Abstract

We introduce a new hybrid of the Laplace transform method and new homotopy perturbation method (LTNHPM) that efficiently solves nonlinear two-dimensional Burgers’ equation. Three examples are given to demonstrate the efficiency of the new method.

1. Introduction

The system of partial differential equation of the following form 𝜕𝑢𝜕𝑡+𝑢𝜕𝑣𝜕𝑥+𝑣𝜕𝑢=1𝜕𝑦𝑅𝜕2𝑢𝜕𝑥2+𝜕2𝑢𝜕𝑦2,𝜕𝑢𝜕𝑡+𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑣=1𝜕𝑦𝑅𝜕2𝑢𝜕𝑥2+𝜕2𝑢𝜕𝑦2,(𝑥,𝑦)∈Ω,𝑡>0,(1) subject to the initial conditions: 𝑣𝑢(𝑥,𝑦,0)=𝑓(𝑥,𝑦),(𝑥,𝑦)∈Ω,(𝑥,𝑦,0)=𝑔(𝑥,𝑦),(𝑥,𝑦)∈Ω,(2) is called the system of two-dimensional Burgers’ equation, where Ω={(𝑥,𝑦)|ğ‘Žâ‰¤ğ‘¥â‰¤ğ‘,ğ‘Žâ‰¤ğ‘¦â‰¤ğ‘}, 𝑢(𝑥,𝑦,𝑡) and 𝑣(𝑥,𝑦,𝑡) are the velocity components to be determined, 𝑓 and 𝑔 are known functions, and 𝑅 is the Reynolds number. The Burgers model of turbulence is a very important fluid dynamic model, and the study of this model and the theory of shock waves have been considered by many authors, both to obtain a conceptual understanding of a class of physical flows and for testing various numerical methods. The mathematical properties of Burgers’ equation have been studied by Burgers [1]. Nonlinear phenomena play a crucial role in applied mathematics and physics. The importance of obtaining the exact or approximate solutions of PDEs in physics and mathematics is still a hot topic as regards seeking new methods for obtaining new exact or approximate solutions [2–5]. For that purpose, different methods have been put forward for seeking various exact solutions of multifarious physical models described using nonlinear PDEs. A well-known model was first introduced by Bateman [6], who found its steady solutions, descriptive of certain viscous flows. It was later proposed by Burgers [1] as one of a class of equations describing mathematical models of turbulence. In the context of gas dynamics, it was discussed by Hopf [7] and Cole [8]. They also illustrated independently that the Burgers equation can be solved exactly for an arbitrary initial condition. Benton and Platzman [9] have surveyed the analytical solutions of the one-dimensional Burgers equation. It can be considered as a simplified form of the Navier-Stokes equation [10] due to the form of the nonlinear convection term and the occurrence of the viscosity term. The numerical solution of the Burgers equation is of great importance due to the application of the equation in the approximate theory of flow through a shock wave, travelling in a viscous fluid [8] and in the Burgers model of turbulence [11]. It can be solved analytically for arbitrary initial conditions [7]. Numerical methods such as finite difference, finite element, and classical ones like Fourier series, Fourier integral, and Laplace transformation commonly used for solving these methods either need a lot of computations and have less convergence speed and accuracy or solve only certain types of problems. Therefore, science and engineering researchers attempt to propose new methods for solving functional equations.

In this paper, we propose a new hybrid of Laplace transform method and new homotopy perturbation method [12] to obtain exact and numerical solutions of the system of two-dimensional Burgers’ equation. Finally, three examples are given to illustrate the proposed approach.

