_{1}

^{*}

Adomian decomposition method is presented as a method for the solution of the Burger’s equation, a popular PDE model in the fluid mechanics. The method is computationally simple in application. The approximate solution is obtained by considering only the first two terms of the decomposition in this paper. Numerical experimentation shows accuracy of a minimum error of order five for various space steps and coefficient of kinematic viscosity. The method is considered high in accuracy.

Burger’s equation is a fundamental partial differential equation in fluid mechanics. It is also a very important model encountered in several areas of applied mathematics such as heat conduction, acoustic waves, gas dynamics and traffic flow [

A decomposition method which provides convergent solutions to nonlinear stochastic operator equations was developed in [

In this paper, presentation of a numerical method for the solution of the nonlinear reaction-diffusion Burger’s equation using Adomian decomposition is made. The organization of this paper is as follows: in Section 2 the theoretical approach is presented. In Section 3 the Adomian’s polynomials for the Burger’s equation are determined. In Section 4 computational results for the Burger’s equation using Adomian’s decomposition are presented while conclusion is presented in Section 5.

Consider the Burger’s equation

Subject to initial condition

And boundary conditions:

Defining the operators:

Equation (1) can be written as:

Solving Equation (6) for

and

where

Operating both sides of Equations (7) and (8) with the inverse operators (9) we obtain:

where

and

Adding Equations (10) and (11) and dividing by 2, gives the canonical form:

The parameterized form of (14) is

where

According to [_{u} are:

A_{n} are the Adomian’s polynomials which can be generated for all types of nonlinearities [

Substituting Equations (17) and (18) into Equation (15) gives

Expanding both sides of Equation (19) gives

By comparing coefficients of both hand sides of Equation (20), it is obtained that:

From which we establish the recursive relation

The Adomian’s polynomials,

Substituting Equation (17) into Equation (18) and expanding the RHS gives

From Equation (22), we establish that the Adomian’s polynomials have the following forms:

The Adomian’s polynomials for Equation (1) are obtained from the recurrent relation

We present numerical results to illustrate the effectiveness of the proposed method. Consider Burger’s Equation (1) with the following initial and boundary conditions.

The exact solution of Equation (1) with the above conditions was given in [

Absolute errors of the method were computed by use of the formula:

where the numerical solution at the grid point

Relative errors were computed by use of the formula:

where the numerical solution at the grid point

In this paper, Adomian decomposition method is used to solve the burger’s equation numerically. From

x | Adomian | Theoretical solution | Error | Relative Errors |
---|---|---|---|---|

0.1 | 0.0006610924400 | 0.0003891849100 | 2.719075297 × 10^{−4 } | 2.718017485 × 10^{−4 } |

0.2 | 0.0013184569550 | 0.0008129882596 | 5.054686954 × 10^{−4 } | 5.050580892 × 10^{−4 } |

0.3 | 0.001968724990 | 0.0013210998120 | 6.47625178 × 10^{−4 } | 6.467707292 ×10^{−4 } |

0.4 | 0.002596743099 | 0.0020106848070 | 5.86058292 × 10^{−4 } | 5.84882278 × 10^{−4 } |

0.5 | 0.0031486447293 | 0.0031366545250 | 1.9922768 × 10^{−5 } | 1.986047255 × 10^{−5 } |

0.6 | 0.003546100003 | 0.005799752849 | 2.253652846 × 10^{−3 } | 2.240657585 × 10^{−3 } |

0.7 | 0.003616277199 | 0.03291862860 | 2.930235148 × 10^{−2 } | 2.836849937 × 10^{−2 } |

0.8 | 0.003425426033 | −0.006782588856 | 1.020801489 × 10^{−2 } | 1.027772447 × 10^{−2 } |

x | Adomian | Theoretical Solution | Error | Relative Errors |
---|---|---|---|---|

0.2 | 0.01342133395 | 0.008119261273 | 5.302072677 × 10^{−3 } | 5.259370475 × 10^{−3 } |

0.4 | 0.02618512105 | 0.02006671613 | 6.11840492 × 10^{−3 } | 5.998043877 × 10^{−3 } |

0.6 | 0.03577004231 | 0.05766474009 | 2.189469778 × 10^{−2 } | 2.070098109 × 10^{−2 } |

0.8 | 0.03132012403 | −0.06901554553 | 1.003356696 × 10^{−1 } | 1.077737325 × 10^{−1 } |

can be improved by considering more terms in the solution approximation. All computations were carried out using Maple 15.

Iyakino P.Akpan, (2015) Adomian Decomposition Approach to the Solution of the Burger’s Equation. American Journal of Computational Mathematics,05,329-335. doi: 10.4236/ajcm.2015.53030