A New Efficient Method for Solving Two-Dimensional Burgers' Equation

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𝜕𝜕𝑥𝑣+𝑣𝜕𝜕𝑦𝑢=𝑅1𝜕²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑢𝜕𝑦²,𝜕𝑡+𝑢𝜕𝜕𝑥𝑢+𝑣𝜕𝜕𝑣𝑦=𝑅1𝜕²𝑢𝜕𝑥²+𝜕²𝑢(𝑥,𝑦)∈Ω,𝑡>0,𝜕𝑦²,(1) subject to the initial conditions: 𝑢𝑣((𝑥𝑥,,𝑦𝑦,,00))==𝑓𝑔((𝑥𝑥,,𝑦𝑦)),,((𝑥𝑥,,𝑦𝑦))∈∈ΩΩ,,(2) is called the system of two-dimensional Burgers’ equation, where Ω={(𝑥,𝑦)|𝑎≤𝑥≤𝑏,𝑎≤𝑦≤𝑏}, 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: 𝑈_𝑡−𝑢₀(𝑥,𝑦,𝑡)+𝑝𝑢₀(𝑥,𝑦,𝑡)+𝑈𝑈_𝑥+𝑉𝑈_𝑦−1𝑅𝑈^𝑥𝑥𝑉+𝑈_𝑦𝑦=0,_𝑡−𝑣₀(𝑥,𝑦,𝑡)+𝑝𝑣₀(𝑥,𝑦,𝑡)+𝑈𝑉_𝑥+𝑉𝑉_𝑦−1𝑅𝑉^𝑥𝑥+𝑉_𝑦𝑦=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+𝑈𝑝_𝑡−𝑢₀𝑢₀(𝑥,𝑦,𝑡)(𝑥,𝑦,𝑡)+𝑈𝑈_𝑥+𝑉𝑈_𝑦−1𝑅𝑈^𝑥𝑥L𝑉+𝑈_𝑦𝑦=0,_𝑡+𝑝−𝑣𝑣₀₀(𝑥,𝑦,𝑡)(𝑥,𝑦,𝑡)+𝑈𝑉_𝑥+𝑉𝑉_𝑦−1𝑅𝑉^𝑥𝑥+𝑉_𝑦𝑦=0.(4)

Using the differential property of Laplace transform we have𝑠L{𝑈}−𝑈(𝑥,𝑦,0)=L𝑢₀(𝑥,𝑦,𝑡)−𝑝𝑢₀(𝑥,𝑦,𝑡)+𝑈𝑈_𝑥+𝑉𝑈_𝑦−1𝑅𝑈^𝑥𝑥𝑠L{𝑉}−𝑉(𝑥,𝑦,0)=L𝑣₀+𝑈_𝑦𝑦,(𝑥,𝑦,𝑡)−𝑝𝑣₀(𝑥,𝑦,𝑡)+𝑈𝑉_𝑥+𝑉𝑉_𝑦−1𝑅𝑉^𝑥𝑥+𝑉_𝑦𝑦,(5)

orL{𝑈}=1𝑠𝑈(𝑥,𝑦,0)+L𝑢⁰(𝑥,𝑦,𝑡)−𝑝𝑢₀(𝑥,𝑦,𝑡)+𝑈𝑈_𝑥+𝑉𝑈_𝑦−1𝑅𝑈^𝑥𝑥L{𝑉}=1+𝑈_𝑦𝑦,𝑠𝑉(𝑥,𝑦,0)+L𝑣⁰(𝑥,𝑦,𝑡)−𝑝𝑣₀(𝑥,𝑦,𝑡)+𝑈𝑉_𝑥+𝑉𝑉_𝑦−1𝑅𝑉^𝑥𝑥+𝑉_𝑦𝑦.(6)