2. Analysis of the Method

For solving system of two-dimensional Burgers’ equation by LTNHPM, we construct the following homotopy: 𝑈𝑡−𝑢0𝑢(𝑥,𝑦,𝑡)+𝑝0(𝑥,𝑦,𝑡)+𝑈𝑈𝑥+𝑉𝑈𝑦−1𝑅𝑈𝑥𝑥+𝑈𝑦𝑦𝑉=0,𝑡−𝑣0𝑣(𝑥,𝑦,𝑡)+𝑝0(𝑥,𝑦,𝑡)+𝑈𝑉𝑥+𝑉𝑉𝑦−1𝑅𝑉𝑥𝑥+𝑉𝑦𝑦=0,(3) where 𝑝∈[0,1] is an embedding parameter and 𝑢0(𝑥,𝑦,𝑡) and 𝑣0(𝑥,𝑦,𝑡) are initial approximation of solution of (1).

By applying Laplace transform on both sides of (3), we have L𝑈𝑡−𝑢0𝑢(𝑥,𝑦,𝑡)+𝑝0(𝑥,𝑦,𝑡)+𝑈𝑈𝑥+𝑉𝑈𝑦−1𝑅𝑈𝑥𝑥+𝑈𝑦𝑦L𝑉=0,𝑡−𝑣0𝑣(𝑥,𝑦,𝑡)+𝑝0(𝑥,𝑦,𝑡)+𝑈𝑉𝑥+𝑉𝑉𝑦−1𝑅𝑉𝑥𝑥+𝑉𝑦𝑦=0.(4)

Using the differential property of Laplace transform we have𝑢𝑠L{𝑈}−𝑈(𝑥,𝑦,0)=L0𝑢(𝑥,𝑦,𝑡)−𝑝0(𝑥,𝑦,𝑡)+𝑈𝑈𝑥+𝑉𝑈𝑦−1𝑅𝑈𝑥𝑥+𝑈𝑦𝑦,𝑣𝑠L{𝑉}−𝑉(𝑥,𝑦,0)=L0𝑣(𝑥,𝑦,𝑡)−𝑝0(𝑥,𝑦,𝑡)+𝑈𝑉𝑥+𝑉𝑉𝑦−1𝑅𝑉𝑥𝑥+𝑉𝑦𝑦,(5)

or1L{𝑈}=𝑠𝑢𝑈(𝑥,𝑦,0)+L0𝑢(𝑥,𝑦,𝑡)−𝑝0(𝑥,𝑦,𝑡)+𝑈𝑈𝑥+𝑉𝑈𝑦−1𝑅𝑈𝑥𝑥+𝑈𝑦𝑦,1L{𝑉}=𝑠𝑣𝑉(𝑥,𝑦,0)+L0𝑣(𝑥,𝑦,𝑡)−𝑝0(𝑥,𝑦,𝑡)+𝑈𝑉𝑥+𝑉𝑉𝑦−1𝑅𝑉𝑥𝑥+𝑉𝑦𝑦.(6)

By applying inverse Laplace transform on both sides of (6), we have𝑈(𝑥,𝑦,𝑡)=L−11𝑠𝑢𝑈(𝑥,𝑦,0)+L0𝑢(𝑥,𝑦,𝑡)−𝑝0(𝑥,𝑦,𝑡)+𝑈𝑈𝑥+𝑉𝑈𝑦−1𝑅𝑈𝑥𝑥+𝑈𝑦𝑦,𝑉(𝑥,𝑦,𝑡)=L−11𝑠𝑣𝑉(𝑥,𝑦,0)+L0𝑣(𝑥,𝑦,𝑡)−𝑝0(𝑥,𝑦,𝑡)+𝑈𝑉𝑥+𝑉𝑉𝑦−1𝑅𝑉𝑥𝑥+𝑉𝑦𝑦.(7)

