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In this paper, an improved algorithm for the solution of Generalized Burger-Fisher’s Equation is presented. A Maple code is generated for the algorithm and simulated. It was observed that the algorithm gives the solution with less computation. The solution gives a better result when compared with the numerical solutions in the existing literature.

Generalized Burger-Fisher equation, being a nonlinear partial differential equation, is of great importance for describing the interaction between reaction mechanisms, convection effects, and diffusion transports. Since there exists no general technique for finding analytical solution of nonlinear diffusion equations so far, numerical solutions of nonlinear equations are of great importance in physical problems.

Many researchers [

Unlike some previous methods that used various transformations and several iterations, we present a new Modified Variational Iteration Method (MVIM) for the numerical solutions of generalized Burger-Fisher equation.

In

In

The idea of variational iteration can be traced to Inokuti [

Table 1. The Absolute error for.

x | T | Exact solution | MVIM solution | MVIM (error) | ADM (error) [3] | VIM (error) [3] |
---|---|---|---|---|---|---|

0.1 | 0.005 | 4.9998900000E−01 | 4.9998875030E−01 | 2.4970000001E−07 | 9.6876300000E−06 | 1.0164970000E−04 |

0.001 | 5.0001300000E−01 | 4.9998775010E−01 | 2.5249900000E−05 | 1.9375300000E−06 | 3.4664990000E−04 | |

0.01 | 4.9999000000E−01 | 4.9999000060E−01 | 5.9999999413E−10 | 1.9375200000E−05 | 1.1780600000E−05 | |

0.5 | 0.005 | 4.9993900000E−01 | 4.9993875030E−01 | 2.4970000001E−07 | 9.6869100000E−06 | 2.7914970000E−04 |

0.001 | 4.9996300000E−01 | 4.9993775010E−01 | 2.5249900000E−05 | 1.9373800000E−06 | 9.8702499000E−03 | |

0.01 | 4.9994000000E−01 | 4.9994000060E−01 | 5.9999999413E−10 | 1.9373800000E−05 | 3.7400600000E−05 | |

0.9 | 0.005 | 4.9988900000E−01 | 4.9988875030E−01 | 2.4970000001E−07 | 9.6861900000E−06 | 2.7149700000E−05 |

0.001 | 4.9991300000E−01 | 4.9988775010E−01 | 2.5249900000E−05 | 1.9372400000E−06 | 9.4249900000E−05 | |

0.01 | 4.9989000000E−01 | 4.9989000060E−01 | 5.9999999413E−10 | 1.9372400000E−05 | 6.3789999994E−08 |

Table 2. The Absolute error for.

x | t | Exact solution | MVIM solution | MVIM (error) | ADM (error) [3] | VIM (error) [3] |
---|---|---|---|---|---|---|

0.1 | 0.0005 | 6.9542600000E−01 | 6.9542575300E−01 | 2.4700000001E−07 | 1.4017700000E−03 | 2.5700000001E−07 |

0.0001 | 6.9526600000E−01 | 6.9526613430E−01 | 1.3429999990E−07 | 2.8039600000E−04 | 7.0939600000E−04 | |

0.001 | 6.9562500000E−01 | 6.9562523130E−01 | 2.3129999993E−07 | 2.8030100000E−03 | 2.8031223000E−03 | |

0.5 | 0.0005 | 6.4629700000E−01 | 6.4629716130E−01 | 1.6130000002E−07 | 1.3452600000E−03 | 9.7961300000E−05 |

0.0001 | 6.4613000000E−01 | 6.4612989020E−01 | 1.0979999998E−07 | 2.6909400000E−04 | 6.1166980000E−04 | |

0.001 | 6.4650600000E−01 | 6.4650622380E−01 | 2.2379999998E−07 | 2.6900000000E−03 | 1.0022380000E−04 | |

0.9 | 0.0005 | 5.9548100000E−01 | 5.9548126730E−01 | 2.6729999991E−07 | 1.2769900000E−03 | 1.7717300000E−05 |

0.0001 | 5.9531000000E−01 | 5.9531045390E−01 | 4.5390000003E−07 | 2.5543800000E−04 | 7.5383900000E−05 | |

0.001 | 5.9569500000E−01 | 5.9569477720E−01 | 2.2280000000E−07 | 2.5534600000E−03 | −2.8407720000E−04 |

Table 3. The Absolute error for.

X | t | Exact solution | MVIM solution | MVIM (error) | ADM (error) | VIM (error) |
---|---|---|---|---|---|---|

0.1 | 0.0005 | 7.8367000000E−01 | 7.8367007490E−01 | 7.4899999980E−08 | 4.4532000000E−04 | 1.3729000000E−06 |

0.0001 | 7.8366000000E−01 | 7.8365991220E−01 | 8.7800000048E−08 | 4.4637900000E−04 | 4.7808780000E−04 | |

0.001 | 7.8368300000E−01 | 7.8368277800E−01 | 2.2200000005E−07 | 4.4399700000E−04 | 7.8652220000E−03 | |

0.5 | 0.0005 | 7.4129600000E−01 | 7.4129553480E−01 | 4.6519999997E−07 | 1.8547400000E−03 | 6.2595200000E−05 |

0.0001 | 7.4128500000E−01 | 7.4128455150E−01 | 4.4849999992E−07 | 1.8605700000E−03 | 1.4364850000E−04 | |

0.001 | 7.4130900000E−01 | 7.4130926350E−01 | 2.6350000004E−07 | 1.8474600000E−03 | 5.3236350000E−04 | |

0.9 | 0.0005 | 6.9616900000E−01 | 6.9616894960E−01 | 5.0400000062E−08 | 9.1958200000E−04 | 7.3303040000E−04 |

0.0001 | 6.9615700000E−01 | 6.9615741750E−01 | 4.1750000002E−07 | 9.3180300000E−04 | 7.6593400000E−05 | |

0.001 | 6.9618300000E−01 | 6.9618336460E−01 | 3.6459999997E−07 | 9.0429700000E−04 | 7.1159460000E−04 |

Graph of Burger-Fisher for

Graph of gBF when

Graph of Exact /MVIM against

Graph of gBF when

presented for the solution of the generalized Burger-Fisher equation.

To illustrate the basic concept of the MVIM, we consider the following general nonlinear partial differential equation:

where L is a linear time derivative operator, R is a linear operator which has partial derivative with respect to

where

The following generalized Burger-Fisher (gBF) equation problems arising in various field of science is considered.

with the initial condition

And the boundary conditions

We used Maple to code (1.1 - 1.2) for the solution of (1.3 - 1.6) and the following results were obtained after one iteration:

When

Tables 1-3 shows that the MVIM is the best approximant when compared with VIM and ADM.

There are some important points to note here. First, the MVIM provides the solutions in terms of convergent series with easily computable components. Second, it is clear and remarkable that approximate solutions using MVIM are in good agreement. Third, the MVIM technique requires less computational work than many existing approaches. The MVIM was used in a direct way without using linearization, perturbation or restrictive assumptions.

The MVIM provides more realistic series solutions, very high accuracy, fast transformation and possibility of implementation of algorithm. The Algorithm makes it easier for the system to predict the next series.