform technique. In [15], the researchers applied the conformable double Laplace transform method and obtained the solution for the fractional partial differential equations. The conformable double Laplace decomposition method [16] has been proposed to obtain exact and approximate solutions of regular and singular one-dimensional conformable fractional coupled burgers’ equations. The exact solutions of the time-fractional Burgers’ equations were established by using the first integral method (see [9]). We introduce a new method called the conformable double Sumudu decomposition method (CDSDM) for solving the nonlinear equations in the present paper. The proposed method is an elegant combination of the conformable double Sumudu transform method and the decomposition method. This paper considers the effectiveness of the conformable double Sumudu decomposition method (CDSDM) in solving regular and singular one-dimensional conformable fractional coupled burgers’ equations, and the advantage of this method is easy to apply for solving fractional nonlinear partial differential equations. In the following, we list some definitions from the conformable derivatives which are used further in this paper.
Definition 1 (see [13, 17–19]). Given a function , then the conformable fractional derivative of of order is defined by
Definition 2 (see [20]). Given a function . Then, the conformable space fractional partial derivative of order a function is defined as follows:
Definition 3 (see [20]). Given a function . Then, the conformable time fractional partial derivative of order a function is defined as follows:
The following are the conformable fractional derivative of certain functions:
Example 4 (see [16]). We introduce the conformable double Sumudu transform and solve the conformable fractional coupled burgers’ equation by using this new definition.
Definition 5. Over the set of function the conformable Sumudu transform is given by
Definition 6. Let be function we defined conformable double Sumudu transform of function , is given by or where represent the conformable double Sumudu transform. The relation between the usual and the conformable double Sumudu transforms is given below.
Example 7. The double conformable Sumudu transform for certain functions are given by
Now, we present the conditions for the existence of the conformable double Sumudu transform:
If is an exponential order and as , and if , then for all and ,
and we write
equivalently,
where and The function does not grow faster than as , .
Theorem 8. The function is defined on and and of exponential order ; then the conformable Sumudu transform of exists for all .
Proof. From equation (29), we obtain For , from equation (13), we get or Now the next result deals with the multiplication of the function by .
Theorem 9. If conformable double Sumudu transform of the function is given by , then the conformable double Sumudu transforms of the functions are given by respectively.
Proof. By using the definition of conformable double Sumudu transform for equation (17), we get
by calculating the partial derivative inside brackets, we can easily get
substituting equation (23) into equation (22), we obtain
by applying equation (7), we get equation (17)
For equation (19), by using second partial derivative for with respect to and definition of conformable double Sumudu transform, we have
Applying equations (7) and (19), we obtain equation (19)
Similarly, in the same way, one can prove equations (18), (20), and (21).
The conformable double Sumudu transform of the first partial derivative with respect to and is given by
and the conformable double Sumudu transform of second partial derivative concerning and is denoted by
The following theorem gives the conformable double Sumudu transform of the fractional partial derivatives and .
Theorem 10. If conformable double Sumudu transform of the fractional partial derivatives and are given by respectively.
Proof. By using the definition of double Sumudu transform, we have we calculate the partial derivative inside brackets as follows: substituting equation (33) into equation (32), we get applying equation (7), we obtain and on using equation (28), we have In a similar manner, one can prove equation (31).
2. One-Dimensional Fractional Coupled Burgers’ Equation
In this section, the solution of regular and irregular one-dimensional conformable fractional coupled burgers’ equations is discussed by using conformable double Sumudu decomposition methods (CDSDM). By taking and in the following problems, one can get the problems which were discussed in [21]:
The first problem is as follows: we consider the following one-dimensional conformable fractional coupled burgers’ equations: appropriate initial conditions where , , , and are known functions and , and are arbitrary constants depending on the system parameters. Employing the conformable double Sumudu transform for both sides of equation (37) and conformable single Sumudu transform for equation (38), we obtain
The solution of one-dimensional conformable fractional coupled burgers’ equations is defined by the infinite series as follows:
The nonlinear operators , , and defined by Adomian polynomials are given as follows:
The few components of the Adomian polynomials are given as follows:
Applying the inverse double Sumudu transform to equation (39), equation (40) and using equation (42), we get
For the case , we set
Now, we can obtain the following general form:
Here, we provide double inverse Sumudu transform with respect to and existing for each term in the right-hand side of the above equations. To show the method of coupled one-dimensional conformable fractional coupled burgers’ equations, the following example is considered.
Example 11. Consider the following homogeneous form of a one-dimensional conformable fractional coupled burgers’ equations:
with initial condition
According to equations (47), (48), and (49), the first three terms of the Sumudu decomposition series are derived as follows:
and so on for other components. Using equation (41), the series solutions are therefore given by
Therefore, the exact solution takes the form of
By taking and , the fractional solution become
The exact solution of equation (50) for = is shown in Figure 1 (see also Figures 2(a)–2(c)) for .
The second problem is as follows: we consider the singular one-dimensional conformable fractional coupled burgers’ equations of the form:
with initial conditions
where is a conformable Bessel operator and , , and are real constants. In order to obtain the solution of equation (56). First, multiplying equation (56) by to obtain the following:
Second, applying the conformable double Sumudu transform on both sides of equation (58) and after using the differentiation property of conformable Sumudu transform, we get
On using Theorems 9 and 10, we have
and arranging equation (60), we obtain
Third, by applying the integral for both sides of equation (61) from to with respect to , we have
Applying the inverse double Sumudu transform, to both sides of equation (62) yields the following:
The conformable double Sumudu decomposition method proposes a series solution of the functions and by the infinite series
where , , and are given by equation (42). Consequently,
Assume that double inverse Sumudu transform with respect and exists for each term in the right-hand side of equations (65), (66), and (67).
Example 12. Consider singular one-dimensional conformable fractional coupled burgers’ equations subject to Using the above steps, we get where the nonlinear terms , , and are denoted in equations (43), (44), and (45), respectively. In view of the recursive relations (65), (66), and (67), we obtained other components as follows: So the solution of singular one-dimensional conformable fractional coupled burgers’ equations in the series form is given by Therefore, the exact solution is obtained as follows: By using and , the fractional solution becomes The exact solution of equation (68) for = is shown in Figure 3 (see also Figures 4(a)–4(c)) for .
3. Conclusion
In this paper, we introduce the conformable double Sumudu decomposition method and give the conformable double Sumudu transform of the fractional partial derivatives. Then we discuss the solution of regular and irregular one-dimensional conformable fractional coupled burgers’ equations by using a conformable double Sumudu decomposition method. Further properties and problems can be considered by using conformable double Sumudu decomposition. We sketched all the graphs by using Matlab. In the future, we are going to generalize conformable Laplace transform to solve more general nonlinear partial differential equations.
Data Availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
The authors read and approved the final manuscript.
Acknowledgments
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this research group (No. RG-1440-030).