According to the HPM, we use the embedding parameter 𝑝 as a small parameter and assume that the solutions of (7) can be represented as a power series in 𝑝 as 𝑈=𝑈0+𝑝𝑈1+𝑝2𝑈2+⋯,𝑉=𝑉0+𝑝𝑉1+𝑝2𝑉2+⋯.(8) Substituting (8) into (7) and equating the terms with the identical powers of 𝑝 lead to𝑝0âˆ¶âŽ§âŽªâŽ¨âŽªâŽ©ğ‘ˆ0=L−11𝑠𝑢𝑈(𝑥,𝑦,0)+L0,𝑉(𝑥,𝑦,𝑡)0=L−11𝑠𝑣𝑉(𝑥,𝑦,0)+L0,𝑝(𝑥,𝑦,𝑡)1âˆ¶âŽ§âŽªâŽ¨âŽªâŽ©ğ‘ˆ1=L−1−1𝑠L𝑢0(𝑥,𝑦,𝑡)+𝑈0𝑈0𝑥+𝑉0𝑈0𝑦−1𝑅𝑈0𝑥𝑥+𝑈0𝑦𝑦,𝑉1=L−1−1𝑠L𝑣0(𝑥,𝑦,𝑡)+𝑈0𝑉0𝑥+𝑉0𝑉0𝑦−1𝑅𝑉0𝑥𝑥+𝑉0𝑦𝑦,𝑝2âˆ¶âŽ§âŽªâŽ¨âŽªâŽ©ğ‘ˆ2=L−1−1𝑠L𝑈0𝑈1𝑥+𝑈1𝑈0𝑥+𝑉0𝑈1𝑦+𝑉1𝑈0𝑦−1𝑅𝑈1𝑥𝑥+𝑈1𝑦𝑦,𝑉2=L−1−1𝑠L𝑈0𝑉1𝑥+𝑈1𝑉0𝑥+𝑉0𝑉1𝑦+𝑉1𝑉0𝑦−1𝑅𝑉1𝑥𝑥+𝑉1𝑦𝑦,â‹®ğ‘î‚î‚‡î‚î‚‡ğ‘—âˆ¶âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©ğ‘ˆğ‘—=L−1−1𝑠L𝑗−1∑𝑘=0𝑈𝑘𝑈𝑗−𝑘−1𝑥+𝑉𝑘𝑈𝑗−𝑘−1𝑦−1𝑅𝑈𝑗−1𝑥𝑥+𝑈𝑗−1𝑦𝑦,𝑉𝑗=L−1−1𝑠L𝑗−1∑𝑘=0𝑈𝑘𝑉𝑗−𝑘−1𝑥+𝑉𝑘𝑉𝑗−𝑘−1𝑦−1𝑅𝑉𝑗−1𝑥𝑥+𝑉𝑗−1𝑦𝑦,⋮(9)Suppose that the initial approximation has the form 𝑈(𝑥,𝑦,0)=𝑢0(𝑥,𝑦,𝑡)=𝑓(𝑥,𝑦) and 𝑉(𝑥,𝑦,0)=𝑣0(𝑥,𝑦,𝑡)=𝑔(𝑥,𝑦); therefore the exact solution may be obtained as follows 𝑢(𝑥,𝑦,𝑡)=lim𝑝→1𝑈(𝑥,𝑦,𝑡)=𝑈0(𝑥,𝑦,𝑡)+𝑈1(𝑥,𝑦,𝑡)+⋯,𝑣(𝑥,𝑦,𝑡)=lim𝑝→1𝑉(𝑥,𝑦,𝑡)=𝑉0(𝑥,𝑦,𝑡)+𝑉1(𝑥,𝑦,𝑡)+⋯.(10)

3. Examples

Example 1. Consider the following homogeneous form of a coupled Burgers equation [13]: 𝑢𝑡+𝑢𝑢𝑥+𝑣𝑢𝑦=1𝑅𝑢𝑥𝑥+𝑢𝑦𝑦,𝑣𝑡+𝑢𝑣𝑥+𝑣𝑣𝑦=1𝑅𝑣𝑥𝑥+𝑣𝑦𝑦,(11) subject to the initial condition 𝑣𝑢(𝑥,𝑦,0)=𝑥+𝑦,(𝑥,𝑦,0)=𝑥−𝑦.(12) The exact solution of this equation is 𝑢(𝑥,𝑦,𝑡)=(𝑥+𝑦−2𝑥𝑡)/(1−2𝑡2) and 𝑣(𝑥,𝑦,𝑡)=(𝑥−𝑦−2𝑦𝑡)/(1−2𝑡)2.
Starting with 𝑈(𝑥,𝑦,0)=𝑢0=𝑥+𝑦, 𝑉(𝑥,𝑦,0)=𝑣0=𝑥−𝑦 and using (9), we obtain

