Abstract

The variational homotopy perturbation method VHPM is used for solving -dimensional Burgers’ system. Some examples are examined to validate that the method reduced the calculation size, treating the difficulty of nonlinear term and the accuracy.

1. Introduction

The variational iteration method VIM and the homotopy perturbation method HPM were proposed by He in [1–6]. Many researchers used these methods in a variety of scientific fields of partial differential equations PDEs including Burgers’ equation which arises in many of physically important phenomena [7–9]. It was shown that the methods are stronger than other techniques such as the Adomian decomposition method [10–18]. In our work -dimensional Burgers’ equation is solved by the variational homotopy perturbation method VHPM which is combination of VIM and HPM. The VHPM was proposed in [19–21]. Vector Burgers’ system is given by [22]where are the velocity components and is the kinematic viscosity. is time and and are Equation (1) can be written as

2. Variational Iteration Method

According to the variational iteration method [2, 3, 10–14] we can write the correction functional for (3) as where , , is a general Lagrangian multiplier which can be found via variational theory, and are restricted variation which means . The solution is given by

3. Homotopy Perturbation Method

Applying HPM according to [4–6, 15–17] for (3), we construct the following homotopy: or where , is an embedding parameter, while are initial approximations of (3). Assume the solution of (3) has the form Now, substituting from (8) in (7) and comparing coefficients of terms with identical powers of we get The solution of (7) is

4. Variational Homotopy Perturbation Method

Consider (3) according to [19–21]. In HPM, assume that the solution of (3) has the form From (11), (3) can be written as In VIM, from the correction functional for (12) we can write where , ; from (11) in (13) and by comparing the coefficients of like powers of , we get The approximations solution is given by To demonstrate the efficiency of the methods we have solved some examples by VHPM as ()-dimensional, ()-dimensional, ()-dimensional, and 2-dimensional. Then we can generalize it for ()-dimensional or -dimensional.

5. Application

Example 1. Consider ()-dimensional Burgers’ equation [17] with the initial condition The correction functional for (16) isThe general Lagrangian multiplier can be found as follows: Then, .
Equation (11) can be written as Applying VHPM, we have Comparing the coefficient of like powers of , we get The approximations solution is given by Exact solution is ().
The results are in Table 1.

Table 1 
Comparison of VHPM solutions with exact solution at (Example 1).

Example 2. Consider ()-dimensional Burgers’ equation [17] with the initial condition As above, we have Comparing the coefficient of like powers of , we get The approximations solution is given by Exact solution is ().
The results are in Table 2.

Table 2 
Comparison of VHPM solutions with exact solution at and (Example 2).

Example 3. Consider ()-dimensional Burgers’ equation [17] with the initial condition We have Comparing the coefficient of like powers of , we get The approximations solution is given by Exact solution is ().
The results are in Table 3.

Table 3 
Comparison of VHPM solutions with exact solution at , , and (Example 3).

Example 4. Consider two-dimensional Burgers’ equations [23] with the initial conditions: The correction functional for (34) isThe general Lagrangian multipliers are

Equation (11) can be written as By VHPM, we have Comparing the coefficient of like powers of , we get The approximations solution is given by Exact solution ; .

The results are in Table 4.

Table 4 
Comparison of VHPM solutions with exact solution at , , and (Example 4).

6. Conclusion

In this work, the approximate solutions of -dimensional Burgers’ equations are obtained by combination of two powerful methods VIM and HPM in VHPM. The examples have shown the efficiency and accuracy of the VHPM; it reduces the size of computation without the restrictive assumption to handle nonlinear terms and it gives the solutions rapidly.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This paper was funded by King Abdulaziz City for Science and Technology (KACST) in Saudi Arabia. The authors therefore thank them for their full collaboration.