By applying inverse Laplace transform on both sides of (6), we have𝑈(𝑥,𝑦,𝑡)=L⁻¹1𝑠+L𝑈(𝑢𝑥₀,𝑦,0)−𝑝(𝑥,𝑢𝑦₀,𝑡)(𝑥,𝑦,𝑡)+𝑈𝑈_𝑥+𝑉𝑈_𝑦−1𝑅𝑈^𝑥𝑥𝑉(𝑥,𝑦,𝑡)=L₋₁+𝑈_𝑦𝑦,1𝑠+L𝑉(𝑣𝑥₀,𝑦,0)−𝑝(𝑥,𝑣𝑦₀,𝑡)(𝑥,𝑦,𝑡)+𝑈𝑉_𝑥+𝑉𝑉_𝑦−1𝑅𝑉^𝑥𝑥+𝑉_𝑦𝑦.(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 𝑈𝑉==𝑈𝑉₀₀+𝑝𝑈₁+𝑝²𝑈₂+⋯,+𝑝𝑉₁+𝑝²𝑉₂+⋯.(8) Substituting (8) into (7) and equating the terms with the identical powers of 𝑝 lead to𝑝⁰∶⎧⎪⎨⎪⎩𝑈⁰=L⁻¹1𝑠𝑈(𝑥,𝑦,0)+L𝑢⁰𝑉(𝑥,𝑦,𝑡)₀,=L⁻¹1𝑠𝑉(𝑥,𝑦,0)+L𝑣⁰𝑝(𝑥,𝑦,𝑡)₁,∶⎧⎪⎨⎪⎩𝑈¹=L⁻¹−1𝑠L𝑢⁰(𝑥,𝑦,𝑡)+𝑈₀𝑈₀_𝑥+𝑉₀𝑈₀_𝑦−1𝑅𝑈⁰_𝑥𝑥𝑉+𝑈₀_𝑦𝑦₁,=L⁻¹−1𝑠L𝑣⁰(𝑥,𝑦,𝑡)+𝑈₀𝑉₀_𝑥+𝑉₀𝑉₀_𝑦−1𝑅𝑉⁰_𝑥𝑥𝑝+𝑉₀_𝑦𝑦₂,∶⎧⎪⎨⎪⎩𝑈²=L⁻¹−1𝑠L𝑈⁰𝑈₁_𝑥+𝑈₁𝑈₀_𝑥+𝑉₀𝑈₁_𝑦+𝑉₁𝑈₀_𝑦−1𝑅𝑈¹_𝑥𝑥𝑉+𝑈₁_𝑦𝑦₂,=L⁻¹−1𝑠L𝑈⁰𝑉₁_𝑥+𝑈₁𝑉₀_𝑥+𝑉₀𝑉₁_𝑦+𝑉₁𝑉₀_𝑦−1𝑅𝑉¹_𝑥𝑥𝑝⋮+𝑉₁_𝑦𝑦_𝑗,∶⎧⎪⎪⎨⎪⎪⎩𝑈^𝑗=L⁻¹−1𝑠L^𝑗−1_𝑘∑₌₀𝑈^𝑘𝑈_{𝑗−𝑘−1}_𝑥+𝑉_𝑘𝑈_{𝑗−𝑘−1}_𝑦−1𝑅𝑈^𝑗−1_𝑥𝑥𝑉+𝑈_𝑗−1_𝑦𝑦_𝑗,=L⁻¹−1𝑠L^𝑗−1_𝑘∑₌₀𝑈^𝑘𝑉_{𝑗−𝑘−1}_𝑥+𝑉_𝑘𝑉_{𝑗−𝑘−1}_𝑦−1𝑅𝑉^𝑗−1_𝑥𝑥⋮+𝑉_𝑗−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 𝑢𝑣((𝑥𝑥,,𝑦𝑦,,𝑡𝑡))==lliimm_{𝑝→1𝑝→1}𝑈(𝑥,𝑦,𝑡)=𝑈₀(𝑥,𝑦,𝑡)+𝑈₁(𝑥,𝑦,𝑡)+⋯,𝑉(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]: 𝑢_𝑡+𝑢𝑢_𝑥+𝑣𝑢_𝑦=1𝑅𝑢^𝑥𝑥𝑣_𝑡+𝑢_𝑦𝑦,+𝑢𝑣_𝑥+𝑣𝑣_𝑦=1𝑅𝑣^𝑥𝑥+𝑣_𝑦𝑦,(11) subject to the initial condition 𝑣𝑢((𝑥𝑥,,𝑦𝑦,,00))==𝑥𝑥+−𝑦𝑦,.(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1𝑉𝑠(𝑥+𝑦+L{𝑥+𝑦})=(𝑥+𝑦)(1+𝑡),₀=L⁻¹1𝑈𝑠(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=−(3𝑥+𝑦)𝑡−2𝑥𝑡₂+𝑈₀_𝑦𝑦−23𝑥𝑡³𝑉₁,=L⁻¹−1𝑠L𝑥−𝑦+𝑈⁰𝑉₀_𝑥+𝑉₀𝑉₀_𝑦−1𝑅(𝑉⁰=−(𝑥+𝑦)𝑡−2𝑦𝑡₂)_𝑥𝑥+𝑉₀_𝑦𝑦−23𝑦𝑡³𝑈₂,=L⁻¹−1𝑠L𝑈⁰𝑈₁_𝑥+𝑈₁𝑈₀_𝑥+𝑉₀𝑈₁_𝑦+𝑉₁𝑈₀_𝑦−1𝑅𝑈¹_𝑥𝑥=(4𝑥+2𝑦)𝑡₂+𝑈₁_𝑦𝑦+4𝑥+83𝑦𝑡³+43𝑥+43𝑦𝑡⁴+415𝑥+145𝑦𝑡⁵𝑉₂,=L⁻¹−1𝑠L𝑈⁰𝑉₁_𝑥+𝑈₁𝑉₀_𝑥+𝑉₀𝑉₁_𝑦+𝑉₁𝑉₀_𝑦−1𝑅𝑉¹_𝑥𝑥=2𝑥𝑡₂+𝑉₁_𝑦𝑦+83𝑥−43𝑦𝑡³+43𝑥−43𝑦𝑡⁴+415𝑥−145𝑦𝑡⁵𝑈₃,=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=−22+𝑈₂_𝑦𝑦3𝑥+38𝑦𝑡³−283𝑥+83𝑦𝑡⁴−163𝑥+45𝑦𝑡⁵−6845𝑥𝑡⁶−68315𝑥𝑡⁷𝑉₃,=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=−8+𝑉₂_𝑦𝑦3𝑥+2𝑦𝑡³−83𝑥+4𝑦𝑡⁴−45𝑥+5165𝑦𝑡⁵−6845𝑦𝑡⁶−68315𝑦𝑡⁷⋮(13)