𝑈0=L−11𝑠𝑉(𝑥+𝑦+L{𝑥+𝑦})=(𝑥+𝑦)(1+𝑡),0=L−11𝑠𝑈(𝑥−𝑦+L{𝑥−𝑦})=(𝑥−𝑦)(1+𝑡),1=L−1−1𝑠L𝑥+𝑦+𝑈0𝑈0𝑥+𝑉0𝑈0𝑦−1𝑅𝑈0𝑥𝑥+𝑈0𝑦𝑦=−(3𝑥+𝑦)𝑡−2𝑥𝑡2−23𝑥𝑡3,𝑉1=L−1−1𝑠L𝑥−𝑦+𝑈0𝑉0𝑥+𝑉0𝑉0𝑦−1𝑅(𝑉0)𝑥𝑥+𝑉0𝑦𝑦=−(𝑥+𝑦)𝑡−2𝑦𝑡2−23𝑦𝑡3,𝑈2=L−1−1𝑠L𝑈0𝑈1𝑥+𝑈1𝑈0𝑥+𝑉0𝑈1𝑦+𝑉1𝑈0𝑦−1𝑅𝑈1𝑥𝑥+𝑈1𝑦𝑦=(4𝑥+2𝑦)𝑡2+84𝑥+3𝑦𝑡3+434𝑥+3𝑦𝑡4+4415𝑥+𝑦𝑡155,𝑉2=L−1−1𝑠L𝑈0𝑉1𝑥+𝑈1𝑉0𝑥+𝑉0𝑉1𝑦+𝑉1𝑉0𝑦−1𝑅𝑉1𝑥𝑥+𝑉1𝑦𝑦=2𝑥𝑡2+834𝑥−3𝑦𝑡3+434𝑥−3𝑦𝑡4+4415𝑥−𝑦𝑡155,𝑈3=L−1−1𝑠L𝑈0𝑈2𝑥+𝑈1𝑈1𝑥+𝑈2𝑈0𝑥+𝑉0𝑈2𝑦+𝑉1𝑈1𝑦+𝑉2𝑈0𝑦−1𝑅𝑈2𝑥𝑥+𝑈2𝑦𝑦=−2238𝑥+3𝑦𝑡3−2838𝑥+3𝑦𝑡4−1634𝑥+5𝑦𝑡5−6845𝑥𝑡6−68315𝑥𝑡7,𝑉3=L−1−1𝑠L𝑈0𝑉2𝑥+𝑈1𝑉1𝑥+𝑈2𝑉0𝑥+𝑉0𝑉2𝑦+𝑉1𝑉1𝑦+𝑉2𝑉0𝑦−1𝑅𝑉2𝑥𝑥+𝑉2𝑦𝑦8=−3𝑡𝑥+2𝑦3−83𝑡𝑥+4𝑦4−45𝑥+56𝑦𝑡155−6845𝑦𝑡6−68315𝑦𝑡7⋮(13)

