Abstract
In this article, we focus on the new exact solutions of Burger’s equation by using a new technique which is known as the power index method (PIM). In this method, we choose suitable indexes of independent variables and similarity transformation so that the partial differential equation may be converted into ODE. We have obtained analytic solution of the ODE by using symbolic package Maple. We have got exact solution of Burgers’ equation by using analytic solution of ODE and similarity transformation. The proposed method has been effectively employed to find new exact solutions for the nonlinear Burgers’ equation. Finally, the proposed resulting answers are compared with the homotopy perturbation, decomposition, and variational iteration solutions.
1. Introduction
Burgers’ equation is used to label several kinds of phenomena such as turbulence and the estimated theory of stream through a shockwave wandering in a viscous fluid. Burgers [1] first developed this equation primarily to throw light on turbulence described by the interaction of two opposite effects of convection and diffusion. Hopf [2] in 1950 and Cole [3] in 1951 independently introduced a transformation to convert Burgers’ equation into a linear heat equation and solved exactly for an arbitrary initial condition. This transformation is known as Hopf–Cole transformation. Due to its widespread variety of applicability, many researchers have explored Burgers’ equation analytically and numerically. The exact solutions of Burgers’ equation are derived by using several methods like the simplest equation method with the Riccati equation [4, 5], direct integration method, and simplest equation method with the Bernoulli equation [6].
Exact solutions to nonlinear PDEs play a central role in science, especially in complex physical science since they can provide much physical information and more insight into the physical aspects of the problem [7]. Recently, several methods have been used to find exact solutions of the nonlinear PDEs (see [6, 8] for details). Lie symmetries [3] are employed to reduce the NLPDEs to nonlinear ordinary differential equations (NLODEs) that may then be reduced to quadratures [9] if sufficient symmetries exist. A variety of powerful methods, such as tanh-sech method [2], extended tanh method [3], hyperbolic function method [10], and the first integral method [11], have been used to solve different types of nonlinear systems of PDEs. Due to modern computer programs like Maple and Mathematica, these methods have become the most powerful and effective algebraic methods for finding exact solutions of PDEs. In this article, we will apply power index method to find new exact solutions of Burgers’ equation.
In this paper, twelve new exact solutions of general Burgers’ equation are derived by using the power index method. The proposed derived solutions are compared with the homotopy perturbation solution [8] and other existing exact solutions [5] (see Remark 1 for details).
2. General Burgers’ Equation
The standard form of general Burgers’ equation iswhere , are the spatial coordinate and time, is the unknown real function, is nonlinear term coefficient, and is the coefficient kinematic viscosity of fluids and it is equal to reciprocal of Reynolds number. When the Reynolds number is large, equation (1) describes shock wave behavior [12]. If the viscosity or negligible, then (1) becomes inviscid Burgers’ equation, and if , then (1) is called viscous Burgers’ equation. By applying transformation to (1), we get
3. Power Index Method
Consider nonlinear partial differential equation (2), and we want to find its exact solutions; we introduce the variable in general formand the function transform
Figure 1 shows the procedure of transformation of PDE to ODE.
Now we differentiate (3) and (4) according to (2) and we can find the relation of indexes of and in each term by the functions
We observe coefficients of and so that the partial differential equations may be transformed to ordinary differential equations. The best optimal indexes of independent variables and are chosen in such a way that only two indexes vary at a time and others are fixed constants. We continue this process with different indexes of and so that we find all well-defined transformations. The following ODE is our objective in this method.
Solve ODE (6) by using computerized symbolic package like Maple. The exact solution of (1) can be obtained from (4) after replacing unknown function .
4. New Exact Solutions of Burgers’ Equation
4.1. Test Case 1
We introduce the wave variable and transformation
Figure 2 shows the solution of (1) and compares and parameters and .
