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In this paper, the system of Burgers’ equations is solved by the optimal homotopy asymptotic method with Daftardar-Jafari polynomials OHAM-DJ. Two numerical examples are illustrated the efficient of this methods for solving the system of Burgers’ equations.

The Burgers equation was first presented by Bateman [

Recently, the OHAM was proposed by Marinca and Herisaun [

with the initial conditions:

and the boundary conditions:

where

This paper is organized into three sections. In Section 2 methodology of OHAM-DJ is presented. In Section 3 application of this method is solved and absolute error of approximate solutions of proposed method is com- pared with exact solutions. In all cases the proposed method yields better results.

Consider (1.1) and let

where

According to the basic idea of OHAM [

which satisfies

where

where,

The nonlinear terms

are decomposed as

where

Then,

Substiting, (2.5),(2.6), (2.7) and (2.9) into (2.3), and comparing the coefficients of like powers of p, we get

The convergence of (2.6) depend upon the auxiliary constants

Substituting (2.11) into (1.1) we get the residuals

In this section, two numerical examples are used to prove the efficiency and the accuracy of the method which we proposed for the system of Burgers’ equations.

Consider the system of two dimensional of Burgers’ equations with the initial conditions as following [

with the initial conditions:

The exact solutions are

Accordance to the methodology of OHAM-DJ,

Their solutions are

Then,

By substituting (3.6) into (3.1) we get the residuals and using the optimization method we have computed that

t | ||||||
---|---|---|---|---|---|---|

0.01 | 0.6171782017 | 0.6171782030 | 0.8828217983 | 0.8828217970 | ||

0.02 | 0.6171587471 | 0.6171587492 | 0.8828412529 | 0.8828412508 | ||

0.03 | 0.6171392929 | 0.6171392955 | 0.8828607071 | 0.8828607045 | ||

0.04 | 0.6171198391 | 0.6171198418 | 0.8828801609 | 0.8828801582 | ||

0.05 | 0.6171003857 | 0.6171003880 | 0.8828996143 | 0.8828996120 | ||

0.06 | 0.6170809326 | 0.6170809343 | 0.8829190674 | 0.8829190657 | ||

0.07 | 0.6170614799 | 0.6170614806 | 0.8829385201 | 0.8829385194 | ||

0.08 | 0.6170420276 | 0.6170420268 | 0.8829579724 | 0.8829579732 | ||

0.09 | 0.6170225757 | 0.6170225731 | 0.8829774243 | 0.8829774269 | ||

0.10 | 0.6170031242 | 0.6170031194 | 0.8829968758 | 0.8829968806 |

We consider the following two-dimensional Burgers’ equations [

On square domain

for which the exact solution is

in (1.1) are symmetry in this example,

t | ||||||
---|---|---|---|---|---|---|

0.01 | 0.6327828880 | 0.6327828896 | 0.8672171120 | 0.8672171104 | ||

0.02 | 0.6327634323 | 0.6327634358 | 0.8672365677 | 0.8672365642 | ||

0.03 | 0.6327439762 | 0.6327439821 | 0.8672560238 | 0.8672560179 | ||

0.04 | 0.6327245197 | 0.6327245284 | 0.8672754803 | 0.8672754716 | ||

0.05 | 0.6327050628 | 0.6327050746 | 0.8672949372 | 0.8672949254 | ||

0.06 | 0.6326856056 | 0.6326856209 | 0.8673143944 | 0.8673143791 | ||

0.07 | 0.6326661480 | 0.6326661672 | 0.8673338520 | 0.8673338328 | ||

0.08 | 0.6326466900 | 0.6326467134 | 0.8673533100 | 0.8673532866 | ||

0.09 | 0.6326272317 | 0.6326272597 | 0.8673727683 | 0.8673727403 | ||

0.10 | 0.6326077730 | 0.6326078060 | 0.8673922270 | 0.8673921940 |

t | ||||||
---|---|---|---|---|---|---|

0.01 | 0.6288854666 | 0.6288854681 | 0.8711145334 | 0.8711145319 | ||

0.02 | 0.6288659542 | 0.6288659574 | 0.8711340458 | 0.8711340426 | ||

0.03 | 0.6288464415 | 0.6288464466 | 0.8711535585 | 0.8711535534 | ||

0.04 | 0.6288269287 | 0.6288269358 | 0.8711730713 | 0.8711730642 | ||

0.05 | 0.6288074156 | 0.6288074251 | 0.8711925844 | 0.8711925749 | ||

0.06 | 0.6287879024 | 0.6287879143 | 0.8712120976 | 0.8712120857 | ||

0.07 | 0.6287683890 | 0.6287684035 | 0.8712316110 | 0.8712315965 | ||

0.08 | 0.6287488754 | 0.6287488928 | 0.8712511246 | 0.8712511072 | ||

0.09 | 0.6287293616 | 0.6287293820 | 0.8712706384 | 0.8712706180 | ||

0.10 | 0.6287098477 | 0.6287098713 | 0.8712901523 | 0.8712901287 |

Their solutions are

Substituting (3.11) into (3.7) we get the residuals and using the optimization method we have computed that

In this work, the OHAM-DJ is applied to obtain numerical solutions of the system of Burgers’ equations. The method is efficient and easy to implement where the first or second order solutions rapidly converges to the exact solutions. Furthermore, OHAM-DJ does not need any discretization in time or in space. Thus the solutions of system of Burgers’ equations are not influenced by computer round off errors. The method can be easily

t | |||
---|---|---|---|

0.01 | 0.00004772535612 | 0.00004770016206 | |

0.02 | 0.00005017216468 | 0.00005000245540 | |

0.03 | 0.00005274441090 | 0.00005230474874 | |

0.04 | 0.00005544852472 | 0.00005460704209 | |

0.05 | 0.00005829126566 | 0.00005690933545 | |

0.06 | 0.00006127973961 | 0.00005921162878 | |

0.07 | 0.00006442141667 | 0.00006151392213 | |

0.08 | 0.00006772414960 | 0.00006381621547 | |

0.09 | 0.00007119619382 | 0.00006611850883 | |

0.10 | 0.00007484622751 | 0.00006842080217 |

t | |||
---|---|---|---|

0.01 | 0.000003917707418 | 0.000003915638782 | |

0.02 | 0.000004118571745 | 0.000004104638279 | |

0.03 | 0.000004329734519 | 0.000004293637776 | |

0.04 | 0.000004551723744 | 0.000004482637274 | |

0.05 | 0.000004785094494 | 0.000004671636770 | |

0.06 | 0.000005030430303 | 0.000004860636268 | |

0.07 | 0.000005288344616 | 0.000005049635766 | |

0.08 | 0.000005559482332 | 0.000005238635264 | |

0.09 | 0.000005844521423 | 0.000005427634761 | |

0.10 | 0.000006144174603 | 0.000005616634259 |

extended to other nonlinear equations. Nutshell, OHAM-DJ is a better numerical method for solving nonlinear equations.

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.

Fatheah Ahmad Alhendi,Bothayna Saleh Kashkari,Aisha Abdullah Alderremy, (2016) Application of OHAM-DJ to the System of Burgers’ Equations. American Journal of Computational Mathematics,06,212-223. doi: 10.4236/ajcm.2016.63022