Therefore we gain the solution of (11) as 𝑢(𝑥,𝑦,𝑡)=𝑈0(𝑥,𝑦,𝑡)+𝑈1(𝑥,𝑦,𝑡)+𝑈3(𝑥,𝑦,𝑡)+⋯=𝑥+𝑦−2𝑥𝑡+2𝑥𝑡2+2𝑦𝑡2−4𝑥𝑡3+4𝑥𝑡4+4𝑦𝑡4−8𝑥𝑡5+⋯=𝑥1+2𝑡2+4𝑡4+⋯+𝑦1+2𝑡2+4𝑡4+⋯−2𝑥𝑡1+2𝑡2+4𝑡4=+⋯𝑥+𝑦−2𝑥𝑡1−2𝑡2,𝑣(𝑥,𝑦,𝑡)=𝑉0(𝑥,𝑦,𝑡)+𝑉1(𝑥,𝑦,𝑡)+𝑉3(𝑥,𝑦,𝑡)+⋯=𝑥−𝑦−2𝑦𝑡+2𝑥𝑡2−2𝑦𝑡2−4𝑦𝑡3+4𝑥𝑡4−4𝑦𝑡4−8𝑦𝑡5+⋯=𝑥1+2𝑡2+4𝑡4+⋯−𝑦1+2𝑡2+4𝑡4+⋯−2𝑦𝑡1+2𝑡2+4𝑡4=+⋯𝑥−𝑦−2𝑦𝑡1−2𝑡2(14) which is exact solution.

Example 2. Let us consider system of Burgers’ equations (1), with the following initial conditions [14]: 3𝑢(𝑥,𝑦,0)=4−14,𝑣31+exp(𝑦−𝑥)𝑅/8(𝑥,𝑦,0)=4+14,1+exp(𝑦−𝑥)𝑅/8(15) for which exact solutions are 3𝑢(𝑥,𝑦,𝑡)=4−14,𝑣31+exp(4𝑦−4𝑥−𝑡)𝑅/32(𝑥,𝑦,𝑡)=4+14.1+exp(4𝑦−4𝑥−𝑡)𝑅/32(16) To solve system (1) by LTNHPM, following the same procedure discussed in Section 2 and Example 1, we obtain the iterative relations (9); in this example we take initial approximations (15). The accuracy of LTNHPM for the system of two-dimensional Burgers’ equation agrees good with the exact solution, and absolute errors are very small for the present choice of 𝑥,𝑦, and 𝑡. These results are listed in Tables 1, 2, 3, and 4 for 𝑅=0.5 and 𝑅=1.

Example 3. Let us consider system of Burgers’ equations (8), with the following initial conditions [14]: 𝑢(𝑥,𝑦,0)=−4𝜋cos(2𝜋𝑥)sin(𝜋𝑦),𝑅(2+sin(2𝜋𝑥)sin(𝜋𝑦))𝑣(𝑥,𝑦,0)=−2𝜋sin(2𝜋𝑥)cos(𝜋𝑦),𝑅(2+sin(2𝜋𝑥)sin(𝜋𝑦))(17) for which exact solutions are 𝑢(𝑥,𝑦,𝑡)=−4𝜋exp−5𝜋2𝑡/𝑅cos(2𝜋𝑥)sin(𝜋𝑦)𝑅2+exp−5𝜋2,𝑡/𝑅sin(2𝜋𝑥)sin(𝜋𝑦)𝑣(𝑥,𝑦,𝑡)=−2𝜋exp−5𝜋2𝑡/𝑅sin(2𝜋𝑥)cos(𝜋𝑦)𝑅2+exp−5𝜋2.𝑡/𝑅sin(2𝜋𝑥)sin(𝜋𝑦)(18) To solve system (1) by LTNHPM, following the same procedure discussed in Section 2 and Example 1, we obtain the iterative relations (9); in this example we take initial approximations (17). The accuracy of LTNHPM for the system of two-dimensional Burgers’ equation agrees good with the exact solution, and absolute errors are very small for the present choice of 𝑥,𝑦, and 𝑡. These results are listed in Tables 5, 6, 7, and 8 for 𝑅=100 and 𝑅=500.

4. Conclusions

In this work, we considered a new hybrid of Laplace transform method and homotopy perturbation method (LTNHPM) for solving system of two-dimensional Burgers’ equation. Using this method we obtained new efficient relations to solve these systems. New method is a powerful straightforward method. The LTNHPM is apt to be utilized as an alternative approach to current techniques being employed to a wide variety of mathematical problems.