In this case, we have considered single index of variable ; using (8), we get the following ODE:
To find the optimal index value of the independent variable so that PDE is converted to ODE in new single independent variable , we have selected two values like or in which PDE can be transformed into ODE. For , equation (9) can be reduced to the form
The analytic solution of (10) is given below:
Equations (8) and (11) are used to find the general solution of Burgers’ equation (1):
4.2. Test Case 2
Now we choose of , and using (8), equation (9) can be reduced to the form
Figure 3 displays graphical illustration of the analytical solution of (16).
The solution of (13) is given by
By using (8) and (14) and the variable , we get the solution of (2):
The solution of Burgers’ equation is
4.3. Test Case 3
Now we choose the new variable and transformation ; (2) can be transformed into
Since the index power of is 2, index powers of variable must have the following relations or .
When , the variable is . Using (8), equation (17) can be reduced to the form
Figure 4 compares and parameter for fixed in (20).
The analytic solution of (18) is given below:
Using (19), the general solution of Burgers’ equation (1) is
4.4. Test Case 4
For , we take new variable and transformation which give us the following differential equation:
The solution of differential equation (21) iswhich is the same solution given in (16).
Further, we consider and the transformation , and equation (2) has the form
Figure 5 compares and parameter for fixed in (20).
In (23), to choose optimal values of index power of variable , we can choose appropriate values which are 0 and 2. When , the variable is . Using (8), equation (23) can be reduced to the form
The analytic solution of (24) is given below:
The general solution of Burgers’ equation (1) is
4.5. Test Case 5
Now we consider in general form and the transformation ; then, equation (2) can be expressed in the form
This differential equation has the following solution:and using (28), we have obtained the following wave transformation:
The above transformation gives the same solution of Burgers’ equation (1), which is given in (16).
4.6. Test Case 6
Now we consider the wave variable and transformation . We obtained the following ordinary differential equation:
The exact solution of (30) is given below:where , and are integration constants. By using (31), we have obtained exact solution of (1) of the form
4.7. Test Case 7
We introduce the wave variable and transformation
We get the following differential equation:
Using (33), we get the following solutions:
4.8. Test Case 8
We choose the variable and transformation
The following differential equation is obtained:
For particular value of in (37), we get the differential equation
The analytic solution of (38) is given by
4.9. Test Case 9
We introduce the wave variable , so that
We have obtained the differential equation
For particular value of in (41), we get the differential equation
The analytic solution of (42) is given by
Using (43), we get the solution of Burgers’ equation:
4.10. Test Case 10
We choose the wave variable , so that
Equation (2) is transformed into the differential equation
The analytic solution of (46) is given by
The solution of (2) is given by
We get the exact solution of Burgers’ equation (1):
For in transformation (45), we get the ODE
The corresponding analytic solutions are
Remark 1. Note that if we choose and initial condition in solution (49), we obtain the solution of Burgers’ equation:which was constructed in [5] (see Example 1). If we choose and initial condition in solution (49), we obtain same solution of Burgers’ equation which is given in [8] (Example 1).
Figure 6 shows the graphical diagram of the analytical solution of (49).
4.11. Test Case 11
We choose the variable , so that
Equation (2) is transformed into the differential equation
For in transformation (53), we get the ODE
The analytic solution of (55) is given by
The exact solution of (2) and Burgers’ equation (1) is given by
For in transformation (53), we get the ODE
We get the following analytic solution of the ODE:
The exact solution of Burgers’ equation (1) is given by
Figure 7 shows the graphical illustration of solution (60).
5. Conclusion
In this study, we have applied PIM for the development of theoretical solutions of Burgers’ equation. The test cases and results show that PIM can solve linear and nonlinear PDEs with parameters. We have discussed the choices of indexes of independent variables in new variable and transform and obtained exact solutions of Burgers’ equation for suitable values of indexes. We have plotted some solutions of Burgers’ equation and also showed graphical effect of parameters involved in Burgers’ equation in its exact solution.
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the study and read and approved the final manuscript.
Acknowledgments
Khalil Ahmad is grateful to Air Marshal Javed Ahmed, HI(M) (Retd), Vice Chancellor, Air University, Islamabad, Pakistan, for providing excellent research facilities.