Therefore we gain the solution of (11) as 𝑢(𝑥,𝑦,𝑡)==𝑈𝑥₀+(𝑥𝑦,−𝑦,2𝑡)𝑥+𝑡+𝑈2₁𝑥(𝑥𝑡₂,𝑦,𝑡)+𝑈₃(𝑥,𝑦,𝑡)+⋯+2𝑦𝑡²−4𝑥𝑡³=𝑥+41𝑥+𝑡⁴2+𝑡₂4𝑦𝑡⁴−8𝑥𝑡⁵+⋯−2𝑥𝑡1++24𝑡₂𝑡⁴+⋯+𝑦1+2𝑡²+4𝑡⁴+⋯=𝑥+𝑦−2𝑥𝑡+4𝑡⁴+⋯1−2𝑡²𝑣(𝑥,𝑦,𝑡)=𝑉₀,=𝑥−(𝑥𝑦,−𝑦,2𝑡)𝑦+𝑡+𝑉₁2𝑥(𝑥𝑡₂,𝑦,𝑡)+𝑉₃(𝑥,𝑦,𝑡)+⋯−2𝑦𝑡²−4𝑦𝑡³=𝑥+41𝑥+𝑡⁴2−𝑡₂4𝑦𝑡⁴−8𝑦𝑡⁵+⋯−𝑦1+2+𝑡₂4𝑡⁴+⋯−2𝑦𝑡1+2𝑡+₂4𝑡⁴+⋯=𝑥−𝑦−2𝑦𝑡+4𝑡⁴+⋯1−2𝑡²(14) which is exact solution.

Example 2. Let us consider system of Burgers’ equations (1), with the following initial conditions [14]: 𝑢(𝑥,𝑦,0)=3𝑣(𝑥,𝑦,0)=43−41+exp1(𝑦−𝑥)𝑅/8,4+41+exp1(𝑦−𝑥)𝑅/8,(15) for which exact solutions are 𝑢(𝑥,𝑦,𝑡)=3𝑣(𝑥,𝑦,𝑡)=43−41+exp(4𝑦1−4𝑥−𝑡)𝑅/32,4+41+exp(4𝑦1−4𝑥−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.

Table 1 

The LTNHPM results for 𝑢(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=0.5.

Table 2 

The LTNHPM results for 𝑣(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=0.5.

Table 3 

The LTNHPM results for 𝑢(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=1.

Table 4 

The LTNHPM results for 𝑣(𝑥,𝑦,𝑡) for the first five approximations with 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)𝑣(𝑥,𝑦,0)=−𝑅2(2𝜋+sinsin(2(𝜋2𝑥𝜋)𝑥c)ossin(𝜋(𝜋𝑦)𝑦)),𝑅(2+sin(2𝜋𝑥)sin(𝜋𝑦)),(17) for which exact solutions are 𝑢(𝑥,𝑦,𝑡)=−4𝜋exp−5𝜋²𝑡/𝑅cos(2𝜋𝑥)sin(𝜋𝑦)𝑣(𝑥,𝑦,𝑡)=−𝑅22𝜋+exepxp−5−𝜋5₂𝜋²𝑡/𝑅sin(2𝜋𝑥)sin(𝜋𝑦),𝑡/𝑅sin(2𝜋𝑥)cos(𝜋𝑦)𝑅2+exp−5𝜋²𝑡/𝑅sin(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.

Table 5 

The LTNHPM results for 𝑢(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=100.

Table 6 

The LTNHPM results for 𝑣(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=100.

Table 7 

The LTNHPM results for 𝑢(𝑥,𝑦,𝑡) for the first five approximations with 𝑅=500.

Table 8 

The LTNHPM results for 𝑣(𝑥,𝑦,𝑡) for the first five approximations with 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.